Topological horseshoes for surface homeomorphisms

TL;DR

This paper introduces a new topological criterion for detecting horseshoes in surface homeomorphisms, with applications to rotation theory, surface dynamics, and structure of transitive sets, advancing understanding of surface dynamics with zero entropy.

Contribution

It develops a purely topological criterion for horseshoes using equivariant Brouwer foliations, and applies it to analyze surface homeomorphisms with zero entropy, including annuli and sphere maps.

Findings

Existence of rotation numbers for points with nonempty omega limits in annuli.

Single rotation vector for generalized regions of instability.

Structure theorem for sphere homeomorphisms without horseshoes, limiting possible dynamics.

Abstract

In this work we develop a new criterion for the existence of topological horseshoes for surface homeomorphisms in the isotopy class of the identity. Based on our previous work on forcing theory, this new criterion is purely topological and can be expressed in terms of equivariant Brouwer foliations and transverse trajectories. We then apply this new tool in the study of the dynamics of homeomorphisms of surfaces with zero genus and null topological entropy and we obtain several applications. For homeomorphisms of the open annulus $\mathbb{A}$ with zero topological entropy, we show that rotation numbers exists for all points with nonempty omega limits, and that if $\mathbb{A}$ is a generalized region of instability then it admits a single rotation vector. We also offer a new proof of a recent result of Passegi, Potrie and Sambarino, showing that zero entropy dissipative homeomorphisms of the annulus having as an atractor a circloid have a single rotation number. Our work also studies homeomorphisms of the sphere without horseshoes. For these maps we present a structure theorem in terms of fixed point free invariant sub-annuli, as well as a very restricted description of all possible dynamical behavior in the transitive subsets. This description ensures, for instance, that transitive sets can contain at most $2$ distinct periodic orbits and that, in many cases, the restriction of the homeomorphism to the transitive set must be an extension of an odometer. In particular, we show that any nontrivial and stable transitive subset of a dissipative diffeomorphism of the plane is always infinitely renormalizable in the sense of Bonatti-Gambaudo-Lion-Tresser.

Equations

$$rh(f)=h(f^r)⩾h(f^r{|}_Y)=h(g)⩾h(\sigma )=\log q,$$

$$\underset{k\to +\infty }{\lim }\frac{1}{n_k}\left({\pi }_1({\check{f}}^{n_k}(\check{z}))-{\pi }_1(\check{z})\right)={\mathrm{rot}}_{\check{f}}(z),$$

$${\mathrm{rot}}_{\check{h}\circ \check{f}\circ {\check{h}}^{-1}}(h(z))=\{\begin{array}{ll}{\mathrm{rot}}_{\check{f}}(z)& \mathrm{if}h\mathrm{induces}\mathrm{the}\mathrm{identity}\mathrm{on}{H}_1(\mathbb{A},\mathbb{Z}),\\ -{\mathrm{rot}}_{\check{f}}(z)& otherwise.\end{array}$$

$${\mathrm{rot}}_{\check{f},-\kappa }(z)=-{\mathrm{rot}}_{\check{f},\kappa }(z).$$

$${\mathrm{rot}}_{f,\kappa }(z)={\mathrm{rot}}_{\check{f},\kappa }(z)+\mathbb{Z}\in \mathbb{R}/\mathbb{Z}.$$

$${\mathrm{rot}}_{{\check{f}}^{\pm }}(z)=\{\begin{array}{ll}{\mathrm{rot}}_{\check{f}}(z)& \mathrm{if}z\in \Omega (f)\cap {\mathrm{ne}}^{+}(f),\\ -{\mathrm{rot}}_{{\check{f}}^{-1}}(z)& \mathrm{if}z\in \Omega (f)\cap {\mathrm{ne}}^{-}(f).\end{array}$$

$${\mathrm{ne}}^{+}(f)={\mathbb{T}}^1\times [-1,1],{\mathrm{ne}}^{-}(f)=\mathbb{A},\Omega (f)={\mathbb{T}}^1\times \{-1,0,1\}.$$

$$\mathrm{rot}_{\check{f}}(z)=\begin{cases}-1&\mathrm{if}\enskip z\in\mathbb{T}^{1}\times\{-1\},\\ 0&\mathrm{if}\enskip z\in\mathbb{T}^{1}\times(-1,1),\\ 1&\mathrm{if}\enskip z\in{\color[rgb]{0,0,0}\mathbb{T}^{1}\times\{1\}},\\ \end{cases}$$

$${\mathrm{rot}}_{{\check{f}}^{-1}}(z)=\{\begin{array}{ll}1& \mathrm{if}z\in {\mathbb{T}}^1\times (-\infty ,0),\\ 0& \mathrm{if}z\in {\mathbb{T}}^1\times \{0\},\\ -1& \mathrm{if}z\in {\mathbb{T}}^1\times (0,+\infty ).\end{array}$$

$$\{\begin{array}{ll}{\mathrm{rot}}_{\check{f}}(z)=\rho & \mathrm{if}z\in \mathcal{B}\cap {\mathrm{ne}}^{+}(f),\\ {\mathrm{rot}}_{{\check{f}}^{-1}}(z)=-\rho & \mathrm{if}z\in \mathcal{B}\cap {\mathrm{ne}}^{-}(f).\end{array}$$

$$\{\begin{array}{ll}{\mathrm{rot}}_{\check{f}}(z_i)=\rho & \mathrm{if}{z}_i\in {\mathrm{ne}}^{+}(f),\\ {\mathrm{rot}}_{{\check{f}}^{-1}}(z_i)=-\rho & \mathrm{if}{z}_i\in {\mathrm{ne}}^{-}(f).\end{array}$$

$$\check{z}\mapsto \frac{{\pi }_1({\check{f}}^n(\check{z}))-{\pi }_1(\check{z})}{n}$$

$${\Omega }^{\prime }(f)=\{z\in \Omega (f)|\alpha (z)\cup \omega (z)\not\subset \mathrm{fix}(f)\}.$$

$$I^n(z)=\prod _{0⩽k<n}I(f^k(z)).$$

$$I^{\mathbb{N}}(z)=\prod _{k⩾0}I(f^k(z)),I^{-\mathbb{N}}(z)=\prod _{k<0}I(f^k(z)),I^{\mathbb{Z}}(z)=\prod _{k\in \mathbb{Z}}I(f^k(z)),$$

$$I_{\mathcal{F}}^n(z)=\prod _{0⩽k<n}{I}_{\mathcal{F}}(f^k(z)).$$

$$I_{\mathcal{F}}^{\mathbb{N}}(z)=\prod _{k⩾0}{I}_{\mathcal{F}}(f^k(z)),I_{\mathcal{F}}^{-\mathbb{N}}(z)=\prod _{k<0}{I}_{\mathcal{F}}(f^k(z)),I_{\mathcal{F}}^{\mathbb{Z}}(z)=\prod _{k\in \mathbb{Z}}{I}_{\mathcal{F}}(f^k(z)),$$

$$|{\pi }_1({\check{h}}_i\circ {\check{f}}_i^{n_k}(\check{z}))-{\pi }_1({\check{h}}_i(\check{z}))-m_k|⩽M.$$

$${\mathrm{rot}}_{{\check{f}}_i}(z)=\underset{k\to +\infty }{\lim }\frac{m_k}{n_k}=\underset{k\to +\infty }{\lim }\frac{{\delta }_k(z_{i+1})-{\delta }_k(z_{i+2})}{n_k}.$$

$$z=z_0\underset{f}{\to }{z}_1\underset{f}{\to }\dots \underset{f}{\to }{z}_{p-1}\underset{f}{\to }{z}_p=z^{\prime }.$$

$$z=z_0\underset{f}{\to }{z}_1\underset{f}{\to }\dots \underset{f}{\to }{z}_{p-1}\underset{f}{\to }{z}_p=z.$$

$$z\underset{f}{\to }f(z)\underset{f}{\to }f(z_1)\underset{f}{\to }\dots \underset{f}{\to }f(z_{i_0})\underset{f}{\to }{z}_{i_0+1}\underset{f}{\to }\dots \underset{f}{\to }{z}_{p-1}\underset{f}{\to }{z}_p=z,$$

$$z=z_0\underset{f}{\to }{z}_1\underset{f}{\to }\dots \underset{f}{\to }{z}_{i_0}\underset{f}{\to }{z}^{\prime }\underset{f}{\to }{z}_{i_0+1}\underset{f}{\to }\dots \underset{f}{\to }{z}_{p-1}\underset{f}{\to }{z}_p=z,$$

$$W_1\cap f^{-qs}(W_2)\subset W_1^0\cap f^{-qs}(W_2^0)=\varnothing ,$$

$$W_1\cap f^{-qs-r}(W_2)\subset W_1^{q-r}\cap f^{-q(s+1)}(W_2^{q-r})=\varnothing ,$$

$$z=z_0\underset{f}{\to }{z}_1\underset{f}{\to }\dots \underset{f}{\to }{z}_{p-1}\underset{f}{\to }{z}_p=z.$$

$$z=z_0\underset{f^q}{\to }{f}^{s_1}(z_1)\underset{f^q}{\to }\dots \underset{f^q}{\to }{f}^{s_{p-1}}(z_{p-1})\underset{f^q}{\to }{f}^{s_p}(z_p)=f^{s_p}(z),$$

$$z=z_0\underset{f^q}{\to }{f}^{s_1}(z_1)\underset{f^q}{\to }\dots \underset{f^q}{\to }{f}^{s_{p-1}}$$