Prime Ends Dynamics in Parametrised Families of Rotational Attractors

TL;DR

This paper explores complex boundary dynamics of invariant domains on the 2-sphere, constructing examples with multiple attractors and rotation numbers, advancing understanding of topological and dynamical properties in parametrized families.

Contribution

It introduces new examples of rotational attractors with complex boundary dynamics, answering open questions and extending previous theoretical results in sphere dynamics.

Findings

Existence of Lakes of Wada rotational attractors close to identity

Construction of parametrized Birkhoff-like cofrontier attractors with multiple rotation numbers

Demonstration of non-transitive Birkhoff-like attractors with specific prime ends rotation properties

Abstract

We provide several new examples in dynamics on the $2$-sphere, with the emphasis on better understanding the induced boundary dynamics of invariant domains in parametrized families. First, motivated by a topological version of the Poincar\'e-Bendixson Theorem obtained recently by Koropecki and Passeggi, we show the existence of homeomorphisms of $\mathbb{S}^2$ with Lakes of Wada rotational attractors, with an arbitrarily large number of complementary domains, and with or without fixed points, that are arbitrarily close to the identity. This answers a question of Le Roux. Second, from reduced Arnold's family we construct a parametrised family of Birkhoff-like cofrontier attractors, where at least for uncountably many choices of the parameters, two distinct irrational prime ends rotation numbers are induced from the two complementary domains. This example complements the resolution of Walker's Conjecture by Koropecki, Le Calvez and Nassiri from 2015. Third, answering a question of Boyland, we show that there exists a non-transitive Birkhoff-like attracting cofrontier which is obtained from a BBM embedding of inverse limit of circles, such that the interior prime ends rotation number belongs to the interior of the rotation interval of the cofrontier dynamics. There exists another BBM embedding of the same attractor so that the two induced prime ends rotation numbers are exactly the two endpoints of the rotation interval.

Equations

$$\underset{ϵ\to 0}{\lim }{\rho }_{\text{ex}}({\Phi }_{ϵ},K_{ϵ})=0.$$

$$\underset{ϵ\to 0}{\lim }{h}_{\text{top}}({\Phi }_{ϵ}{|}_{K_{ϵ}})=0.$$

$$\underleftarrow{\lim}\{X,f\}:=\{\big{(}\ldots,x_{-1},x_{0}\big{)}\in X^{-\mathbb{N}_{0}}\big{|}x_{i}\in X,x_{i}=f(x_{i-1}),\text{ for any }i\leq 0\}.$$

$$\sigma_{f}(\underline{x}):=\big{(}\ldots,x_{-1},x_{0},f(x_{0})\big{)}.$$

$$\lim_{n\to+\infty}\frac{1}{n}\text{pr}_{1}\big{(}\widetilde{f}^{n}(\widetilde{p})-\widetilde{p}\big{)}=\lim_{n\to+\infty}\frac{1}{n}\text{pr}_{1}\big{(}\widetilde{p}-\widetilde{f}^{-n}(\widetilde{p})\big{)},$$

$$\rho (\tilde{f},x):=\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}({\tilde{f}}^n(\tilde{x})-\tilde{x}).$$

$$\rho(\widetilde{f}):=\{\rho(\widetilde{f},x)\big{|}x\in\mathbb{S}^{1}\},$$

$$\rho_{\text{pp}}(\widetilde{f})=\{\lim_{n\to\infty}\frac{1}{n}(\widetilde{f}^{n}(\widetilde{x})-\widetilde{x})\big{|}\text{ when the limit exist.}\}$$

$${\rho }_{\text{pp}}(\tilde{f})=\rho (\tilde{f}).$$

$$\widetilde{w}_{0}:=\sup\{\widetilde{x}\big{|}\widetilde{x}\in[\widetilde{z}_{0},\widetilde{z}_{0}+1],\widetilde{f}(\widetilde{x})=\widetilde{f}(\widetilde{z}_{0})\}.$$

$$\widetilde{w}_{0}^{\prime}=\inf\{\widetilde{x}\big{|}\widetilde{x}\in[\widetilde{y}_{0},\widetilde{z}_{0}+1],\widetilde{f}(\widetilde{x})=\widetilde{f}(\widetilde{y}_{0})+1\}.$$

$$\sup \rho (\tilde{f})=\rho (\tilde{f},y_0)=\underset{n\to +\infty }{\lim }\frac{1}{n}({\tilde{f}}^n({\tilde{y}}_0)-{\tilde{y}}_0).$$

$${{\rm Y}}_{ϵ}(\alpha (x,s))=\alpha (x,{\gamma }_{ϵ}(s)).$$

$$\Omega_{\epsilon}:=\{\alpha(x,s)\big{|}x\in\partial(D_{2}),s\in[1-\epsilon,1]\}.$$

$$\varphi_{\epsilon}\big{(}\alpha(x,s)\big{)}:=\alpha(x,\epsilon s+1-\epsilon).$$

$$\phi_{\epsilon}=\Upsilon_{\epsilon}\circ\overline{\phi}_{\epsilon}\big{|}_{G_{2}}.$$

$${\overline{f}}_{ϵ}:={\phi }_{ϵ}\circ {\overline{\varphi }}_{ϵ}\circ {\phi }_{ϵ}^{-1}.$$

$${\psi }_{ϵ}:={{\rm Y}}_{ϵ}\circ {\overline{f}}_{ϵ},$$

$${\Phi }_{ϵ}:={\sigma }_{{\psi }_{ϵ}}.$$

$${\psi }_{{ϵ}_0}(Q_0)=Q_1.$$

$${\psi }_{{ϵ}_0}(Q_1)=Q_0.$$

$$\underline{Q}:=\big{(}\ldots,Q_{1},Q_{0},Q_{1}\big{)}.$$

$$\lim_{n\to+\infty}\frac{1}{n}\big{(}(\widetilde{\phi}_{\epsilon}^{\ast})^{n}(\widetilde{p})-\widetilde{p}\big{)}=\lim_{n\to+\infty}\frac{1}{n}\big{(}\widetilde{p}-\widetilde{p}_{-k}\big{)}=\sup\rho\big{(}\widetilde{\phi}_{\epsilon}^{\ast}\big{)}.$$

$$\widetilde{\rho}_{\text{ex}}(\widetilde{\Phi}_{\epsilon},K_{\epsilon})=\sup\rho\big{(}\widetilde{\phi}_{\epsilon}^{\ast}\big{)}.$$

$$U_{\tau ,m,ϵ}$$

$$L_{\tau ,ϵ}$$

$$\underleftarrow{\lim }\{D_2,{\psi }_{ϵ}\}=K_{ϵ}\bigsqcup L_{\text{out},ϵ}\bigsqcup L_{0,ϵ}\bigsqcup L_{1,ϵ},$$

$$\underline{\Gamma }:=\bigcup _{n\in Z}{\Phi }_{ϵ}^n(\underline{\lambda })\subset L_{\text{out},ϵ}.$$

$${\underline{q}}_1:=(\dots ,q_2,q_3,q_4,q_1).$$

$${\underline{q}}_1:=(\dots ,q_3,q_4,q_1,q_2).$$

$${\underline{q}}_3:=(\dots ,q_4,q_1,q_2,q_3).$$

$${\underline{q}}_4:=(\dots ,q_1,q_2,q_3,q_4).$$

Full text

Jan P. Boroński111National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic e-mail: jan.boronski@osu.cz., Jernej Činč222National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic – and –AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland. e-mail: jernej.cinc@osu.cz., and Xiao-Chuan Liu333Instituto de Matemática e Estatística da Universidade de São Paulo, R. do Matão, 1010 - Vila Universitaria, São Paulo, Brasil. e-mail: lxc1984@gmail.com.

1. Introduction

The prime ends rotation number induced by surface homeomorphisms restricted to an invariant disk is one of the important invariants in the study of boundary dynamics. Parametrised families of dynamical systems can provide a clearer view of both the surface dynamics and the boundary dynamics in many situations. In this paper, our study serves as a contribution in this direction, by providing new examples in various interesting contexts. Let us postpone to Subsection 2.2 for several standard terminologies needed below.

We were initially motivated by the following recent result, referred to as topological version of the Poincaré-Bendixson Theorem.

Theorem A. (Koropecki and Passeggi, [18])

• Let $\Gamma$ be a translation line for an orientation-preserving homeomorphism of $\mathbb{S}^{2}$. Then either the omega limit set $\omega(\Gamma)$ contains a fixed point, or $\Gamma$ is a topologically embedded line, and its filled $\omega$-limit set $\widetilde{\omega}(\Gamma)$ is a rotational attractor, disjoint from $\Gamma$.*

In connection with Theorem A, we will construct a parametrised family of Lakes of Wada continua, which are the omega limit sets of translation lines, and whose filled continua are rotational attractors. We will also study the corresponding exterior prime ends rotation numbers. The approach we take is the so-called Brown-Barge-Martin (BBM) embeddings of inverse limits of topological graphs (see [5] and [13]). This tool has proven to play an important role in constructing surprising new examples for topological dynamical systems. A particularly useful extension of this method is provided by the parametrised version of BBM embedding, proved recently by Boyland, de Carvalho and Hall [9]. This generalized version makes it possible to know precisely how in a family of maps, the rotation set changes. The same authors used this method as a tool to find new rotation sets for torus homeomorphisms (see [10]) as well as to study prime ends of natural extensions of unimodal maps (see [11]). However, for our purposes in this paper, some adaptations of this technique is necessary.

1.1. Statements of the Results

A continuum is a compact and connected metric space. A continuum $X$ in the sphere $\mathbb{S}^{2}$ (or the plane) is said to be $n$-separating if $\mathbb{S}^{2}\setminus X$ has $n$ connected components. A Lakes of Wada continuum is an $n$-separating continuum in $\mathbb{S}^{2}$ which is the common boundary of each of the $n>2$ components of its complement. There are well known examples of attractors that are Lakes of Wada continua arising as projections of DA-attractors from the torus onto the sphere, as well as those constructed directly on the sphere by Plykin [29]. Other examples of Lakes of Wada continua, as well as their relations with physical phenomena, were given in [16] and bifurcations of basins of attraction from the view point of prime ends were studied for planar diffeomorphisms in [27]. Lakes of Wada property for trapping regions was studied in [26]. In the present paper the study of Lakes of Wada attractors is motivated by Theorem A, which in turn can be viewed as a particular generalization of the classical Poincaré-Bendixon Theorem. In view of Theorem A, during the conference Surfaces in Luminy, held at Centre International de Rencontres Mathématiques, October 3 - 7, 2016, Frédéric Le Roux asked if the boundary of the rotational attractor $\widetilde{\omega}(\Gamma)$ could be a Lakes of Wada continuum. Possible existence of such examples was conjectured in [18]. Clearly, in order to construct such an example, it is necessary to understand the corresponding exterior prime ends dynamics better. However, the literature suggests that the prime ends rotation numbers for Lakes of Wada attractors have not been systematically studied yet. Another feature is as follows. The planar attractors we obtain in Theorem 1.1 and Theorem 1.2 share the Lakes of Wada topology with the Plykin attractors. However, our attractors arise for dynamical systems which are arbitrarily small perturbations of identity, and they are not expansive (see Remark 4.4 for details), contrasting with expansivity of the Plykin attractors resulting from the global stretching and bending on the sphere. Up to our knowledge, ours is the first example of such phenomena in close vicinity of the identity.

Let $d_{H}$ denote the Hausdorff distance between two compact subsets of $\mathbb{S}^{2}$, and let $d_{C^{0}}$ denote the usual $C^{0}$ topology in the space of homeomorphisms of some metric space. The following is the main result of this paper.

Theorem 1.1**.**

For each $n>2$, there exists a family of homeomorphisms $\Phi_{\epsilon}:\mathbb{S}^{2}\to\mathbb{S}^{2}$, and a family of $\Phi_{\epsilon}$-invariant continua $K_{\epsilon}$, where $\epsilon\in(0,\epsilon_{0}]$ for some $\epsilon_{0}>0$, such that:

1. (i)

$\underset{\delta\to 0}{\lim}d_{H}(K_{\epsilon},K_{\epsilon+\delta})=0$, for each $\epsilon\in(0,\epsilon_{0}]$. 2. (ii)

the external prime ends rotation numbers, denoted by $\rho_{\text{ex}}(\Phi_{\epsilon},K_{\epsilon})$, is a strictly increasing function of $\epsilon$. Moreover,

[TABLE] 3. (iii)

for any $\epsilon\in(0,\epsilon_{0}]$, the continuum $K_{\epsilon}$ is an $n$-separating Lakes of Wada continuum, which is also a rotational attractor. 4. (iv)

$\underset{\epsilon\to 0}{\lim}d_{C^{0}}(\Phi_{\epsilon},id)=0$. 5. (v)

the topological entropy $h_{\text{top}}(\Phi_{\epsilon}|_{K_{\epsilon}})$ is strictly decreasing as $\epsilon\to 0$, and

[TABLE] 6. (vi)

for every $\epsilon\in(0,\epsilon_{0}]$, there exists a translation line $\underline{\Gamma}\subset\mathbb{S}^{2}$ so that $\omega(\underline{\Gamma})=K_{\epsilon}$. The filled omega limit of $\underline{\Gamma}$, denoted by $\widetilde{\omega}(\underline{\Gamma})$, is a rotational attractor, which is disjoint from $\underline{\Gamma}$.

The continua $K_{\epsilon}$ from Theorem 1.1 have a $\Phi_{\epsilon}$-fixed point. However, one can also obtain examples without fixed points, as we will show in the next theorem, thus providing examples for both cases stated in Theorem A. For the purpose of the discussion that follows Theorem 1.2, we state this next result on the plane.

Theorem 1.2**.**

For any $n\geq 2$, there exists a $2n$-separating Lakes of Wada rotational attractor $K$, and a homeomorphism $\Phi:\mathbb{R}^{2}\to\mathbb{R}^{2}$ such that $\Phi|_{K}$ is fixed-point free. There exists a translation line $\Gamma\subset\mathbb{R}^{2}$ so that $\omega(\Gamma)=K$ and $\widetilde{\omega}(\Gamma)$ is a rotational attractor disjoint from $\Gamma$.

We are in an unfortunate position to have to point out an error in a research announcement [12], where in the Introduction it is stated, that corollaries of methods proposed in [12] imply the following: every homeomorphism of the Lakes of Wada continuum, that is extendable to the plane, must have a fixed point in the composant accessible from the unbounded complementary domain. Theorem 1.2 implies that there exist homeomorphisms of the Lakes of Wada continua that are extendable to the plane where no composants contain any fixed points of the homeomorphism. Therefore, it follows that composants which are accessible from the unbounded complementary domain have no fixed point of the homeomorphism as well, implying an error in the results announced in [12].

We now turn our attention to another class of examples. Suppose some orientation-preserving homeomorphism on $\mathbb{S}^{2}$ admits an invariant separating continuum, which is a common boundary of two complementary domains (such a continuum is called a cofrontier). This provides another interesting context, where prime ends rotation number comes into play. A conjecture of Walker [30] stated that it is not possible for the induced dynamics on the two prime ends circles to be conjugate to two rigid rotations with different irrational rotation numbers. Recently, this conjecture was proved by Koropecki, Le Calvez and Nassiri (see Theorem F of [17]). However, in the spirit of this conjecture, it is still interesting to find out if it is possible that two primes ends rotation numbers are two different irrationals. The following result gives an affirmative answer to this question through a family of examples. To state the result, assuming that we are working with invariant cofrontiers, in what follows let $\rho_{\text{in}}$ denote the prime ends rotation number of the induced dynamics on the interior circle of prime ends, defined analogously as in the exterior prime ends rotation number. A Birkhoff-like attractor for an orientation-preserving homeomorphism $g:\mathbb{S}^{2}\to\mathbb{S}^{2}$ is an attractor $\Lambda\subset\mathbb{S}^{2}$ such that the rotation set $\rho(g|\Lambda)$ is a nondegenerate interval.

Theorem 1.3**.**

There exists a parametrised family of homeomorphisms $\{\Phi_{t}\}_{t\in I}:\mathbb{S}^{2}\to\mathbb{S}^{2}$, which is parametrised by a closed interval $I$, such that each $\Phi_{t}$ preserves a cofrontier attractor $K_{t}\subset\mathbb{S}^{2}$, with the following properties.

1. (i)

$K_{t}$* is a Birkhoff-like attractor for $\Phi_{t}$, for each $t\in I$,* 2. (ii)

for each $t\in I$ and for each $\epsilon>0$, there exists a sphere homeomorphism $\Phi^{\prime}_{t}$ which preserves a Birkhoff-like attractor $K^{\prime}_{t}$ homeomorphic to the pseudo-circle, such that $d_{H}(K_{t},K^{\prime}_{t})<\epsilon$, and $d_{C^{0}}(\Phi_{t},\Phi^{\prime}_{t})<\epsilon$ 3. (iii)

there are uncountably many choices of the parameter $t\in I$, such that the lifted interior and exterior prime ends rotation numbers $\rho_{\text{ex}}(\widetilde{\Phi}_{t},K_{t})=\eta_{t}\neq\eta^{\prime}_{t}=\rho_{\text{in}}(\widetilde{\Phi}_{t},K_{t})$, with $\eta_{t},\eta^{\prime}_{t}\notin\mathbb{Q}$.

Note that Theorem F of [17] implies that when prime ends rotation numbers from Theorem 1.3 are irrational, the induced dynamics on two circles of prime ends are Denjoy homeomorphisms. In view of item (iii) Theorem 1.3, we can ensure that each cofrontier attractor $K_{t}$ is indecomposable, but it is yet to be determined whether such $K_{t}$ can all be homeomorphic to Bing’s pseudo-circle [6]. Note that the homeomorphism group of the pseudo-circle does not contain any non-degenerate continuum [21], so in a potential family the embeddings would need to change continuously, in such a way as to produce continuosly varying prime ends rotation numbers, but the homeomorphisms could not change in such a way.

Question 1**.**

Can the interior and exterior prime ends rotation numbers be distinct irrational numbers for a cofrontier that is a pseudo-circle?

Question 2**.**

Does there exist a parameterized family of homeomorphisms $\{\Phi_{t}:\mathbb{S}^{2}\to\mathbb{S}^{2}\}_{t\in I}$ such that for any $t\in I$, $\Phi_{t}$ preserves a cofrontier attractor $K_{t}$, which is homeomorphic to the pseudo-circle?

Another problem that can be dealt with proper embeddings of the inverse limit dynamics is as follows. During the Workshop on Dynamical Systems and Continuum Theory, held at University of Vienna, June 29 - July 3, 2015, Philip Boyland raised the following question:

Suppose that $\widetilde{f}$ is a lift of a circle endomorphism $f:\mathbb{S}^{1}\to\mathbb{S}^{1}$ of degree $1$, and assume that its rotation interval $\rho(\widetilde{f})=[a,b]$. Then is it true that, the exterior and interior prime ends rotation numbers of the unwrapping of $f$ via inverse limit method in the annulus are exactly $a$ and $b$, respectively?

For Birkhoff attractors, it is always true that the prime ends rotation numbers are the endpoints of the rotation interval (see [20]). Related to this, Boyland also asked if complicated inverse limit spaces (of the interval or the circle) can be embedded in $\mathbb{S}^{2}$ in multiple ways, which was already answered for unimodal inverse limits in [2]. Here we answer these two questions by showing the following result.

Theorem 1.4**.**

There exists an orientation-preserving and homology-preserving annulus homeomorphism $\Phi:\mathbb{A}\to\mathbb{A}$, with an attracting non-transitive Birkhoff-like cofrontier $K$, such that the lifted interior prime ends rotation number $\rho_{\text{in}}(\widetilde{\Phi},K)$ is contained in the interior of the interval $\rho(\widetilde{\Phi})$. There exists another embedding of the pair (K,\Phi\big{|}_{K}) as an attractor in $\mathbb{A}$, so that the two induced lifted prime ends rotation numbers are exactly the two endpoints of the rotation interval of the corresponding natural extension.

The paper is organized as follows. In Section 2, we present some standard notation and state some related results that we will use in the paper, in particular a parametrised version of the BBM method. In Section 3, we will prove a lemma about circle endomorphisms needed for future use. In Section 4, we give a proof of Theorem 1.1, which is the core of this paper. In Section 5 we show Theorem 1.2. Finally, in Section 6, we prove Theorem 1.3 and Theorem 1.4.

2. Preliminaries

2.1. Inverse Limit Spaces and the BBM Embedding

Let $X$ be a metric space. Two homeomorphisms $f,g:X\to X$ are called topologically conjugate, if there exists a homeomorphism $h:X\to X$ so that $h\circ g\circ h^{-1}=f$. Denote by $\mathcal{C}(X,X)$ (respectively, $\mathcal{H}(X,X)$) the set of all continuous mappings on a metric space $X$ (respectively, the set of all homeomorphisms of $X$).

Let $\mathbb{N}_{0}$ denote the set of non-negative integers $\{0,1,\cdots\}$. Our main tool for constructing the examples are inverse limit spaces. For $f\in\mathcal{C}(X,X)$, we denote

[TABLE]

We equip $\underleftarrow{\lim}\{X,f\}$ with the subspace metric induced from the product metric in $X^{-\mathbb{N}_{0}}$, where $f$ is called the bonding map. The inverse limit space $\underleftarrow{\lim}\{X,f\}$ also comes with a natural homeomorphism, called the natural extension of $f$, or the shift homeomorphism $\sigma_{f}:\underleftarrow{\lim}\{X,f\}\to\underleftarrow{\lim}\{X,f\}$, defined as follows. For any \underline{x}:=\big{(}\ldots,x_{-2},x_{-1},x_{0}\big{)}\in\underleftarrow{\lim}\{X,f\},

[TABLE]

By $\pi_{-k}$ we shall denote the $(-k)$-th projection from $\underleftarrow{\lim}\{X,f\}$ to the $(-k)$-th coordinate. Now we fix some notation useful for constructing parametrised families of examples. We will follow mainly [9], and we refer to it for the more general setting.

For $n\geq 1$, let $D_{n}\subset\mathbb{S}^{2}$ denote a closed topological disk with $n$ open holes. A subset $G\subset D_{n}$ is called a boundary retract of $D_{n}$ if there is a continuous map $\alpha:\partial D_{n}\times[0,1]\to D_{n}$ which decomposes $D_{n}$ into a continuously varying family of arcs $\{\alpha(x,\cdot)\}_{x\in\partial D_{n}}\subset\mathcal{C}([0,1],D_{n})$, so that $\alpha(x,\cdot)([0,1])$ are pairwise disjoint except perhaps at the endpoints $\alpha(x,1)$, where $\alpha(x,1)\in G$. We can then associate a retraction $r:D_{n}\to G$ defined by $r(\alpha(x,s))=\alpha(x,1)$ for every $x\in\partial D_{n}$ corresponding to the given decomposition. We say a continuous map $f:G\to G$ unwraps in $D_{n}$ if there is a near-homeomorphism $\bar{f}:D_{n}\to D_{n}$ such that $r\circ\bar{f}=f$. The near-homeomorphism $\bar{f}$ is called the unwrapping of $f$. Let $I=[0,\epsilon_{0}]$ for some $\epsilon_{0}>0$. A continuous family $\{f_{t}\}_{t\in I}$ is said to unwrap in $D_{n}$ if there exists a continuous family of unwrappings $\{\bar{f_{t}}\}_{t\in I}$ associated to it. We are now ready to state the parametrised BBM technique, which we will use in our construction later. The following lemma is an adaptation of Theorem 3.1 from [9].

Lemma 2.1**.**

For $n\geq 1$, let $D_{n}$ denote the closed set obtained by removing from a closed disk interiors of $n$ disjoint closed disks. Let $G\subset D_{n}$ denote a boundary retract of $D_{n}$ and suppose a family $\{f_{t}\}_{t\in I}\subset\mathcal{C}(G,G)$ unwraps in $D_{n}$. Moreover, suppose that there exists $m>0$ such that $f^{m+1}_{t}(G)=f^{m}_{t}(G)$ for all $t\in I$. Then, there is a continuous family $\{F_{t}\}_{t\in I}$ in $\mathcal{H}(D_{n},D_{n})$, such that:

• (a)

For each $t\in I$ there is a compact $F_{t}$-invariant set $K_{t}\subset D_{n}$ so that:

• (i)

$F_{t}|_{K_{t}}:K_{t}\to K_{t}$* is topologically conjugate to $\sigma_{f_{t}}:\underleftarrow{\lim}\{G,f_{t}\}\to\underleftarrow{\lim}\{G,f_{t}\}$.*

• (ii)

If $x\in D_{n}\setminus\partial D_{n}$, then the omega limit set $\omega(x,F_{t})\subset K_{t}$.

• (b)

The attractors $K_{t}$ vary continuously in Hausdorff metric with $t\in I$.

Sketch of proof.

The proof follows the proof of Theorem 3.1 in the paper [9]. Our version is a bit more general, in that, we allow our unwrapping to be a near-homeomorphism (i.e., the uniform limit of homeomorphisms), instead of just a homeomorphism (this is allowed due to Brown’s approximation theorem from [13]). The smash mapping will be also chosen carefully, instead of being fixed as in the original proof. In particular, our specific choices of the unwrapping and smash mappings will imply that the dynamics of $F_{t}$ in $D_{n}$ of examples from Theorem 1.1 is indeed close to identity in the $C^{0}$ topology as $t$ is sufficiently close to [math]. However, the same conclusion of the above mentioned theorem holds true with our choices of unwrapping and smash mappings with proofs unchanged, so we do not repeat them. Instead, we will stress this point again during the proofs of the main theorem, and give precise definitions of the choices. ∎

Let us remark that whenever we will apply Lemma 2.1, the condition $f^{m+1}_{t}(G)=f^{m}_{t}(G)$ for all $t\in I$ from the statement of Lemma 2.1 will be satisfied for $m=1$, so we are not repeating this condition again.

2.2. Surface Dynamics and Prime Ends Rotation Numbers

We will mainly work with $\mathbb{S}^{2}$ due to its compactness, and due to the fact that any planar homeomorphism can be extended to the sphere by compactifying the plane by a point at infinity, and setting it as a fixed point. Recall that $\Gamma$ is called a translation line for a homeomorphism $h:\mathbb{S}^{2}\to\mathbb{S}^{2}$, if there exists a continuous injective map $\gamma:\mathbb{R}\to\mathbb{S}^{2}$, onto its image $\Gamma=\gamma(\mathbb{R})$, such that $\Gamma$ is $h$-invariant, and the restriction $h|\Gamma$ is fixed point free. Equivalently, it means that the composition $\gamma^{-1}\circ h\circ\gamma$ is topologically conjugate to the translation given by $T(x)=x+1$ for all $x\in\mathbb{R}$. Define the $\omega$-limit of a translation line $\Gamma=\gamma(R)$ as $\omega(\Gamma)=\bigcap_{t>0}\overline{\gamma([t,\infty))}$. If $\Gamma$ is disjoint from $\omega(\Gamma)$, define the filled $\omega$-limit set, written as $\widetilde{\omega}(\Gamma)$, as the union of $\omega(\Gamma)$ with all the connected components of $\mathbb{S}^{2}\setminus\omega(\Gamma)$ which does not contain $\Gamma$. Note that the set $\widetilde{\omega}(\Gamma)$ is a continuum. which does not separate $\mathbb{S}^{2}$. Following [18] we call a continuum $K$ a rotational attractor if it is a topological attractor for $f$, and the corresponding external prime ends rotation number is nonzero (modulo $1$).

We now introduce the terminology from the prime end theory that we will use in the paper. For a comprehensive introduction to the prime end theory the reader is referred to e.g. [23]. It is well known that, for a domain in $\mathbb{S}^{2}$ which is homeomorphic to an open topological disk $\mathbb{D}$, one can define the so-called prime ends circle, so that its union with $\mathbb{D}$ is homeomorphic to the closed unit disk with a proper topology. If $h:\mathbb{S}^{2}\to\mathbb{S}^{2}$ preserves orientation and $h(\mathbb{D})=\mathbb{D}$ then $h$ induces an orientation preserving homeomorphism of the prime ends circle, and therefore it gives a natural prime ends rotation number. It is yet to be completely determined how exactly the prime ends rotation number is related with the actual dynamics on $\mathbb{D}$. We refer to the recent paper [17] for more details and state of the art in this topic. Here we are mostly interested in knowing how the prime ends rotation number changes along a parametrised family of attractors and homeomorphisms, obtained from BBM embeddings.

Our more specific context is as follows. For a dynamical system $f:\mathbb{S}^{2}\to\mathbb{S}^{2}$ preserving a non-separating invariant continuum $K$, we will consider the so called exterior prime ends rotation number $\rho_{\text{ex}}(f,K)$. The complement $K^{c}$ is $f$-invariant, and is a topological disk in $\mathbb{S}^{2}$. By definition, the exterior prime ends rotation number $\rho_{\text{ex}}(f,K)$ is the rotation number of the induced prime ends circle homeomorphism. In order to relate the value of prime ends rotation number with the actual dynamics, one has to understand the sets of accessible points of $K$. For this purpose we will need a recent result obtained by Hernández-Corbato from [14], which we recall below.

Let $K$ be a plane non-separating continuum in an interior of a closed disk $D$. If $f$ is a homeomorphism of $D$ with $f(K)=K$, then there exists a $\xi\in K$ so that $f(\xi)=\xi$. Then $D\setminus\{\xi\}$ is homeomorphic to a half-open annulus $\mathbb{A}:=\mathbb{S}^{1}\times(−\infty,0]$. We define a universal cover $\widetilde{\mathbb{A}}:=\mathbb{R}\times(\infty,0]$ of $\mathbb{A}$ as $\pi:\widetilde{\mathbb{A}}\to\mathbb{A}$ given by $\pi((\theta,r))=(e^{2\pi i\theta},r)$. Then the lower boundary of $\widetilde{\mathbb{A}}\backslash\pi^{-1}(K)$ induces a prime ends line. If we fix a lift $\widetilde{f}:\widetilde{\mathbb{A}}\to\widetilde{\mathbb{A}}$ then the induced dynamics on this prime ends line has a rotation number, denoted as $\widetilde{\rho}_{\text{ex}}(\widetilde{f},K)\in\mathbb{R}$. Clearly, $\widetilde{\rho}_{\text{ex}}(\widetilde{f},K)$ modulo $\mathbb{Z}$ is the exterior prime ends rotation number $\rho_{\text{ex}}(f,K)$. Therefore, we call $\widetilde{\rho}_{\text{ex}}(\widetilde{f},K)$ the lifted exterior prime ends rotation number. A point $p\in K$ is accessible if there is an arc $\lambda:[0,1]\to\mathbb{S}^{2}$, such that $\lambda([0,1))\subset\mathbb{S}^{2}\backslash K$ and $\lambda(1)=p\in K$.

Lemma 2.2** (Theorem 1.1 and Theorem 1.2 in [14]).**

Let $\widetilde{f}$ be defined as above. Let $p\in K$ denote an accessible point of $K$ from exterior. Suppose one of the following conditions hold.

1. (1)

either $p$ is $f$-periodic for some period $n$. 2. (2)

or, the forward rotation number of $p$ equals the backward rotation number, i.e., for any lifted point $\widetilde{p}$ corresponding to $p$,

[TABLE]

where $\text{pr}_{1}$ denotes the first coordinate projection.

Then, the lifted exterior prime ends rotation number $\widetilde{\rho}_{\text{ex}}(\widetilde{f},K)$ equals the point-wise rotation number of the point $p$.

We will also need the following result by Barge [3].

Lemma 2.3** (Proposition 2.2 in [3]).**

Suppose that $\{\Phi_{t}\}_{t\in I}$ is a continuous family of orientation-preserving homeomorphisms on $\mathbb{S}^{2}$. For every $t\in I$ let $K_{t}$ be a non-degenerate sphere non-separating continuum, invariant under $\Phi_{t}$, and assume that $\{K_{t}\}_{t\in I}$ vary continuously with $t$ in Hausdorff metric. Then the exterior prime ends rotation numbers $\rho_{\text{ex}}(\Phi_{t},\Lambda_{t})$ vary continuously with $t$.

3. Auxiliary Lemma on Circle Endomorphisms with Two Turns

In this section, we recall a useful lemma concerning rotation sets of circle endomorphisms with two turns. This should be already known from [15] and [25] (see also [8] for this special class of endomorphisms). For reader’s convenience we include the proofs. Denote by $\text{End}_{1}(\mathbb{S}^{1})$ the space of circle endomorphisms of degree $1$, and by $\widetilde{\text{End}}_{1}(\mathbb{S}^{1})$ the set of lifts of elements in $\text{End}_{1}(\mathbb{S}^{1})$ to $\mathbb{R}^{1}$. For any $\widetilde{f}\in\widetilde{\text{End}}_{1}(\mathbb{S}^{1})$, we define a rotation set $\rho(\widetilde{f})$ as follows. First let

[TABLE]

Note that the expression above does not depend on the choice of the lifted point $\widetilde{x}$ corresponding to $x$. We set

[TABLE]

which is a closed interval by the main result of [15]. Now let us consider a family of endomorphisms of a special form, which was also considered in Section 2 of [8] for other purposes. Suppose some interval $[\widetilde{z}_{0},\widetilde{z}_{0}+1]$ is subdivided into two subintervals, namely, $\widetilde{I}_{1}=[\widetilde{z}_{0},\widetilde{y}_{0}]$, $\widetilde{I}_{2}=[\widetilde{y}_{0},\widetilde{z}_{0}+1]$. Assume $\widetilde{f}\in\widetilde{\text{End}}_{1}(\mathbb{S}^{1})$ is such that \widetilde{f}\big{|}_{\widetilde{I}_{1}} is decreasing, and \widetilde{f}\big{|}_{\widetilde{I}_{2}} is increasing. Whenever there exist $\widetilde{z}_{0},\widetilde{y}_{0}$ as above, we call such a $\widetilde{f}$ the lift of a circle endomorphism with two turns. For such an endomorphisms it was proved in [8] that every rotation number $\rho\in\rho(\widetilde{f})$ can always be realized by some point. More precisely, we can replace the original definition (7) with the more naturally defined pointwise rotation number.

[TABLE]

By Proposition 2.3 of [8], for an endomorphism with two turns we have

[TABLE]

We shall need this observation in the arguments below. Let us define the point

[TABLE]

Then, we call the interval $\widetilde{J}:=[\widetilde{w}_{0},\widetilde{z}_{0}+1]$ an efficient climbing interval. Note that it is uniquely defined up to an integer translation. We also define the lower climbing interval as the interval $[\widetilde{y}_{0},\widetilde{w}_{0}^{\prime}]$, where

[TABLE]

Lemma 3.1**.**

Let $\widetilde{f}\in\widetilde{\text{End}}_{1}(\mathbb{S}^{1})$ be the lift of a circle endomorphism with two turns, and denote by $\widetilde{J}$ an efficient climbing interval of $\widetilde{f}$. Then there exists a point $\widetilde{y}_{0}\in\widetilde{J}$, with the following two properties.

• •

It is possible to choose backward iterates of $\widetilde{y}_{0}$, namely, $\widetilde{y}_{-k}\in\widetilde{f}^{-k}(\widetilde{y}_{0})$, such that each $\widetilde{y}_{k}$ belongs to some integer translate of $\widetilde{J}$.

• •

The forward rotation number of $\widetilde{y}_{0}$ coincides with the backward rotation number of $\widetilde{y}_{0}$, which is the supremum of the rotation segment $\rho(\widetilde{f})$.

Proof.

For endomorphisms with two turns, since the definitions (7) and (8) are equivalent, there always exists some point $y_{0}$ with a lift $\widetilde{y}_{0}$, such that,

[TABLE]

Under the assumptions, the image of $\widetilde{J}$ is the interval $[\widetilde{f}(\widetilde{z}_{0}),\widetilde{f}(\widetilde{z}_{0}+1)]$, which has length $1$, because $f$ has degree $1$. It follows that $\underset{m\in\mathbb{Z}}{\bigcup}\widetilde{f}(\widetilde{J}+m)=\mathbb{R}$. Thus for any $\widetilde{z}\in\mathbb{R}$, there exists an integer $m$, such that $\widetilde{f}^{-1}(\widetilde{z})\cap(\widetilde{J}+m)\neq\emptyset$. In particular, for $\widetilde{y}_{0}\in\widetilde{J}$, we can choose a sequence of its backward iterates by $\widetilde{f}$, each of which lies in the translates of $\widetilde{J}$. Therefore, up to taking one backward iterate, we can choose backward iterates $\{\widetilde{y}_{-k}\}_{k\geq 0}$ such that for all $k\geq 0$, $\widetilde{y}_{-k}$ belongs to the translates of $\widetilde{J}$, and $\widetilde{f}(\widetilde{y}_{-(k+1)})=\widetilde{y}_{-k}$. Now we modify our dynamics. Define the function $\widetilde{g}:\mathbb{R}\to\mathbb{R}$, which coincides with $\widetilde{f}$ when restricted to $\widetilde{J}=[\widetilde{w}_{0},\widetilde{z}_{0}+1]$, and takes constant value $\widetilde{f}(\widetilde{z}_{0})$ when restricted to the interval $[\widetilde{z}_{0},\widetilde{w}_{0}]$ (the map $\widetilde{g}$ is the so-called ”water pouring map“, see [1], page 143 for more details). Note that, $\widetilde{g}$ is a monotone function, so its rotation number is uniquely defined, independent of any starting point. Since the backward iterates of $\widetilde{y}_{0}$ all lie in translates of $\widetilde{J}$, we know in particular that the backward rotation number of $\widetilde{y}_{0}$ exists and it is equal to $\rho(\widetilde{g})$. Clearly, it follows that the forward rotation number of $\widetilde{y}_{0}$ equals the backward rotation number of $\widetilde{y}_{0}$. The proof is complete now. ∎

Remark 3.2**.**

Lemma 3.1 will be used for showing accessibility of certain points in BBM embeddings of some circle-like attractors, and this in turn will serve as a way of determining their exterior prime ends rotation numbers. The definition of the efficient climbing interval is designed for this purpose. Note that, with the definition of lower climbing intervals, one can similarly study the interior prime ends rotation numbers, and obtain a similar statement of Lemma 3.1. We omit the repetition of the proofs. However, we will use both notions in Subsection 6.1.

4. A parametric family near the identity

In this section we prove Theorem 1.1. Let us first introduce the following notion, which will be used in the present and the subsequent sections. Call a connected topological graph $G_{k}$ a chain of $k$ circles if $G_{k}$ is a union of $k>1$ circles $S_{0}\cup\ldots\cup S_{k-1}$ where for any $i,j\in\{0,1,\ldots,k-1\}$, $S_{i}\cap S_{j}$ is a point if and only if $|i-j|=1$ and if $|i-j|>1$, then $S_{i}\cap S_{j}=\emptyset$.

Proof of Theorem 1.1.

We start with the case $n=3$, i.e. we construct a $3$-separating Lakes of Wada continuum.

Step 1. Boundary Retracts and the choice of $\epsilon_{0}$.

Let $G_{2}$ be a chain of $2$ circles $S_{0}$ and $S_{1}$ with the same radius whose intersection contains a single point $x_{0}$. Then $G_{2}$ is the spine of a pair of pants $D_{2}$. Topologically, $D_{2}$ is obtained from a closed topological disk, with boundary $C_{2}$, by removing two disjoint open disks from it, with boundaries $C_{0}$ and $C_{1}$ respectively, see Figure 1. Moreover, $G_{2}$ is the image of a boundary retract of $D_{2}$. More precisely, we can define the function $\alpha:\partial D_{2}\times[0,1]\to D_{2}$ such that, for any $x\in\partial D_{2}$, $\alpha(x,\cdot)$ restricted to $[0,1]$ is one-to-one and $\alpha(x,0)=x$ and $\alpha(x,1)\in G_{2}$. For our convenience, we will always choose $\alpha$, such that the image of the arc $\alpha(x,\cdot)([0,1])$ is straight line segments connecting $\partial D_{2}$ to $G_{2}$ (see Figure 1). We refer these as the radial directions.

For each $i=0,1$, we parametrise each circle clockwise with $\beta_{i}:[0,1]\to S_{i}$ in the uniformly scaling way, so that, $\beta_{i}(t_{i})=x_{0}$ if and only if $t_{i}\in\{0,1\}$. Fix $\epsilon\in(0,\frac{1}{2})$. Now we define a continuous map $\phi_{\epsilon}:G_{2}\to G_{2}$ as follows. For $i=0,1$:

1. (1)

Consider the arc $\beta_{i}([0,1-2\epsilon])$. The restriction of $\phi_{\epsilon}$ to this arc is a uniform scaling map, whose scaling factor is $\frac{1}{1-2\epsilon}$. Note that the image is the whole circle $S_{i}$. 2. (2)

Consider the arc $\beta_{i}([1-2\epsilon,1-\epsilon])$. The restriction of $\phi_{\epsilon}$ maps this arc to the arc $\beta_{1-i}([0,\frac{\epsilon}{1-2\epsilon}])$, in a uniform scaling way. 3. (3)

Consider the arc $\beta_{i}([1-\epsilon,1])$. We require that $\phi_{\epsilon}$ maps this arc to the arc $\beta_{1-i}([\frac{\epsilon}{1-2\epsilon},0])$, in a uniform scaling way. Note that this restriction reverses the orientation.

In particular, for $\epsilon_{0}=1-\frac{\sqrt{2}}{2}$, the two points $q_{i}=\beta_{i}(\frac{\sqrt{2}}{2})$, with $i=0,1$, form a periodic orbit for $\phi_{\epsilon_{0}}$ whose period is $2$. We denote by $I:=[0,\epsilon_{0}]$.

Step 2. The Choice of the Smash Mapping and the Unwrapping.

Fix some $\epsilon\in(0,\epsilon_{0}]$. Let $\alpha:\partial D_{2}\times[0,1]\to D_{2}$ be as given in the previous step. Define $\gamma_{\epsilon}:[0,1]\to[0,1]$, such that $\gamma_{\epsilon}(s)=\frac{s}{1-\epsilon}$ for $s\in[0,1-\epsilon]$, and $\gamma_{\epsilon}(s)=1$ for $s\in[1-\epsilon,1]$. Then we define the smash mapping $\Upsilon_{\epsilon}:D_{2}\to D_{2}$ as follows. For every $x\in\partial(D_{2})$ and $s\in[0,1]$ let

[TABLE]

Note that the “smashing region” for $\Upsilon_{\epsilon}$ is the set

[TABLE]

There is a natural homeomorphism $\varphi_{\epsilon}:D_{2}\to\Omega_{\epsilon}$, defined by

[TABLE]

Note $\gamma_{\epsilon}$ is a near-homeomorphism of $[0,1]$, i.e., a uniform limit of homeomorphisms. It follows that $\Upsilon_{\epsilon}$ is a near-homeomorphism of $D_{2}$. The function $\Upsilon_{\epsilon}$ represents the smash in the BBM construction and the smashing region $\Omega_{\epsilon}$ converges to $G_{2}$ as $\epsilon$ tends to [math]. In particular, $\Omega_{\epsilon}$ is sufficiently thin if $\epsilon$ is sufficiently small. We stress again that there is a difference between this setup and the original proof of the Theorem 3.1 of [9], where the definition of the smash mapping $\Upsilon$ was fixed for the whole parametrised family. Nevertheless, the parametrised family $\{\phi_{\epsilon}\}_{\epsilon\in I}\subset\mathcal{C}(G_{2},G_{2})$ unwraps in $D_{2}$. We proceed by describing the specific choice of the definition of the unwrapping $\overline{\phi}_{\epsilon}$.

Definition 4.1**.**

Define $\overline{\phi}_{\epsilon}\in\mathcal{C}(D_{2},D_{2})$ as a near-homeomorphism, satisfying the following conditions.

1. (a)

$\overline{\phi}_{\epsilon}(x)=x$* for $x\in C_{0}\cup C_{1}$. * 2. (b)

$\alpha(C_{0}\cup C_{1},[0,1))$* is a homeomorphism onto its image. In particular, $\overline{\phi}_{\epsilon}|_{G_{2}}$ is a homeomorphism onto its image.* 3. (c)

$\overline{\phi}_{\epsilon}({G_{2}})\subset\alpha(C_{2},[0,1))$. 4. (d)

The restriction of $\overline{\phi}_{\epsilon}$ to $G_{2}$ satisfies

[TABLE] 5. (e)

Denote $J_{i,\epsilon}=\beta_{i}([\epsilon,1-\epsilon])$, for $i=0,1$, and then denote $J_{\epsilon}=J_{0,\epsilon}\cup J_{1,\epsilon}$. For any $y\in J_{\epsilon}$, let $x\in C_{2}$ be such that $\alpha(x,1)=y$. Denote by $z\in C_{2}$ a point so that $\alpha(z,1)=\phi_{\epsilon}(y)$. Then $\overline{\phi}_{\epsilon}$ restricted to the radial arc $\alpha(x,[0,1])$ is a monotone map, whose image is contained in a radial arc connecting $\phi_{\epsilon}(y)$ to $z$.

See Figure 2 for certain arcs and their images under $\overline{\phi}_{\epsilon}$.

Remark 4.2**.**

In item (e) of the Definition 4.1, the choice of the interval $J_{\epsilon}$ is related to the efficient climbing interval that we have defined in Section 3 for circle endomorphisms with two turns. Rigorous proof that $\overline{\phi}_{\epsilon}$ is indeed a near-homeomorphism, is left to the reader. Let us note that $\overline{\phi}_{\epsilon}$ can be defined to be a homeomorphism as well (see Figure 3), but for our study of prime ends it is more convenient to use a near-homeomorphism that we define above.

As we already remarked in the proof of Lemma 2.1, although the unwrapping $\overline{\phi}_{\epsilon}$ is by the definition only a near-homeomorphism, the conclusions of Lemma 2.1 still hold and were used in this form in e.g. [11] for studying unimodal inverse limit spaces as attractors of sphere homeomorphims. Thus, similarly as in the beginning of page 1081 of the paper [9] (but with our choice of $\Upsilon_{\epsilon}$), we can define the mapping

[TABLE]

Then we extend it to the whole $D_{2}$ radially. Finally, consider the composition

[TABLE]

and denote the induced shift homeomorphism by

[TABLE]

By Brown’s approximation theorem from [13], the inverse limit space $\underleftarrow{\lim}\{D_{2},\psi_{\epsilon}\}$ is homeomorphic to $D_{2}$ itself, via a homeomorphism $h_{\epsilon}$. Note that it follows from item (b) of Definition 4.1 that \phi_{\epsilon}\big{|}_{G}=\psi_{\epsilon}\big{|}_{G}.

From now on, we will identify via $h_{\epsilon}$ the spaces $\underleftarrow{\lim}\{D_{2},\psi_{\epsilon}\}$ with $D_{2}$. Denote $K_{\epsilon}:=h_{\epsilon}(\underleftarrow{\lim}\{G,\phi_{\epsilon}\})$. Then, in order to shorten the cumbersome notations, we will neglect the homeomorphism $h_{\epsilon}$ and work with $K_{\epsilon}$ by identifying it with $\underleftarrow{\lim}\{G,\phi_{\epsilon}\}$. Finally, we observe that $\psi_{\epsilon}$ extends to $\mathbb{S}^{2}$, and we are not concerned about what happens outside $D_{2}$. So in what follows, it suffices for us to only consider $\psi_{\epsilon}$ and $\Phi_{\epsilon}$ restricted to $D_{2}.$

**Step 3. Main arguments of the proof.

**

We start to check the assertions of Theorem 1.1. As explained in the previous paragraph, by Lemma 2.1, the $\Phi_{\epsilon}$-invariant continuum $K_{\epsilon}$ is an attractor, and $\Phi_{\epsilon}|_{K_{\epsilon}}$ is topologically conjugate to the shift homeomorphism $\sigma_{\phi_{\epsilon}}$ for every $\epsilon\in I\setminus\{0\}$. Furthermore, by (b) from Lemma 2.1, $K_{\epsilon}$ vary continuously in the Hausdorff distance with parameter $\epsilon\in I$. This shows item (i) of Theorem 1.1.

Next, in order to understand the prime ends rotation number, we need to first study accessible points of these attractors. First we show how this is done when $\epsilon=\epsilon_{0}$.

Lemma 4.3**.**

Recall the choice of $q_{0}$ and $q_{1}$ at the end of Step 1 of the proof. Let $\underline{q}:=(\ldots,q_{0},q_{1},q_{0}),\underline{q}^{\prime}:=(\ldots,q_{1},q_{0},q_{1})\in K_{\epsilon_{0}}$. Then, the points $\underline{q}$ and $\underline{q}^{\prime}$ are accessible points of $K_{\epsilon_{0}}$.

Proof.

Choose the smash mapping $\Upsilon_{\epsilon_{0}}$ and the unwrapping $\overline{\phi}_{\epsilon_{0}}$, and obtain the near-homeomorphism $\psi_{\epsilon_{0}}$ as in Step 2. Denote $Q_{0},Q_{1}\subset D_{2}$ for the radial arcs from $q_{0},q_{1}$ to the boundary component $C_{2}$, respectively. Then the following holds:

[TABLE]

Let us focus on $i=1$, and consider the inverse limit set

[TABLE]

Note that, \psi_{\epsilon_{0}}\big{|}_{Q_{i}} is a near-homeomorphism for every $i=0,1$. This shows that $\underline{Q}$ is an arc in the inverse limit space $\underleftarrow{\lim}\{D_{2},\psi_{\epsilon_{0}}\}$, which is identified with $D_{2}$. Moreover, for any $\underline{x}\in\underline{Q}\backslash\{\underline{q}^{\prime}\}$, by definition, for some $k>0$, $\pi_{-k}(\underline{x})\notin G_{2}$. This implies that, for all $m\geq k$, $\pi_{-m}(\underline{x})\notin G_{2}$. Therefore, $\underline{Q}\cap K_{\epsilon_{0}}=\underline{q}^{\prime}$, and so $\underline{q}^{\prime}$ is accessible by $\underline{Q}$. The argument for the point $\underline{q}$ follows analogously. ∎

For any fixed $\epsilon\in(0,\epsilon_{0}]$, let $\bar{K}_{\epsilon}$ denote the union of the attractor $K_{\epsilon}$ with its two complementary domains with boundaries $C_{0}$ and $C_{1}$ respectively (i.e. we are considering filled disk $D_{2}$). Then, $\bar{K}_{\epsilon}$ is a plane non-separating continuum whose boundary is just $K_{\epsilon}$. So we can talk about the exterior prime ends rotation number of $\bar{K}_{\epsilon}$, and denote it as $\rho_{\text{ex}}(\Phi_{\epsilon},K_{\epsilon})$. For the parameter $\epsilon_{0}$, since we found two accessible periodic points $\underline{q}_{0}$ and $\underline{q}_{1}$ of $K_{\epsilon_{0}}$, Lemma 2.2 implies that the exterior prime ends rotation number satisfies $\rho_{\text{ex}}(\Phi_{\epsilon_{0}},K_{\epsilon_{0}})=\frac{1}{2}\text{ mod }\mathbb{Z}$.

In fact, Lemma 4.3 is a simpler version of what we will do next. The new difficulty is that, we do not have accessible periodic orbit in general.

Recall the choices of the smash mapping $\Upsilon_{\epsilon}$ and the unwrapping $\overline{\phi}_{\epsilon}$ in Step 2. Let us mark an arbitrarily point $q_{\ast}\in G_{2}$, and let the radial arc $Q_{\ast}$ connect $q_{\ast}$ to a point in $C_{2}$ where $Q_{\ast}\cap G_{2}=\{q_{\ast}\}$. We can then choose a sequence of backward iterates, namely, $q_{-k}\in\phi_{\epsilon}^{-k}(q_{\ast})$, such that for all $k\geq 1$, $q_{-k}\in J_{\epsilon}$ (the set $J_{\epsilon}$ was defined in item (d) of Definition 4.1). Then we claim that the element $\underline{q}_{\ast}:=(\ldots,q_{-2},q_{-1},q_{\ast})$ is an accessible point of $K_{\epsilon}$.

To show the claim, we note that for any $k\geq 1$, $q_{-k}$ is the endpoint of a radial arc $Q_{-k}$, and $Q_{-k}\cap G_{2}=\{q_{-k}\}$. Observe also that, for all $k\geq 1$, $\psi_{\epsilon}$ restricted to $Q_{-k}$ is a near-homeomorphism onto $Q_{-(k-1)}$. Consider the inverse limit space, obtained by these arcs, $\underline{Q}_{\ast}:=(\ldots,Q_{-2},Q_{-1},Q_{\ast})$. Then, $\underline{Q}_{\ast}$ is an arc in the inverse limit space $\underleftarrow{\lim}\{D_{2},\psi_{\epsilon}\}$, which in turn is homeomorphic via $h_{\epsilon}$ to $D_{2}$. Similar argument as in the proof of Lemma 4.3 shows that, $\underline{Q}_{\ast}\cap K_{\epsilon}=\{\underline{q}_{\ast}\}$, and clearly $\underline{Q}_{\ast}\backslash\{\underline{q}_{\ast}\}$ is contained in the exterior complementary domain of $K_{\epsilon}$. Thus, $\underline{q}_{\ast}$ is accessible by $\underline{Q}_{\ast}$ from the exterior complementary domain of $\bar{K}_{\epsilon}$.

Now we are ready to show item (ii) and (iii) of Theorem 1.1. Observe first the case $\epsilon=0$. Define $\phi_{0}=\text{id}$, and then $K_{0}=G_{2}$ and all the construction becomes trivial, in the sense that the smash mapping and the unwrapping are all identity (note that we do not have an attractor at this instance yet). In particular, $\rho_{\text{ex}}(\Phi_{0},K_{0})=0\text{ mod }\mathbb{Z}$. Then by Proposition 2.3, we note the exterior prime ends rotation numbers $\rho_{\text{ex}}(\Phi_{\epsilon},K_{\epsilon})$ vary continuously with $\epsilon\in I$.

Both the smash mapping and the unwrapping can be defined in a symmetric way, with respect to $S_{0}$ and $S_{1}$, due to symmetricity in the definition of the map $\phi_{\epsilon}$. Let us consider the continuum $\text{Fill}(S_{0}\bigcup S_{1})$, obtained by the union of $G_{2}$ with the interior disks bounded by two circles $S_{0}$ and $S_{1}$. The boundary $G_{2}$ of the continuum $\text{Fill}(S_{0}\bigcup S_{1})$ can be for our purposes regarded as a single circle 444Rigorously, the circle $\mathbb{T}^{1}$ is exactly the exterior prime ends circle of the continuum $\text{Fill}(S_{0}\cup S_{1})$. Therefore, the point $x_{0}\in S_{0}\cap S_{1}$ corresponds to two antipodal points in the circle $\mathbb{T}^{1}$. , denoted by $\mathbb{T}^{1}$. Naturally, the map $\phi_{\epsilon}$ induces a map $\phi_{\epsilon}^{\ast}$ on $\mathbb{T}^{1}$. Observe that, $\phi_{\epsilon}^{\ast}$ is a two cover of a circle endomorphism with two turns (see the notation in Section 3).

Fix any $\epsilon\in(0,\epsilon_{0}]$, and choose a proper lift $\widetilde{\phi}_{\epsilon}^{\ast}$. We see the interval $J_{\epsilon}$ corresponds to exactly two copies of the efficient climbing intervals for the lifted map. The restriction of $\widetilde{\phi}_{\epsilon}^{\ast}$ to $[0,\frac{1}{2}]$ consists of one increasing interval and one decreasing interval. We now apply Lemma 3.1 to the lifted map $\widetilde{\phi}_{\epsilon}^{\ast}$, obtaining some point $p\in J_{\epsilon}$, such that for any lifted point $\widetilde{p}$ corresponding to $p$, we can choose its backward iterates $\widetilde{p}_{-k}\in(\widetilde{\phi}_{\epsilon}^{\ast})^{-k}(\widetilde{p})$, with the following properties.

1. (1)

Each $\widetilde{p}_{-k}$ belongs to $\pi^{-1}(J_{\epsilon})$ (recall the definition of $\pi$ in the paragraph preceding Lemma 2.2). 2. (2)

The backward rotation number equals the forward rotation number, which realises the upper endpoint of the rotation interval \rho\big{(}\widetilde{\phi}_{\epsilon}^{\ast}\big{)}:

[TABLE]

Then, as we have already shown, the point $(\ldots,p_{-2},p_{-1},p)\in K_{\epsilon}$ is a point accessible from the complement of $\bar{K}_{\epsilon}$. Thus, by Lemma 2.2, we can consider the lifted prime ends rotation number $\widetilde{\rho}_{\text{ex}}(\widetilde{\Phi}_{\epsilon},K_{\epsilon})$, and it follows that

[TABLE]

Now, the monotonicity of the function $\frac{1-\epsilon}{1-2\epsilon}$ implies the monotonicity of $\widetilde{\phi}_{\epsilon}^{\ast}$. Therefore, we conclude that $\rho_{\text{ex}}(\widetilde{\Phi}_{\epsilon},K_{\epsilon})$ is a strictly increasing function as $\epsilon\to\epsilon_{0}$. This combining with continuity of the function shows item (ii) of Theorem 1.1.

Now, for any $\epsilon\in(0,\epsilon_{0}]$, we can choose some parameter $\epsilon^{\prime}<\epsilon$, such that there exists some periodic orbit $p^{\prime}\in J_{\epsilon^{\prime}}$ with positive rational rotation number. Thus, $\widetilde{\rho}_{\text{ex}}(\widetilde{\Phi}_{\epsilon},K_{\epsilon})\geq\widetilde{\rho}_{\text{ex}}(\widetilde{\Phi}_{\epsilon^{\prime}},K_{\epsilon^{\prime}})>0$ for the proper lift. So $\rho_{\text{ex}}(\Phi_{\epsilon},K_{\epsilon})\neq 0\text{ mod }\mathbb{Z}$ for $\epsilon\in(0,\epsilon_{0}]$, and thus the $K_{\epsilon}$ is a rotational attractor.

We next argue that attractors $\{K_{\epsilon}\}_{\epsilon\in I}$ are indeed Lakes of Wada continua. Denote by $U_{\text{out}}$ the complementary domain of $G_{2}$ bounded by $C_{2}$ and by $U_{0}$ and $U_{1}$ the complementary domains bounded by the circles $C_{0}$ and $C_{1}$, respectively. Suppose $\underline{x}\in\underleftarrow{\lim}\{D_{2},\psi_{\epsilon}\}$. If for $\tau\in\{\text{out},0,1\}$, and for a positive integer $m$, we have $\pi_{-m}(\underline{x})\in U_{\tau}$, then it can be verified by the definition of $\psi_{\epsilon}$ in (18), that, for all $k\geq m$, $\pi_{-k}(\underline{x})\in U_{\tau}$.

Thus, for $\tau\in\{\text{out},0,1\}$, and $m\geq 0$, one defines the following.

[TABLE]

[TABLE]

Clearly,

[TABLE]

where $L_{\text{out},\epsilon},L_{0,\epsilon},L_{1,\epsilon}$ are three disjoint $\Phi_{\epsilon}$-invariant domains. Now, for any $\underline{y}=(\ldots,y_{-1},y_{0})\in K_{\epsilon}$, choose arbitrarily $\tau\in\{\text{out},0,1\}$, and integer $k\geq 1$. We pick $\underline{x}\in U_{\tau,k+1,\epsilon}$, such that, for all $j\leq k$, $\pi_{-j}(\underline{x})=\pi_{-j}(\underline{y})$, and $\pi_{-(k+1)}(\underline{x})\in U_{\tau}$. By the arbitrary choice of $k\geq 1$, and by the topology of $\underleftarrow{\lim}\{D_{2},\psi_{\epsilon}\}$, this shows that for any small neighbourhood $B$ of $\underline{y}$ in $\underleftarrow{\lim}\{D_{2},\psi_{\epsilon}\}$, there is some point $\underline{x}\in L_{\tau,\epsilon}\cap B$. In other words, $\underline{y}$ is a boundary point of the domain $L_{\tau,\epsilon}$. Thus $K_{\epsilon}$ is the common boundary of each of the domains $L_{\text{out},\epsilon},L_{0,\epsilon}$ and $L_{1,\epsilon}$. The proof of item (iii) is thus complete.

Now we address (iv). For any $\delta>0$, we can choose sufficiently small $\epsilon$, with $d_{C^{0}}(\phi_{\epsilon},\text{id})<\delta.$ Thus, by the construction of both $\Upsilon_{\epsilon}$ and $\overline{\phi}_{\epsilon}$ from Step 2, by reducing $\epsilon$ one more time if necessary, we can ensure that for all sufficiently small $\epsilon>0$, $d_{C^{0}}(\Phi_{\epsilon},\text{id})\leq\delta,$ which proves item (iv).

To address (v), note that for every $\epsilon\in I$, the topological entropy of $\phi_{\epsilon}$ is $\log\frac{1}{1-2\epsilon}$, since $\phi_{\epsilon}$ is a piecewise monotone map with constant slope $\pm\lambda=\pm\frac{1}{1-2\epsilon}$ (see [24]). This is also equal to h_{\text{top}}(\Phi_{\epsilon}\big{|}_{K_{\epsilon}}), since the natural extension of an endomorphism has the same topological entropy, see [22]. In particular, h_{\text{top}}(\Phi_{\epsilon}\big{|}_{K_{\epsilon}}) decreases and converges to [math] as $\epsilon\to 0$.

Lastly, we check (vi). Choose a point $z_{0}$ contained in the intersection of $U_{\text{out}}$ with $\psi_{\epsilon}^{-1}(G_{2})$, and choose some pre-image $z_{-1}\in\psi_{\epsilon}^{-1}(z_{0})$. We can find an arc $\lambda$ connecting $z_{-1}$ to $z_{0}$, which is contained in $U_{\text{out}}$. Inductively, for any $k\geq 1$, we can define an arc $\lambda_{-(k+1)}\subset U_{\text{out}}$ connecting some point $z_{-(k+1)}$ and $z_{-k}$, such that $\psi_{\epsilon}(\lambda_{-(k+1)})=\lambda_{-k}$. Note that for all $k\geq 1$, $z_{-k}\in\psi_{\epsilon}^{-k}(z_{0})$. Now we consider the inverse limit set $\underline{\lambda}:=(\cdots,\lambda_{-2},\lambda_{-1},\lambda)$. Moreover, we define $\underline{\Gamma}$ as follows.

[TABLE]

From the above argument, it is clear that $\underline{\Gamma}$ is a translation line for $\Phi_{\epsilon}$.

Clearly, $\omega(\underline{\Gamma})=K_{\epsilon}$. Then, the filled set $\bar{K}_{\epsilon}$ is a rotational attractor, which is disjoint from $\underline{\Gamma}$. This shows item (vi).

We finally remark that the cases for $n>3$ can be dealt with analogously. The only difference is that the maps $\phi_{\epsilon,n}:G_{n}\to G_{n}$ where $G_{n}=S_{0}\cup S_{1}\cup\ldots S_{n-1}$ are defined separately for the upper and lower halves of the circles $S_{i}$ where $i\in\{1,\ldots,n-2\}$ (see Figure 4). We omit the details. ∎

Remark 4.4**.**

*Recall that a homeomorphism $h:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is expansive if for some $c>0$, $\sup_{n\in\mathbb{Z}}\text{dist}(h^{n}(x),h^{n}(x))>c$ for any pair $x,y\in\mathbb{R}^{2}$. Here $\Phi_{\epsilon}$ restricted to $K_{\epsilon}$ is not expansive. We can choose two points $x\neq x^{\prime}\in S_{0}$ so that $\phi_{\epsilon}(x)=\phi_{\epsilon}(x^{\prime})\in S_{1}$. Note that such points exist for every $\epsilon>0$ and they can be chosen so that their distance on $S_{0}$ is arbitrarily small. Furthermore, we can choose $\mathrm{dist}(\phi^{-i}_{\epsilon}(x)-\phi^{-i}_{\epsilon}(x^{\prime}))\to 0$ as $i\to\infty$. Therefore, the induced shift homeomorphism is not expansive. Similar argument works for the cases when $n>3$ as well. *

5. Wada Lakes Rotational Attractors without fixed points

In this section, we prove Theorem 1.2, which provides a family of Lakes of Wada rotational attractors without fixed points in the boundary of the attractor.

Proof of Theorem 1.2.

Set $n=2$, so we aim to construct $4$-separating Lakes of Wada rotational attractor with no fixed points in its boundary. Denote by $D_{3}$ a closed topological disk minus a union of three disjoint open disks. Let $G_{3}=S_{0}\cup S_{1}\cup S_{2}\subset D_{3}\subset\mathbb{S}^{2}$ be a chain of three circles. Suppose $S_{0}$ and $S_{1}$ intersect at a single point $a_{1}$. Let $S_{1}$ and $S_{2}$ intersect at a single point $a_{3}$. Denote points $q_{4}\in S_{0}$, $q_{1},q_{3}\in S_{1}$ and $q_{2}\in S_{2}$ as depicted in Figure 4.

Similar to the definition of the map $\phi_{\epsilon}$ in the proof of Theorem 1.1, we define a piecewise uniform scaling map $\xi:G_{3}\to G_{3}$, satisfying the following conditions (see Figure 4).

1. (1)

Restricted to $S_{0}$, $\xi$ maps the arc from $a_{1}$ to $q_{5}$ to the whole circle $S_{0}$, uniformly scaled and counterclockwise; it maps the arc from $q_{5}$ to $q_{4}$ to the the arc from $a_{1}$ to $q_{3}$ in a uniform scaling way; it maps the arc from $q_{4}$ to $a_{1}$ to the arc from $q_{3}$ to $a_{1}$, reversing the orientation. 2. (2)

Restricted to the upper half circle of $S_{1}$, $\xi$ maps the arc from $a_{1}$ to $q_{1}$ to the upper arc from $a_{1}$ to $q_{4}$ in a uniform scaling way. It maps the arc from $q_{1}$ to $q_{6}$ to the upper arc from $q_{4}$ to $a_{1}$, uniformly scaled. It maps the arc from $q_{6}$ to $a_{3}$ to the upper arc from $a_{1}$ to $a_{3}$, uniformly scaled. 3. (3)

Finally, restricted to the lower half circle of $S_{1}$ and restricted to $S_{2}$, the definition of $\xi$ is given in a symmetric way. We omit the precise description and refer to Figure 4.

Let $R:G_{3}\to G_{3}$ be the rotation of $G_{3}$ by the angle $\pi$ around the central point of the circle $S_{1}$. Define $g:=R\circ\xi:G_{3}\to G_{3}$. Observe that, there exists a $4$-periodic orbit of $g$, namely, $\{q_{1},q_{2},q_{3},q_{4}\}$.

Recall that we denote by $\sigma_{g}$ the corresponding shift homeomorphism for the inverse limit space $\underleftarrow{\mathrm{lim}}\{G_{3},g\}$. Similar to the case in the previous section, $G_{3}$ is a boundary retract of $D_{3}$. We define the smash mapping $\Upsilon$ and the unwrapping $\overline{g}$ in a much simpler way than in Section 4, in particular, here we only need to define one embedding, instead of a parametrised family and thus we just follow a construction from [5]; see Figure 4. Now we can apply Lemma 2.1, to obtain an extension $\psi:D_{3}\to D_{3}$ of the unwrapping $\overline{g}$, as well as a homeomorphism $\Phi=\sigma_{\psi}$ of $D_{3}$ (which is homeomorphic to $\underleftarrow{\mathrm{lim}}\{D_{3},\psi\}$), for which $K=\underleftarrow{\mathrm{lim}}\{G_{3},g\}$ is an attractor. $K$ is homeomorphic to a Lakes of Wada continuum. The argument to show this follows exactly the lines of the proof of item (iii) of Theorem 1.1.

What remains to be checked for Theorem 1.2 is that $K$ has no fixed points and that its external prime ends rotation number is non-zero.

Claim 5.1**.**

$\Phi|_{K}=\sigma_{g}$* has no fixed points.*

Proof of Claim 5.1.

To prove this it is enough to show that the map $g$ has no fixed points. But this is obvious from the definition of $g$. Therefore the induced homeomorphism $\sigma_{g}$ has no fixed points by [22] as well. ∎

Claim 5.2**.**

The external prime ends rotation number $\rho_{\text{ex}}(\Phi,K)\neq 0\text{ mod }\mathbb{Z}$.

Proof of Claim 5.2.

Note that we have already specified a $4$-periodic orbit of $g$ being $\{q_{1},q_{2},q_{2},q_{4}\}\subset G_{3}$. Then we obtain elements in the inverse limit space, namely,

[TABLE]

These form a $4$-periodic orbit for $\Phi$ in $\underline{\lim}\{G_{3},g\}$. We proceed as in the proof of Lemma 4.3. By attaching arcs, we can show that these four points are indeed all accessible points of the continuum $K$. Clearly, again by Lemma 2.2, the dynamics over this periodic orbit of accessible points imply that the external prime ends rotation number $\rho_{\text{ex}}(\Phi,K)$ is $\frac{3}{4}\text{ mod }\mathbb{Z}$, which is nonzero. This concludes the proof of Claim 5.2. ∎

The part of the statement about translation line $\Gamma$ is argued in the same way as in the proof of Theorem 1.1. Arguments concerning $n>2$ follow analogously as described above for $n=2$, working on $G_{2n-1}$ being the chain of $(2n-1)$-circles. ∎

6. Cofrontier Dynamics and Embeddings

We include two more applications of the BBM technique in this section, by proving Theorem 1.3 and Theorem 1.4.

6.1. Conjecture of Walker revisited

As we have already mentioned in the introduction, in the spirit of Walker’s conjecture, it is still a question if certain cofrontier dynamics can induce two different irrational prime ends rotation numbers in the two complementary domains. Walker’s paper [30] did not contain results with such properties. In this section, we answer this question affirmatively by proving Theorem 1.3.

Proof of Theorem 1.3.

Let us begin by defining a parametrised family $\{\widetilde{g}_{t}\}_{t\geq 0}\subset\widetilde{\text{End}}_{1}(\mathbb{S}^{1})$ as follows.

[TABLE]

This is a reduced Arnold’s family (see [8] for more). It is known that when $t\geq t_{\ast}$ for some $t_{\ast}>1$, the rotation set $\rho(\widetilde{g}_{t})$ is a nondegenerate interval. Note that this family consists of odd functions. It follows immediately that, $\rho(\widetilde{g}_{t})$ is of the form $[-\eta_{t},\eta_{t}]$ for some $\eta_{t}>0$. On the other hand, these rotation intervals change continuously (Lemma 3.1 of [8]). Clearly, we can choose $I_{\ast}=[t_{\ast},t_{\ast}^{\prime}]$, such that $\eta_{t_{\ast}^{\prime}}>\eta_{t_{\ast}}$.

In what follows, we will work with $D_{1}$ which is the closed annulus. The round circle $G_{1}=\mathbb{S}^{1}$ can be regarded as the spine of $D_{1}$.

The main idea is to again apply Lemma 2.1 to study appropriate inverse limit spaces. Denote the outer boundary and the inner boundary of $D_{1}$ by $C_{1}$ and $C_{0}$, respectively. Therefore, we obtain the radial decomposition $\alpha:C_{0}\bigcup C_{1}\times[0,1]\to D_{1}$ as before. Note that, for all $t\in I_{\ast}$, the bonding map $g_{t}$ is a circle endomorphism with two turns.

Define the smash mapping $\Upsilon_{t}$ as in [9]. Then we define the unwrapping $\overline{g}_{t}$, similar to the definition of $\overline{\phi}_{\epsilon}$ in Definition 4.1; see Figure 2. More precisely, the definition of the unwrapping $\overline{g}_{t}$ is given as follows (c.f. Definition 4.1).

1. (1)

$\overline{g}_{t}$ is a near-homeomorphism. $\overline{g}_{t}$ restricted to $G_{1}$ is a homeomorphism onto its image. 2. (2)

\Upsilon_{t}\circ\overline{g}_{t}\big{|}_{G_{1}}=g_{t}. 3. (3)

there exists an interval $J_{\text{out},t}\subset G_{1}$, which corresponds to the efficient climbing interval of $g_{t}$, such that if for any $y\in J_{\text{out},t}$ we denote $x\in C_{1}$ such that $\alpha(x,1)=y$, then $\overline{g}_{t}$ restricted to the radial arc $\alpha(x,[0,1])$ is a monotone map, whose image is contained in the radial arc connecting $g_{t}(y)$ to the corresponding point in $C_{1}$. 4. (4)

there exists an interval $J_{\text{in},t}\subset G_{1}$, which corresponds to the lower climbing interval of $g_{t}$, such that if for any $y^{\prime}\in J_{\text{in},t}$ we denote $x^{\prime}\in C_{0}$ with $\alpha(x^{\prime},1)=y^{\prime}$, then $\overline{g}_{t}$ restricted to the radial arc $\alpha(x^{\prime},[0,1])$ is a monotone map, whose image is contained in the radial arc connecting $g_{t}(y^{\prime})$ to the corresponding point in $C_{0}$.

The full details are left to the reader because of analogy with the proof of Theorem 1.1. We proceed to define the homeomorphism $\psi_{t}$ which extends $g_{t}$ to $D_{1}$, as well as the shift homeomorphism $\Phi_{t}=\sigma_{\psi_{t}}$, exactly the same way as in Step 2 of the proof of Theorem 1.1. By Lemma 2.1, for each $t$, we obtain the inverse limit space $\underleftarrow{\lim}\{D_{1},\psi_{t}\}$, which is homeomorphic to $D_{1}$. Moreover, the inverse limit space $\underleftarrow{\lim}\{G_{1},g_{t}\}$ is homeomorphic to $K_{t}$.

Now, each inverse limit space $K_{t}$ is a circle-like continuum by definition. It follows that every proper subcontinuum of $K_{t}$ is chainable and thus non-separating. Since clearly $K_{t}$ has empty interior, it follows $K_{t}$ is a cofrontier with two complementary domains for each $t\in I_{\ast}$. To proceed we need the following claim.

Claim 6.1**.**

Suppose that $\{\Phi_{t}\}_{t\in I_{\ast}}$ and $\{K_{t}\}_{t\in I_{\ast}}$ are as above. Then for any $t\in I_{\ast}$ the following properties hold.

1. (a)

There exists an accessible point $\underline{p}_{\text{ex}}\in K_{t}$ (respectively, $\underline{p}_{\text{in}}\in K_{t}$), corresponding to some point $p_{\text{ex}}\in G_{1}$ (respectively, $p_{\text{in}}\in G_{1}$), whose forward iterates (respectively, backward iterates) rotation number is equal to the upper endpoint of the rotation interval, $\sup\rho(\widetilde{g}_{t})$ (respectively, to the lower endpoint $\inf\rho(\widetilde{g}_{t})$). 2. (b)

For a certain lift $\widetilde{\Phi}_{t}$, the lifted external prime ends rotation number $\widetilde{\rho}_{\text{ex}}(\widetilde{\Phi}_{t},K_{t})$ equals $\sup\rho(\widetilde{g}_{t})$ and lifted internal prime ends rotation number $\widetilde{\rho}_{\text{in}}(\widetilde{\Phi}_{t},K_{t})$ equals $\inf\rho(\widetilde{g}_{t})$. 3. (c)

The attractor $K_{t}$ is an indecomposable continuum.

Remark 6.2**.**

The proof is a simplified version of a part of the proof of Theorem 1.1, with a few variations. So we only sketch the main points and stress the differences.

Proof of Claim 6.1.

In item (a), for the efficient climbing interval $J_{\text{out},t}$, we can apply Lemma 3.1 to find some point $p_{\text{ex}}\in J_{\text{out},t}$, and choices of its backward iterates, $\{p_{-\ell}\}_{\ell\geq 1}\subset J_{\text{out},t}$, such that the backward rotation number of $p_{\text{ex}}$ and the forward rotation number of $p_{\text{ex}}$ coincide and equal to the upper endpoint of the rotation interval $\eta_{t}=\sup\rho(\widetilde{g}_{t})$. Then we denote $\underline{p}:=(\ldots,p_{-2},p_{-1},p_{\text{ex}})\in K_{t}\simeq\underleftarrow{\lim}(G_{1},g_{t})$.

Now, similar to what we did in the proof of Theorem 1.1, one can denote $Q_{0}$ for the radial arc connecting $p_{\text{ex}}$ to some point in the outer boundary $C_{1}$ of $D_{1}$, and define $Q_{-\ell}$ to be the radial subarc of the arc connecting $p_{-\ell}$ to the corresponding point in $C_{1}$ so that $\psi_{t}(Q_{-\ell})=Q_{-(\ell-1)}$ for all $\ell\geq 1$. It follows that the inverse limit set $\underline{Q}:=(\ldots,Q_{-1},Q_{0})\subset\underleftarrow{\lim}\{D_{1},\psi_{t}\}$ is an arc, and $\underline{Q}\bigcap K_{t}=\{\underline{p}\}$. Therefore, $\underline{p}$ is an accessible point of $K_{t}$ from the exterior domain by the arc $\underline{Q}$. This completes half of the proof of item (a) of Claim 6.1. For the other part, we use the definition of lower climbing interval. In a similar way, one shows that there is point $p_{\text{in}}\in G_{1}$, whose forward rotation number and the backward rotation number coincide and equal $-\eta_{t}=\inf\rho(\widetilde{g}_{t})$. Then we can obtain $\underline{p}_{\text{in}}$ in $K_{t}$, which is accessible by an arc from interior of the annulus. Item (a) is thus complete.

Now we check item (b). Observe that, by the choice of the point $p_{\text{ex}}$, the forward and backward rotation numbers of $p_{\text{ex}}$ coincide. Then by Lemma 2.2, this number is equal to the lifted exterior prime ends rotation number $\widetilde{\rho}_{\text{ex}}(\widetilde{\Phi}_{t},K_{t})$, for a proper lift. Then by item (a), $\widetilde{\rho}_{\text{ex}}(\widetilde{\Phi}_{t},K_{t})$ is equal to $\eta_{t}$. The situation for the interior prime ends rotation number is similar.

For item (c), recall that the rotation set $\rho(\widetilde{g}_{t})$ is a non-degenerate segment. By Theorem 2.7 of [4], $K_{t}$ is an indecomposable continuum for every $t\in I_{\ast}$. ∎

We are now ready to finish the proof of Theorem 1.3. By Claim 6.1, for $t\in I_{\ast}$, the rotation set \rho(\widetilde{\Phi}_{t}\big{|}_{K_{t}}) is a non-trivial segment. By definition, item (i) is proved, namely $K_{t}$ are indeed Birkhoff-like attractors.

To show item (ii), note that for any $t\in I_{\ast}$, the circle map $g_{t}$ is topologically exact. Then by Theorem 22 in [19], for any $\epsilon>0$, there is another map $g_{t}^{\prime}$, with $d_{C^{0}}(g_{t},g^{\prime}_{t})<\epsilon$, such that the inverse limit $\underleftarrow{\lim}\{G_{1},g_{t}^{\prime}\}$ is the pseudo-circle (see Theorem 3.2 of [7] where this argument was applied as well). Therefore, we can apply a similar construction as we did in Theorem 1.1 with a properly chosen smash function and the unwrapping. Therefore, the invariant pseudo-circle Birkhoff-like attractor $K^{\prime}_{t}$ can be embedded such that $d_{H}(K^{\prime}_{t},K_{t})<\epsilon$. This shows item (ii).

To show item (iii), observe that when $t$ varies in $I_{\ast}$ the rotation set $\rho(\widetilde{g}_{t})$ changes continuously in a strictly monotone way from $[-\eta_{t_{\ast}},\eta_{t_{\ast}}]$ to $[-\eta_{t_{\ast}^{\prime}},\eta_{t_{\ast}^{\prime}}]$, with $\eta_{t_{\ast}^{\prime}}>\eta_{t_{\ast}}$. It follows that, there are uncountably many parameters $t\in I_{\ast}$ for which $\eta_{t}\notin\mathbb{Q}$. In particular, for those choices of $t$, the two lifted prime ends rotation numbers are different and both are irrationals. ∎

In the proof of this theorem, we thus already showed the following statement.

Corollary 6.3**.**

For a circle endomorphism with two turns, we can embed the inverse limit space such that, the exterior prime ends rotation number and the interior prime ends rotation number equal the two endpoints of the rotation interval of the shift homeomorphism restricted to it.

6.2. Prime ends rotation number realising

an interior point of the rotation interval

In this subsection, we prove Theorem 1.4. The proof is much simpler and does not require parametrised family of inverse limit embeddings. In fact, we only give one circle endomorphism and we will exhibit two different embeddings by drawing the graph of the unwrapping.

Proof of Theorem 1.4.

Similar to the arguments we have seen in previous sections, we construct the attractor as an inverse limit of a continuous circle endomorphism $g$. However, for our purpose here, the circle map $g$ is not an endomorphism with two turns. By definition, $g$ is affine restricted to the five consecutive subintervals splitting the circle (see left of Figure 5). Then we define the unwrapping $\overline{g}$ as depicted on the right of Figure 5. We omit the precise definitions because they are much simpler than what we did in previous sections. Then, like in the proof of Theorem 1.2, we can define the smash mapping $\Upsilon$ preserving radial segments, as well as the near-homeomorphism $\psi$ which extends $g$ to $D_{1}$.

Then we invoke Lemma 2.1, to obtain the inverse limit space $\underleftarrow{\lim}\{D_{1},\psi\}$, which is homeomorphic to $D_{1}$, and obtain the shift homeomorphism $\Phi=\sigma_{\psi}$ on $\underleftarrow{\lim}\{D_{1},\psi\}$, with the attractor $K=\underleftarrow{\lim}\{\mathbb{S}^{1},g\}$, where $\mathbb{S}^{1}$ is the circle. As we have already argued in the previous subsection, $K$ is a cofrontier.

Note that the lift $\widetilde{g}$ at point $1/8$ and $3/8$ has rotation number equal to $-1$, and at point $1/4$ has rotation number $1$. So the rotation interval $\rho(\widetilde{g})$ is non-trivial (in fact it is equal to $[-1,1]$ by more detailed analysis of the map $g$ which we omit here). Moreover, by the relation (4.1) in [9], we have that $\rho(\widetilde{\Phi},K)=\rho(\widetilde{g})=[-1,1]$ for proper choice of lifts.

On the other hand, note that $\widetilde{g}$ point-wise fixes the arc $[1/2,1]$, where $\widetilde{g}$ has rotation number [math]. Applying similar arguments as in the proof of Lemma 4.3, we show that there is a semi-circle of fixed points that are accessible from interior. By Lemma 2.2 we conclude that the lifted interior prime ends rotation number $\rho_{\text{in}}(\widetilde{\Phi},K)$ equals [math]. Similar considerations show that the lifted exterior prime ends rotation number $\rho_{\text{ex}}(\widetilde{\Phi},K)$ equals $1$.

It remains to be shown that the shift homeomorphism on $K$ (i.e., the restriction of $\Phi$ on $K$) is not transitive. This follows from the fact that $g$ is not transitive, because it contains an arc of fixed points, and from the fact that $\omega((\ldots,x_{2},x_{1}),\sigma_{g})=\underleftarrow{\mathrm{lim}}\{\omega(x_{1},g),g\}$ for any $x_{1}\in\mathbb{S}^{1}$ (see for example [22]).

For the last assertion of the theorem, we will consider a different embedding of $\underleftarrow{\mathrm{lim}}(\mathbb{S}^{1},g)$ that makes the arc of fixed points inaccessible. The idea is to define the unwrapping in a different way. Let us skip the precise definition and only refer to Figure 6 below for the graph of the unwrapping. Then following the same argument invoking Lemma 2.1, one obtains the cofrontier attractor $K^{\prime}$, which is homeomorphic to $K$ by construction. The proof of Theorem 1.4 is now completed. ∎

Remark 6.4**.**

Note that we could (similarly as in previous sections) obtain a parametrised family of rotational attractors by modifying the map $\widetilde{g}$ and move three critical points in $(0,1/2)$ parallely with $y-axis$ uniformly towards the map $\widetilde{id}$. Embedding the spaces similarly as it is suggested on Figure 5 we would obtain a parametrised family of examples answering the question of Boyland above (of course excluding the map $\widetilde{id}$). Choosing an appropriate smash similarly as in Theorem 2.1 we can even obtain a parametrised family of such non-transitive examples arbitrary close to identity.

Remark 6.5**.**

Two BBM embeddings are called equivalent if there is a conjugacy $h$ which restricted to the attractor is identity. In Theorem 2.17 from [11], (based on previous results from [28]) the authors proved that any two BBM embeddings of unimodal inverse limit spaces are equivalent. In Theorem 1.4 we provide a contrast picture for circle-like continua as demonstrated with the two non-equivalent planar embeddings obtained from the BBM construction.

7. Acknowledgements

The first two authors are supported by University of Ostrava subsidy for institutional development IRP201824 “Complex topological structures” and the NPU II project LQ1602 IT4 Innovations excellence in science. JB is grateful to Luis Hernández-Corbato, Jose M.R. Sanjurjo and Francisco R. Ruiz del Portal for some useful conversations and hospitality during author’s visit at Universidad Complutense de Madrid in October 2018. JČ was also supported by FWF Schrödinger Fellowship stand-alone project J-4276. X-C. Liu is supported by Fapesp Pós-Doutorado grant (Grant Number 2018/03762-2). X-C. Liu thanks Ana Anušić for useful conversations. We thank also P. Boyland, A. Koropecki and A. Passeggi for useful remarks, as well as Salvador Addas-Zanata and Fabio Armando Tal for valuable comments on Subsection 6.1, and for suggesting to consider reduced Arnold’s family.