Dimensions and dimension spectra of Non-autonomous iterated function systems

TL;DR

This paper investigates the intermediate dimension spectra of non-autonomous conformal sets, unifying various fractal dimensions, and provides formulas for their Hausdorff, packing, and box dimensions, including for systems with countably many mappings.

Contribution

It introduces a formula for the intermediate dimension spectra of non-autonomous conformal sets using topological pressures, extending understanding of their fractal dimensions.

Findings

Derived the intermediate dimension spectra formula using topological pressures.

Simplified formulas for Hausdorff, packing, and box dimensions of these sets.

Established Hausdorff dimension formulas for systems with countably many conformal mappings.

Abstract

Non-autonomous iterated function systems are a generalization of iterated function systems. If the contractions in the system are conformal mappings, it is called a non-autonomous conformal iterated function system, and its attractor is called a non-autonomous conformal set. In this paper, we study intermediate dimension spectra of non-autonomous conformal sets which provide a unifying framework for Hausdorff and box-counting dimensions. First, we obtain the intermediate dimension spectra formula of non-autonomous conformal sets by using upper and lower topological pressures. As a consequence, we obtain simplified forms of their Hausdorff, packing and box dimensions. Finally, we explore the Hausdorff dimensions of the non-autonomous infinite conformal iterated function systems which consists of countably many conformal mappings at each level, and we provide the Hausdorff dimension formula under certain conditions.

Equations

$${\underline{\text{mbox}\dim }}_{\theta }E$$

$${\overline{\text{mbox}\dim }}_{\theta }E$$

$${\overline{\dim }}_{0}E={\underline{\dim }}_{0}E=\text{mbox}{\dim }_{\mathrm{H}}E,{\overline{\dim }}_{1}E={\overline{\dim }}_{B}E,{\underline{\dim }}_{1}E={\underline{\dim }}_{B}E,$$

$$E=\bigcup _{i\in I}{\phi }_i(E).$$

$$\text{mbox}{\dim }_{\mathrm{H}}E={\overline{\text{mbox}\dim }}_{\mathrm{B}}E=h.$$

$$\text{mbox}{\dim }_{\mathrm{H}}E={\dim }_{B}E=\min \{d,d(T_1,\dots ,T_{\#I})\},$$

$${\overline{\text{mbox}\dim }}_{\theta }E=\max \{h,{\overline{\text{mbox}\dim }}_{\theta }P\}\text{for }\theta \in [0,1],$$

$${\Sigma }_l^k=\{u_l{u}_{l+1}\dots u_k:u_j\in I_j,j=l,l+1,\dots ,k\},$$

$${\Sigma }^{\ast }=\bigcup _{k=0}^{\infty }{\Sigma }^k$$

$$|{\phi }_{k,i}(x)-{\phi }_{k,i}(y)|\le c|x-y|\text{ for all }x,y\in J.$$

$$J_{\mathbf{u}}={\Psi }_{\mathbf{u}}(J)={\phi }_{\mathbf{u}}(J)+{\omega }_{\mathbf{u}},$$

$$E=E(\Phi )=\bigcap _{k=1}^{\infty }\bigcup _{\mathbf{u}\in {\Sigma }^k}{\Psi }_{\mathbf{u}}(J)$$

$$\text{mbox}int(J_{\mathbf{u}i})\cap \text{mbox}int(J_{\mathbf{u}j})=\varnothing ,$$

$$J_{\mathbf{u}i}\cap J_{\mathbf{u}j}=\varnothing ,$$

$$\underset{k\to +\infty }{\lim }\frac{\log {\underline{c}}_k}{\log M_k}=0,$$

$$\text{mbox}{\dim }_{\mathrm{H}}E=s_{\ast }=\underset{k\to \infty }{\mathrm{liminf}}{s}_k,\text{mbox}{\dim }_{\mathrm{P}}E={\overline{\text{mbox}\dim }}_{\mathrm{B}}E=s^{\ast }=\underset{k\to \infty }{\mathrm{limsup}}{s}_k,$$

$$\Vert D\varphi \Vert =\sup \{\Vert D\varphi (x)\Vert :x\in J\}.$$

$$\Vert D{\phi }_{\mathbf{u}}(x)\Vert \le C\Vert D{\phi }_{\mathbf{u}}(y)\Vert$$

$$E=E(\Phi )=\bigcap _{k=1}^{\infty }\bigcup _{\mathbf{u}\in {\Sigma }^k}{J}_{\mathbf{u}}$$

$$\begin{array}{rl}& m_k=\underset{\mathbf{u}\in {\Sigma }^k}{\inf }\{\Vert D{\phi }_{\mathbf{u}}\Vert \},M_k=\underset{\mathbf{u}\in {\Sigma }^k}{\sup }\{\Vert D{\phi }_{\mathbf{u}}\Vert \},\\ & {\underline{c}}_k=\underset{1\le j\le \#I_k}{\inf }\{\Vert D{\phi }_{k,j}\Vert \},{\overline{c}}_k=\underset{1\le j\le \#I_k}{\sup }\{\Vert D{\phi }_{k,j}\Vert \}.\end{array}$$

$$\underset{k\to +\infty }{\lim }\frac{\log {\underline{c}}_k}{\log M_k}=0,$$

$$\underset{k\to \infty }{\lim }\frac{\log \#I_k}{k}=0,$$

$$\underset{k\to +\infty }{\lim }\frac{\log {\overline{c}}_k-\log \#I_k}{\log M_k}=0.$$

$$\underset{k\to +\infty }{\lim }\frac{\log {\underline{c}}_k}{\log M_k}=0\text{ and}\underset{k\to \infty }{\lim }\frac{\log \#I_k}{k}=\log 2.$$

$$\underset{k\to +\infty }{\lim }\frac{\log {\underline{c}}_k}{\log M_k}=2,\underset{k\to \infty }{\lim }\frac{\log \#I_k}{k}=\log 2,\underset{k\to +\infty }{\lim }\frac{\log {\overline{c}}_k-\log \#I_k}{\log M_k}=0.$$

$$\underset{k\to \infty }{\lim }\frac{\log \#I_k}{k}=0\text{ and}\underset{k\to +\infty }{\lim }\frac{\log {\overline{c}}_k-\log \#I_k}{\log M_k}=\frac{1}{2}.$$

$$\mathcal{M}(\delta ,\theta )=\{\mathcal{M}:\mathcal{M}\text{ is a cut set satisfying }{\delta }^{\frac{1}{\theta }}<\Vert D{\phi }_{{\mathbf{u}}^{\ast }}\Vert \text{ and }\Vert D{\phi }_{\mathbf{u}}\Vert \le \delta \text{ for each }\mathbf{u}\in \mathcal{M}\},$$

$$k_{\delta }=\min \{|\mathbf{u}|:\Vert D{\phi }_{\mathbf{u}}\Vert \le \delta <\Vert D{\phi }_{{\mathbf{u}}^{\ast }}\Vert ,\mathbf{u}\in {\Sigma }^{\ast }\}.$$

$$\begin{array}{rl}\overline{P}(t,\theta )& =\underset{\delta \to 0}{\mathrm{limsup}}\frac{1}{k_{\delta }}\log \underset{\mathcal{M}\in \mathcal{M}(\delta ,\theta )}{\min }\{\sum _{\mathbf{u}\in \mathcal{M}}\Vert D{\phi }_{\mathbf{u}}{\Vert }^t\},\\ \underline{P}(t,\theta )& =\underset{\delta \to 0}{\mathrm{liminf}}\frac{1}{k_{\delta }}\log \underset{\mathcal{M}\in \mathcal{M}(\delta ,\theta )}{\min }\{\sum _{\mathbf{u}\in \mathcal{M}}\Vert D{\phi }_{\mathbf{u}}{\Vert }^t\}).\end{array}$$

$$\begin{array}{rl}{s}^{\theta }& =\inf \{t:\overline{P}(t,\theta )<0\}=\sup \{t:\overline{P}(t,\theta )>0\},\\ s_{\theta }& =\inf \{t:\underline{P}(t,\theta )<0\}=\sup \{t:\underline{P}(t,\theta )>0\}.\end{array}$$

$$s_0=s^0=\text{mbox}{\dim }_{\mathrm{H}}E.$$

Full text

1. Introduction

1.1. Dimensions of fractals

Dimensions theory is central to fractal geometry, and Hausdorff and box-counting dimensions are two fundamental notions used in fractal geometry and dynamical systems. It is well-known that Hausdorff and box-counting dimensions are identical for self-similar sets satisfying the open set condition, but there are also many interesting fractal sets with the two dimensions different. For example, the Hausdorff dimensions of many non-typical self-affine sets and Moran sets are strictly less than their (upper) box-counting dimensions; see [7, 9, 32, 34, 42] for details. The reason is because covering sets of widely ranging scales are permitted in the definition of Hausdorff dimensions, whereas covering sets with the same size are essentially used in box-counting dimensions. We refer readers to  [13, 42] for background reading on fractal geometry. .

Recently, there is a large amount of literature on dimension spectra which provides a unifying framework for many dimensions of fractal geometry; see [1, 16, 20, 21] for various studies on dimension spectra. In [16], Falconer, Fraser and Kempton introduced intermediate dimensions to provide a unifying framework for Hausdorff and box-counting dimensions.

Definition 1**.**

Given a subset $E\subset\mathbb{R}^{d}$. For each $0\leq\theta\leq 1$, the lower and upper $\theta$-intermediate dimensions of $E$ are defined respectively by

[TABLE]

If $\underline{\mbox{\rm dim}}_{\rm\theta}\,E=\overline{\mbox{\rm dim}}_{\rm\theta}\,E$, we call it the $\theta$-intermediate dimension of $E$, denoted by $\dim_{\theta}E$.

Moreover, in [16, 20], the authors proved the continuity of the intermediate dimensions.

Proposition 1.1**.**

Given a bounded set $E\subset\mathbb{R}^{d}$, the dimension spectra $\underline{\mbox{\rm dim}}_{\rm\theta}\,E$ and $\overline{\mbox{\rm dim}}_{\rm\theta}\,E$ are continuous functions for $\theta\in(0,1]$.

As we see from the definition, for $\theta=0$, there are no restrictions for the size of the sets used in the covers which gives Hausdorff dimension. On the other hand, for $\theta=1$, the only covers using sets of the same size are allowable which recovers box-counting dimension. Therefore, Hausdorff and box-counting dimensions are the special values of intermediate dimensions $\dim_{\theta}E$, that is,

[TABLE]

It is valuable to study intermediate dimension spectra for which $\theta$ the transition occurs in geometric behavior, and this may deepen our understanding of dimensions and the geometric structure of fractal sets; see [3, 5, 10, 19, 31] for various related studies and applications.

Since intermediate dimension spectra provide a continuum between Hausdorff and box-counting dimensions, it is interesting to explore the dimensions spectra for the fractals with different Hausdorff and box-counting dimensions. In this paper, we study the dimensions and the intermediate dimension spectra of non-autonomous conformal fractals generated by non-autonomous iterated function systems.

1.2. Iterated function systems.

Let $\{\varphi_{i}\}_{i\in I}$ be a finite set of contractions on $\mathbb{R}^{d}$ with $\#I\geq 2$. The set $\{\varphi_{i}\}_{i\in I}$ is called an iterated function system(IFS), and it has a unique attractor, that is, a unique non-empty compact $E\subset\mathbb{R}^{d}$ such that

[TABLE]

If the contractions $\varphi_{i}$ are conformal $($similar or affine$)$ mappings, we call $\{\varphi_{i}\}_{i\in I}$ a conformal $($similar or affine$)$ iterated function system, and call $E$ a self-conformal $($self-similar or self-affine$)$ set. Note that self-similar sets are special cases of self-conformal and self-affine sets. See [13, 42] for details.

Formulae giving the Hausdorff and box-counting dimensions of self-similar sets satisfying the open set condition are well-known. However, calculation of the dimensions of self-conformal and self-affine sets is more awkward. In [35], Mauldin and Urbański studied self-conformal sets $E$, and defined a critical value $h$ by topological pressure $P(t)$ satisfying $h=\inf\{t:\overline{P}(t)<0\}=\sup\{t:\overline{P}(t)>0\}$. They obtained the following

[TABLE]

Moreover, they extended to infinite conformal iterated function systems allowing $\#I=\infty$. By approximating with finite subsystems, they obtained $\mbox{\rm dim}_{\rm H}\,E=h$. In [36], Mauldin and Urbański further studied the box and packing dimensions of infinite conformal iterated function systems. Falconer in [12] studied the dimensions of self-affine sets. He defined a critical value $d(T_{1},\ldots,T_{\#I})$ by using singular value functions, which is often called affine dimension or Falconer dimension, and under certain conditions, he proved

[TABLE]

almost surely. We refer readers to [6, 14, 15, 17, 25, 37] for various related studies on iterated function systems.

Recently, Banaji and Fraser in [3] studied the intermediate dimensions of infinite conformal sets and obtained that

[TABLE]

where $P$ is any subset of $J$ that intersects $\varphi_{i}(J)$ in exactly one point for each $i\in\mathbb{N}$, and $h$ is given by (1.2).

In this paper, we study the fractal dimensions and dimension spectra of non-autonomous finite conformal sets, and we extend our results to non-autonomous infinite conformal iterated function systems.

1.3. Non-autonomous iterated function systems.

Non-autonomous iterated function systems may be regarded as a generalization of iterated function systems. In 1946, Moran first studied non-autonomous similar sets, also called Moran sets which is a special class of fractals generated by non-autonomous iterated function systems; see [24, 38]. First, we recall the definitions of non-autonomous iterated function systems.

Let $\{I_{k}\}_{k\geq 1}$ be a sequence of index sets with $\#I_{k}\geq 2$ which is either finite or countable infinity. Given integers $k\geq l\geq 1$, we write

[TABLE]

and for simplicity, we set $\Sigma^{k}=\Sigma^{k}_{1}$ if $l=1$. We write

[TABLE]

for the set of all finite words with $\Sigma^{0}=\{\emptyset\}$ containing only the empty word $\emptyset$.

Let $J\subset\mathbb{R}^{d}$ be a compact set with non-empty interior, and $J$ satisfies $\overline{\mbox{int}(J)}=J$. For each integer $k>0$, let $\Phi_{k}=\{\varphi_{k,i}\}_{i\in I_{k}}$ be a family of mappings $\varphi_{k,i}:J\to J$. We say the collection $\mathcal{J}=\{J_{\mathbf{u}}:\mathbf{u}\in\Sigma^{*}\}$ of closed subsets of $J$ fulfills the non-autonomous structure with respect to $\{\Phi_{k}\}_{k=1}^{\infty}$ if it satisfies the following conditions:

• (i).

Uniform contraction: There exists $0<c<1$ such that for all integers $k>0$ and all $i\in I_{k}$,

[TABLE]

• (ii).

For all integers $k>0$ and all $\mathbf{u}\in\Sigma^{k-1}$, the elements $J_{\mathbf{u}i},i\in I_{k}$ of $\mathcal{J}$ are subsets of $J_{\mathbf{u}}$. We write $J_{\emptyset}=J$ for the empty word $\emptyset$.

• (iii).

For each $\mathbf{u}=u_{1}\ldots u_{k}\in\Sigma^{*}$, there exists $\omega_{\mathbf{u}}\in\mathbb{R}^{d}$ and $\Psi_{\mathbf{u}}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ such that

[TABLE]

where $\varphi_{\mathbf{u}}=\varphi_{u_{1}}\circ\cdots\circ\varphi_{u_{j}}\cdots\circ\varphi_{u_{k}}(J)$ and $\varphi_{u_{j}}\in\Phi_{j}$.

We call $\Phi=\{\Phi_{k}\}_{k=1}^{\infty}$ a non-autonomous iterated function system and

[TABLE]

the non-autonomous attractor of $\Phi$.

If $\#I_{k}<\infty$ for every $k\in\mathbb{N}_{+}$, we say $\Phi$ is a non-autonomous finite iterated function system(NFIFS) and $E$ is the non-autonomous finite set of $\Phi$. If $\#I_{k}=\infty$ for infinitely many $k\in\mathbb{N}_{+}$, we say $\Phi$ is a non-autonomous infinite iterated function system(NIIFS) and $E$ is the non-autonomous infinite set of $\Phi$. Since the non-autonomous infinite set $E$ of $\Phi$ is generated by iterations of contractions, without loss of generality, we assume that $\#I_{k}=\infty$ for every $k\in\mathbb{N}_{+}$ in $\Phi$. We do not consider the case where $\#I_{k}=\infty$ for only finitely many $k\in\mathbb{N}_{+}$. Note that the attractors of non-autonomous infinite iterated function systems may not be compact.

If for all integers $k\geq 0$ and $\mathbf{u}\in\Sigma^{k},$ we have

[TABLE]

for all $i\not=j\in I_{k+1}$, we say that $\Phi$ (or $E$) satisfies the open set condition (OSC). If for all integers $k\geq 0$ and $\mathbf{u}\in\Sigma^{k},$ we have that

[TABLE]

for all $i\not=j\in I_{k+1}$, we say that $\Phi$ (or $E$) satisfies the strong separation condition (SSC).

If for each integer $k>0$, every mapping $\varphi_{k,j}$ in $\Phi_{k}$ is a similarity with contraction ratio $c_{k,j}$, and $\Phi$ satisfies OSC, then the attractor $E$ is called a Moran set, which was first studied by Moran [38] in 1946. In 1994, Hua first studied the Hausdorff dimension of Moran sets, and later, Hua and others showed that if

[TABLE]

where $\underline{c}_{k}=\min_{1\leq j\leq\#I_{k}}\{c_{k,j}\}$ and $M_{k}=\max_{\mathbf{u}\in\Sigma^{k}}\lvert J_{\mathbf{u}}\rvert$, then

[TABLE]

where $s_{k}$ is the unique real solution to $\prod\nolimits_{i=1}^{k}\left(\sum\nolimits_{j=1}^{\#I_{i}}(c_{i,j})^{s}\right)=1.$ We refer readers to [27, 28, 29, 42] for details. There are various generalizations of such fractals. In [24], Gu and Miao studied the general theory of non-autonomous iterated function systems. In [26], Holland and Zhang studied the Hausdorff dimension of a class of special non-autonomous iterated function systems and the box and packing dimension of a class of generalized non-autonomous conformal sets. We refer readers to [22, 23, 33, 42] for other related works.

1.4. Non-autonomous conformal iterated function systems.

In this paper, we mainly focus on the dimension theory of non-autonomous conformal sets.

Let $V\subset\mathbb{R}^{d}$ be an open set and $\phi:V\to V$ be a conformal map. We denote by $D\phi(x)$ the derivative of $\phi$ at $x$, i.e., $D\phi(x):\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a similarity linear map, and we denote by $\|D\phi(x)\|$ the scaling factor at $x$. For non-autonomous conformal systems, we write

[TABLE]

Let $\Phi$ be a non-autonomous finite (or infinite) iterated function system satisfying that

• (i’).

Conformality: There exists an open connected set $V$ independent of $k$ with $J\subset V$ such that each $\varphi_{k,j}$ extends to a $C^{1}$ conformal diffeomorphism of $V$ into $V$.

• (ii’).

Bounded distortion: There exists a constant $C\geq 1$ such that for all $\mathbf{u}=u_{l}u_{l+1}\ldots u_{k}\in\Sigma_{l}^{k}$, the map $\varphi_{\mathbf{u}}=\varphi_{l,u_{l}}\circ\varphi_{l+1,u_{l+1}}\circ\dots\circ\varphi_{k,u_{k}}$ satisfies

[TABLE]

for all $x,y\in V$.

We say $\Phi=\{\Phi_{k}\}_{k=1}^{\infty}$ is a non-autonomous finite (or infinite) conformal iterated function system(NFCIFS or NICIFS), and we call

[TABLE]

is the * non-autonomous finite (or infinite) conformal set* of $\Phi$.

Let $\Phi$ be the NFCIFS and $E$ the corresponding attractor given by (1.9). Let

[TABLE]

In this paper, we study the non-autonomous finite conformal iterated function systems $\Phi$ satisfying open set condition and

[TABLE]

which plays a fundamental role in the study of Moran fractals; see [42] for details. In [39], Rempe-Gillen and Urbański studied Hausdorff dimensions for non-autonomous conformal sets under the different assumption

[TABLE]

and they provided the Hausdorff dimension formula for such sets. However, the conditions (1.11) and (1.12) do not imply each other; see Example 1. Inspiring by their work, we further relax the condition (1.11) to

[TABLE]

See Proposition 6.2 and Corollary 2.4 for details. Under the cone condition, we make certain generalization of the conclusions in [39]; see Corollary 7.3 and Corollary 8.1.

Next, we provide an example to illustrate the difference to condition (1.11), condition (1.13) and condition (1.12).

Example 1**.**

Let $E_{1}$ be a Moran set with $\#I_{k}=2^{k}$ and $c_{k,j}=\frac{1}{3^{k+1}}$. It is clear that

[TABLE]

Then by (1.8), we have $\mbox{\rm dim}_{\rm H}\,E_{1}=\frac{\log 2}{\log 3}.$

Let $E_{2}$ be a Moran set with $\#I_{k}=2^{k}$ ,$c_{k,1}={\frac{1}{3^{k+1}}}$ and $c_{k,j}={\frac{1}{3^{k(k+1)}}}(2\leq j\leq 2^{k})$. It is straightforward that

[TABLE]

By Theorem 2.4, we have $\mbox{\rm dim}_{\rm H}\,E_{2}=0$.

Let $E_{3}$ be a homogeneous Moran set with $\#I_{k}=2$ ,$c_{1}={\frac{1}{2}}$, $c_{2}={\frac{1}{4}}$ and $c_{k}=c_{1}c_{2}\cdots c_{k-1}(k\geq 3)$. Then it follows that

[TABLE]

By simple calculation, we have $\mbox{\rm dim}_{\rm H}\,E_{3}=0$.

2. Main conclusions

Let $E$ be the non-autonomous conformal set of the non-autonomous finite iterated function system $\Phi$, that is, $\#I_{k}<\infty$ for all $k>0$. First, we give intermediate dimension spectra formula for $E$. For real $\delta>0$ and $\theta\in(0,1]$, let

[TABLE]

and we write

[TABLE]

For $\theta\in(0,1]$, upper and lower pressure functions are given by

[TABLE]

We write their jump points respectively as

[TABLE]

Note that the existence of $s^{\theta}$ and $s_{\theta}$ follows from the monotonicity of $\underline{P}(t)$ and $\overline{P}(t)$; see Lemma 4.2 in section 4. When $\theta=0$, we write

[TABLE]

Theorem 2.1**.**

Let $E$ be the non-autonomous finite conformal set satisfying (1.11) and OSC. Then upper and lower intermediate dimension of $E$ are given by

[TABLE]

where $s^{\theta}$ and $s_{\theta}$ are given by (2.15).

The box and Hausdorff dimensions of non-autonomous finite conformal sets may be given in a simpler way. For non-autonomous finite (or infinite) conformal iterated function systems, we define upper and lower pressures by

[TABLE]

We write their jump points respectively as

[TABLE]

The next conclusion shows that the critical value $s^{*}$ always gives the box and packing dimensions of non-autonomous finite conformal sets.

Theorem 2.2**.**

Let $E$ be the non-autonomous finite conformal set satisfying (1.11) and OSC. Then the upper box and packing dimensions of $E$ are given by

[TABLE]

where $s^{*}$ is given by (2.17).

In Section 6, we change the condition (1.11) in Theorem 2.2 to (1.13), and obtain some partial results. See Proposition 6.2.

Generally, Hausdorff dimensions of non-autonomous finite conformal sets are not easy to calculate. We obtain Hausdorff dimensions under extra conditions.

Theorem 2.3**.**

Let $E$ be the non-autonomous finite conformal set with $\mathcal{L}^{d}(\partial J)=0$ satisfying (1.11) and OSC. Then the Hausdorff dimension is given by

[TABLE]

where $s_{*}$ is given by (2.17).

Next, we weaken the condition (1.11) in Theorem 2.3 to (1.13).

Corollary 2.4**.**

Let $E$ be a non-autonomous finite conformal set with $\mathcal{L}^{d}(\partial J)=0$ satisfying (1.13) and OSC. Then the Hausdorff dimension of $E$ is given by

[TABLE]

where $s_{*}$ is given by (2.17).

Remark 1*.*

If $J$ is a convex set, $\mathcal{L}^{d}(\partial J)=0$ holds, for example $J=[0,1]^{d}$.

Remark 2*.*

If $E$ satisfies the strong separation condition, condition $\mathcal{L}^{d}(\partial J)=0$ may be omitted.

Non-autonomous infinite conformal sets are more awkward to study, and we extend our conclusions to such sets with the following assumption instead of (1.11),

[TABLE]

For a NICIFS $\Phi$, let

[TABLE]

Theorem 2.5**.**

Let $E$ be the non-autonomous infinite conformal set satisfying (2.18) and OSC. Suppose for all $t\in(0,d)$ and all $\epsilon>0$, the following hold

• (1).

The sums $\sum_{j\in I_{k}}\|D\varphi_{k,j}\|^{t}$ are either infinite for all $k$ or finite for all $k$.

• (2).

If $\sum_{j\in I_{k}}\|D\varphi_{k,j}\|^{t}<\infty$, then

[TABLE]

• (3).

If $\sum_{j\in I_{k}}\|D\varphi_{k,j}\|^{t}=\infty$, then $\lim_{n\to\infty}\sum_{k\leq{M_{n}^{-\epsilon}}}\|D\varphi_{n,k}\|^{t}=\infty.$

Let $s^{*}$ and $s_{*}$ be given by (2.17). Then $\overline{\mbox{\rm dim}}_{\rm B}\,E\geq\mbox{\rm dim}_{\rm P}\,E\geq s^{*}$. Furthermore, if $\mathcal{L}^{d}(\partial J)=0$, then $\mbox{\rm dim}_{\rm H}\,E=s_{*}$.

The rest of the paper is organized as follows. In Section 3, we study the general properties of non-autonomous IFS and estimate the dimensions of non-autonomous fractals. In Section 4, we study the properties of pressure functions, which are essential to study the dimensions of non-autonomous conformal sets. In Section 5, we give the proof of Theorem 2.1. Theorem 2.2 is proved in Section 6. In Section 7, we prove Theorem 2.3, and relax the condition (1.11) to (1.13). Finally, we extend our conclusions to non-autonomous infinite conformal fractals in Section 8.

3. Dimension estimate of non-autonomous sets

In this section, we study the properties of non-autonomous iterated function systems defined in Subsection 1.3, and these properties are useful to explore the non-autonomous conformal fractals.

3.1. Symbolic space and pressure functions

We write

[TABLE]

for the set of words with infinity length, and we topologize $\Sigma^{\infty}$ using the metric $d(\mathbf{u},\mathbf{v})=2^{-|\mathbf{u}\wedge\mathbf{v}|}$ for distinct $\mathbf{u},\mathbf{v}\in\Sigma^{\infty}$ to make $\Sigma^{\infty}$ into a compact metric space. Given an integer $n>0$, for every $\mathbf{u}\in\Sigma^{\infty}$, we write $\mathbf{u}|_{n}=u_{1}\ldots u_{n}$. For each $\mathbf{u}=u_{1}\dots u_{k}\in\Sigma^{k}$, we write $\mathbf{u}^{*}=u_{1}\dots u_{k-1}$. Given $\mathbf{u}\in\Sigma^{l}$, for $\mathbf{v}\in\Sigma^{k}$ where $k\geq l$ or $\mathbf{v}\in\Sigma^{\infty}$, we write $\mathbf{u}\prec\mathbf{v}$ if $u_{i}=v_{i}$ for all $i=1,2,\ldots,l$.

We define the cylinders $\mathcal{C}_{\mathbf{u}}=\{\mathbf{v}\in\Sigma^{\infty}:\mathbf{u}\prec\mathbf{v}\}$ for $\mathbf{u}\in\Sigma^{*}$; the set of cylinders $\{\mathcal{C}_{\mathbf{u}}:\mathbf{u}\in\Sigma^{*}\}$ forms a base of open and closed neighborhoods for $\Sigma^{\infty}$. We term a subset $A$ of $\Sigma^{*}$ a cut set if $\Sigma^{\infty}\subset\bigcup_{\mathbf{u}\in A}\mathcal{C}_{\mathbf{u}}$, where $\mathcal{C}_{\mathbf{u}}\bigcap\mathcal{C}_{\mathbf{v}}=\emptyset$ for all $\mathbf{u}\neq\mathbf{v}\in A$. It is equivalent to that, for every $\mathbf{w}\in\Sigma^{\infty}$, there is a unique word $\mathbf{u}\in A$ with $|\mathbf{u}|<\infty$ such that $\mathbf{u}\prec\mathbf{w}$.

Let $\pi_{\Phi}:\Sigma^{\infty}\to J$ by

[TABLE]

where $J_{\mathbf{u}|_{n}}$ is given by (1.6).

We cite König’s lemma in graph theory, and see [41] for details.

Lemma 3.1**.**

Let $T$ be a subset of $\Sigma^{*}$ such that $\#T=\infty$ and $\emptyset\in T$. Suppose that for each $\mathbf{u}\in T$, the set $Pr_{T}(\mathbf{u})=\{\mathbf{v}\in T:\mathbf{v}\prec\mathbf{u}\}$ and the set

[TABLE]

are finite. Then there exists $\mathbf{w}\in\Sigma^{\infty}$ such that $\mathbf{w}|_{n}\in T$ for each $n>0$.

Given a non-autonomous iterated function system $\Phi$, we say that $\Phi$ satisfies the finite overlap condition (FOC) if for all $\mathbf{u}\in\Sigma^{*}$ and all $x\in J_{\mathbf{u}}$, we have

[TABLE]

Next, under the finite overlap condition, we use $\pi_{\Phi}$ to give another representation of non-autonomous attractors.

Lemma 3.2**.**

If a non-autonomous iterated function system $\Phi$ satisfies the finite overlap condition, then the attractor $E$ of $\Phi$ is the image of $\pi$, that is

[TABLE]

Proof.

For each $\mathbf{u}\in\Sigma^{\infty}$, we have $\pi_{\Phi}(\mathbf{u})=\bigcap_{n=1}^{\infty}J_{\mathbf{u}|_{n}}$. Since $J_{\mathbf{u}|_{n}}\subset\bigcup_{\mathbf{v}\in\Sigma^{n}}J_{\mathbf{v}},$ it follows that $\pi_{\Phi}(\mathbf{u})\in E$ and $\pi_{\Phi}(\Sigma^{\infty})\subset E$.

For each $x\in E$, let $T=\Sigma(x)=\{\mathbf{u}\in\Sigma^{*}:x\in J_{\mathbf{u}}\}.$ Since for each $n\in\mathbb{N}$, there exists $\mathbf{u}\in\Sigma^{n}$ such that $x\in J_{\mathbf{u}}$, we have $\#T=\infty$. For each $\mathbf{u}\in T$, it is clear that $Pr_{T}(\mathbf{u})=\{\mathbf{u}|_{k}:1\leq k\leq|\mathbf{u}|\}$ is finite. By finite overlap condition, we have that $\#succ_{T}(\mathbf{u})=\#\{i\in I_{|\mathbf{u}|+1}:x\in J_{\mathbf{u}i}\}<\infty,$ and by Lemma 3.1, there exists $\mathbf{w}\in\Sigma^{\infty}$ such that $\mathbf{w}|_{n}\in T$ for each $n>0$. It implies that $x\in J_{\mathbf{w}|_{n}}$ for every $n\in\mathbb{N}$ and $x=\pi_{\Phi}(\mathbf{w})$, and we have $E\subset\pi_{\Phi}(\Sigma^{\infty})$. ∎

Remark 3*.*

If $\Phi$ is a non-autonomous finite iterated function system, the finite overlap condition automatically holds, but the finite overlap condition is not necessarily satisfied for a non-autonomous infinite iterated function system.

Let $\Phi$ be the non-autonomous finite or infinite iterated function system satisfying OSC, and $E$ the attractor of $\Phi$ given by (1.9). Given $k\geq l\geq 0$, for each $\mathbf{u}\in\Sigma_{l}^{k}$, let

[TABLE]

It is clear that

[TABLE]

We define upper and lower pressures respectively by

[TABLE]

and both of them are strictly monotonous.

Lemma 3.3**.**

For all $t_{1}<t_{2}$, if $\overline{P}(t_{1})$ and $\overline{P}(t_{2})$ are finite, then $\overline{P}(t_{1})>\overline{P}(t_{2})$.

Proof.

Given $t_{1}<t_{2}$ such that $\overline{P}(t_{1})$ and $\overline{P}(t_{2})$ are finite. By (1.5), we have

[TABLE]

where $c$ is given by (1.5), and it follows that $\overline{P}(t_{2})\leq\overline{P}(t_{1})-(t_{2}-t_{1})\log\frac{1}{c}.$ Hence we have $\overline{P}(t_{2})<\overline{P}(t_{1})$ since $0<c<1$. ∎

3.2. Dimensions estimation of non-autonomous fractals

Let $\overline{P}(t)$ and $\underline{P}(t)$ be the pressure functions given by (3.20). By Lemma 3.3, we write their jump points respectively as

[TABLE]

Since these are consistent with the critical values defined in (2.17), we use the same notation for simplicity. Let

[TABLE]

Generally, $s_{*}$ serves as an upper bound for the Hausdorff dimension of non-autonomous attractors, $t^{*}$ as an upper bound for the box dimension of non-autonomous attractors, and $s^{*}$ as a lower bound for the box dimension of non-autonomous attractors. For some special cases, we may have $s^{*}=t^{*}$, and see Lemma 6.1 for details.

Theorem 3.4**.**

Given a non-autonomous finite or infinite iterated function system $\Phi$ satisfying OSC, let $E$ be the corresponding non-autonomous set given by (1.9) and $s_{*}$ given by (3.21). Then

[TABLE]

If $\#I_{k}$ is finite for all $k>0$, and there exists a constant $N>0$ such that

[TABLE]

then

[TABLE]

where $t^{*}$ is given by (3.22).

Proof.

For every $\mathbf{u}\in\Sigma^{*}$, by (3.19), we have $|J_{\mathbf{u}}|\leq\overline{c}_{\mathbf{u}}|J|.$ For each real $t>s_{*}$, we have $\underline{P}(t)<0$, and by (3.20), there exists $\{n_{k}\}$ such that for each $k$,

[TABLE]

Let $\delta_{k}=\max\{|J_{\mathbf{u}}|:\mathbf{u}\in\Sigma^{k}\}$. Thus the Pre-Hausdorff measure of $E$ is bounded by

[TABLE]

and the Hausdorff measure of $E$ is zero. Hence, $\mbox{\rm dim}_{\rm H}\,E\leq t$ for all $t>s_{*}$, and we obtain that $\mbox{\rm dim}_{\rm H}\,E\leq s_{*}$.

Suppose that $\#I_{k}$ is finite for all $k>0$. Next, we prove $\overline{\mbox{\rm dim}}_{\rm B}\,E\leq t^{*}$. For every $s>t^{*}$, by (3.22), it is clear that

[TABLE]

For all sufficiently small $\delta>0$, we write

[TABLE]

Denote by $N_{\delta}(E)$ the smallest number of sets with diameters at most $\delta$ covering the set $E$, and it is clear that $N_{\delta}(E)\leq\#\mathcal{M}^{\prime}(\delta)$.

We write

[TABLE]

and note that $\mathcal{M}^{\prime\prime}(\delta)$ may not be a cut set. Let

[TABLE]

Then we have $\#\mathcal{M}^{\prime}(\delta)\leq N(\delta)\#\mathcal{M}^{\prime\prime}(\delta),$ and it follows that

[TABLE]

Thus

[TABLE]

Since $\delta<\max_{\mathbf{v}\in\Sigma^{|\mathbf{u}|-1}}\{\overline{c}_{\mathbf{v}}\}$ for every $\mathbf{u}\in\mathcal{M}^{\prime}(\delta)$, by (3.23), we obtain that

[TABLE]

By (3.24), this implies that $\limsup_{\delta\to 0}\frac{\log N(\delta)}{-\log\delta}=0$. Therefore, $\overline{\mbox{\rm dim}}_{\rm B}\,E\leq s$ for all $s>t^{*}$, and the conclusion holds. ∎

3.3. Non-autonomous infinite iterated function systems

Given a non-autonomous iterated function system $\Phi=\{\Phi_{k}\}_{k=1}^{\infty}$. In this subsection, we assume that there exists a constant $D>0$ such that for all $k\geq l\geq 0$ and all $\mathbf{u}\in\Sigma_{l}^{k}$,

[TABLE]

where $\overline{c}_{\mathbf{u}}$ and $\underline{c}_{\mathbf{u}}$ are given by (3.19). We say $\Phi^{\prime}=\{\Phi_{k}^{\prime}\}_{k=1}^{\infty}$ is a subsystem of $\Phi$ if $I^{\prime}_{k}\subset I_{k}$ and $\Phi_{k}^{\prime}\subset\Phi_{k}$ for all $k>0$, and we write $\overline{P}^{\prime}(t)$ and $\underline{P}^{\prime}(t)$ for the upper and lower pressure functions of the subsystem $\Phi^{\prime}$, and write $(\Sigma_{l}^{k})^{\prime}$ for the subspace of $\Sigma_{l}^{k}$.

Generally, it is difficult to calculate the dimensions of non-autonomous sets. Therefore, we adopt the method of approximating the infinite system with a finite subsystem to obtain an estimation of the dimension of the non-autonomous infinite iterated function system. The following conclusion is inspired by [39].

Lemma 3.5**.**

Let $\Phi$ be a non-autonomous finite or infinite iterated function system satisfying (3.25). Given a real $t\in[0,d]$, assume that $\sum_{k\in I_{n}}\overline{c}_{n,k}^{t}<\infty$ for all $n$. Given $\delta>0$ and, for each $n$, let $I^{\prime}_{n}\subset I_{n}$ be a finite index set such that

[TABLE]

for all sufficiently large $n$. Let $\Phi^{\prime}$ be the subsystem given by $I^{\prime}_{n}$. Then

[TABLE]

Proof.

Fix $n>0$, and for each $s=0,1,\ldots,n$, we write

[TABLE]

For every $P\in\mathcal{P}_{s}$, we divide $P$ into consecutive parts and write

[TABLE]

and

[TABLE]

For each $\mathbf{u}\in\Sigma^{n}(P)$, let

[TABLE]

Let $k=\#I^{*}_{P}$. It is clear that $k-1\leq n-s$. Recall (3.19), and by (3.25), it follows that for each $\mathbf{u}\in\Sigma^{n}$

[TABLE]

We write ${\Sigma^{\prime}}^{n}=\{u_{1}u_{2}\ldots u_{n}:u_{j}\in I_{j}^{\prime},j=1,2,\ldots,n\}$. By (3.26), this implies that

[TABLE]

Since the number of elements in $\mathcal{P}_{s}$ is $C_{n}^{s}$ and $t\leq d$, it immediately follows that

[TABLE]

By (2.16), we obtain that

[TABLE]

Similarly, we have that $\overline{P}^{\prime}(t)\geq\overline{P}(t)-\delta D^{2d}$, and the conclusion holds. ∎

Under certain conditions, a non-autonomous infinite iterated function system has a finite subsystem with identical pressure functions, and we apply this property to control the number of contractions in subsystem. The following conclusion is inspired by [39].

Proposition 3.6**.**

Let $E$ be a non-autonomous infinite set of $\Phi$ satisfying (3.25). Given a real sequence $\{\beta_{n}\}_{n=1}^{\infty}$ convergent to zero. Suppose that $\{\beta_{n}\}_{n=1}^{\infty}$ satisfies the following for all $t\in(0,d)$ and all $\epsilon>0$:

• (1).

The sum $\sum_{k\in I_{n}}\overline{c}_{n,k}^{t}$ are either infinite for all $n$ or finite for all $n$.

• (2).

If $\sum_{k\in I_{n}}\overline{c}_{n,k}^{t}<\infty$, then

[TABLE]

• (3).

If $\sum_{k\in I_{n}}\overline{c}_{n,k}^{t}=\infty$, then $\lim_{n\to\infty}\sum_{k\leq{\beta_{n}^{-\epsilon}}}\overline{c}_{n,k}^{t}=\infty.$

Then there exists a subsystem $\Phi^{\prime}$ of $\Phi$ such that $\overline{P}^{\prime}(t)=\overline{P}(t)$ and $\underline{P}^{\prime}(t)=\underline{P}(t)$ for all $t\geq 0$ and

[TABLE]

Let $E^{\prime}$ be the non-autonomous set of $\Phi^{\prime}$, and let $s_{*}^{\prime}$ and ${s^{*}}^{\prime}$ be given by (3.21) with respect to $\Phi^{\prime}$. If $\mbox{\rm dim}_{\rm H}\,E^{\prime}=s_{*}^{\prime}$, then $\mbox{\rm dim}_{\rm H}\,E=s_{*}^{\prime}=s_{*}.$ If $\mbox{\rm dim}_{\rm P}\,E^{\prime}\geq{s^{*}}^{\prime}$, then $\overline{\mbox{\rm dim}}_{\rm B}\,E\geq\mbox{\rm dim}_{\rm P}\,E\geq{s^{*}}^{\prime}=s^{*}.$

Proof.

Without loss of generality, we assume $\overline{c}_{k,i}\geq\overline{c}_{k,j}$ for all $i\geq j$ and all $k\in\mathbb{N}$.

Let $h=\min\{d,\sup\{t:\sum_{j\in I_{k}}\overline{c}_{k,j}^{t}=+\infty\}\}$. First, we consider $h<d$. Arbitrarily choose $t>h$, and by $(1)$, we have that $\sum_{j\in I_{k}}\overline{c}_{k,j}^{t}$ is finite for all integers $k>0$.

Fix $\delta\in(0,1)$. For each integer $i>0$, setting $\epsilon=\frac{1}{i}$, by (2), there exists an integer $N_{i}>0$ such that for every $n>N_{i}$, we have

[TABLE]

We choose $\{N_{i}\}_{i=1}^{\infty}$ to be an increasing sequence. Let $I^{t,\delta}_{n}=\{1,2,\ldots,[\beta_{n}^{-\frac{1}{i}}]\}$ for $N_{i}<n\leq N_{i+1}$. Let $\Psi_{t,\delta}$ be the subsystem generated by $I^{t,\delta}_{n}$, and it is clear that

[TABLE]

and $(1+\delta)\sum_{k\in I^{t,\delta}_{n}}\overline{c}_{n,k}^{t}>\sum_{k\in I_{n}}\overline{c}_{n,k}^{t}.$ Let $\delta=\frac{1}{j}$. By (3.27), there exists an increasing sequence $\{K_{j}\}_{j=1}^{\infty}$ such that for $n>K_{j}$ we have

[TABLE]

Let $I^{t}_{n}=I^{t,\frac{1}{j}}_{n}$ for $K_{j}<n\leq K_{j+1}$. Then we have

[TABLE]

Let $\Psi_{t}$ be the subsystem generated by $I^{t}_{n}$. For every $n\in\mathbb{N}$ and $d>t^{\prime}>t>h$, since $\frac{\overline{c}_{n,k}}{\overline{c}_{n,\#I_{n}^{t}}}\geq 1$ for $k\in I^{t}_{n}$ and $\frac{\overline{c}_{n,k}}{\overline{c}_{n,\#I_{n}^{t}}}\leq 1$ for $k\notin I^{t}_{n}$, it follows that

[TABLE]

which implies that for all $t^{\prime}>t$,

[TABLE]

By (1), we have that either $\sum_{k\in I_{n}}\overline{c}_{n,k}^{h}<\infty$ for all $n$ or $\sum_{k\in I_{n}}\overline{c}_{n,k}^{h}=\infty$ for all $n$. If $\sum_{k\in I_{n}}\overline{c}_{n,k}^{h}<\infty$, by the same argument as above, we obtain a subsystem $\Psi_{h}$ and take $\Phi_{h}=\Psi_{h}$. If $\sum_{k\in I_{n}}\overline{c}_{n,k}^{h}=\infty$, fix a decreasing sequence $\{t_{m}\}_{m=1}^{\infty}$ convergent to $h$, and for each $m$, by the above argument, there exists an increasing sequence $\{L_{m}\}_{m=1}^{\infty}$ such that for each $n>L_{m}$,

[TABLE]

Let $I^{h}_{n}=I^{t_{m}}_{n}$ for $L_{m}<n\leq L_{m+1}$, and let $\Phi_{h}$ be the subsystem generated by $I^{h}_{n}$.

In both case, the subsystem $\Phi_{h}$ satisfies that for each $t$ with $\sum_{k\in I_{n}}\overline{c}_{n,k}^{t}<\infty$,

[TABLE]

By Lemma 3.5, we have $\underline{P}^{h}(t)=\underline{P}(t)$ for each $t$ with $\sum_{k\in I_{n}}\overline{c}_{n,k}^{t}<\infty$.

By a similar method, we may choose another sequence $J^{h}_{n}\subset I_{n}$, and let $\Phi^{\prime}_{h}$ be the corresponding subsystem, which satisfies that for each $t$ with $\sum_{k\in I_{n}}\overline{c}_{n,k}^{t}=\infty$,

[TABLE]

which means $\underline{P}^{\prime h}(t)=\infty$ for $t$ with $\sum_{k\in I_{n}}\overline{c}_{n,k}^{t}=\infty$. Let $I^{\prime}_{n}=\max\{I^{h}_{n},J^{h}_{n}\}$. Then the corresponding finite subsystem $\Phi^{\prime}$ satisfies

[TABLE]

and for each $t\in(0,d)$, we have $\underline{P}^{\prime}(t)=\underline{P}(t)$.

For $h=d$ or $h=0$, the conclusion follows by the same argument,.

Furthermore, if $\mbox{\rm dim}_{\rm H}\,E^{\prime}=s_{*}^{\prime}$, by $\underline{P}^{\prime}(t)=\underline{P}(t)$, we have $s_{*}=s_{*}^{\prime}$. Since $E^{\prime}\subset E$, by Lemma 3.4 it follows that

[TABLE]

that is $\mbox{\rm dim}_{\rm H}\,E=\mbox{\rm dim}_{\rm H}\,E^{\prime}=s_{*}.$ If $\mbox{\rm dim}_{\rm P}\,E^{\prime}\geq{s^{*}}^{\prime}$, by $\overline{P}^{\prime}(t)=\overline{P}(t)$, we have $s^{*}={s^{*}}^{\prime}$, which implies $\overline{\mbox{\rm dim}}_{\rm B}\,E\geq\mbox{\rm dim}_{\rm P}\,E\geq s^{*}.$ ∎

Corollary 3.7**.**

Let $E$ be the non-autonomous infinite set of $\Phi$ satisfying (3.25).Given a real sequence $\{\beta_{n}\}_{n=1}^{\infty}$ convergent to zero. Suppose that $\{\beta_{n}\}_{n=1}^{\infty}$ satisfies the following for all $t\in(0,d)$:

• (1).

The sum $\sum_{k\in I_{n}}\overline{c}_{n,k}^{t}$ are either infinite for all $n$ or finite for all $n$.

• (2).

If $\sum_{k\in I_{n}}\overline{c}_{n,k}^{t}<\infty$, then

[TABLE]

• (3).

If $\sum_{k\in I_{n}}\overline{c}_{n,k}^{t}=\infty$, then $\lim_{n\to\infty}\sum_{k\leq{\beta_{n}^{-\frac{1}{2}}}}\overline{c}_{n,k}^{t}=\infty.$

Then there exists a finite subsystem $\Phi^{\prime}$ of $\Phi$ such that $\underline{P}^{\prime}(t)=\underline{P}(t)$ for all $t\geq 0$ and

[TABLE]

Furthermore, let $E^{\prime}$ be the non-autonomous set of $\Phi^{\prime}$, and let $s_{*}^{\prime}$ be given by (3.21) with respect to $\Phi^{\prime}$. If $\mbox{\rm dim}_{\rm H}\,E^{\prime}=s_{*}^{\prime}$, then $\mbox{\rm dim}_{\rm H}\,E=s_{*}^{\prime}=s_{*}.$

For a non-autonomous finite iterated function system, the following lemma shows that we may approximate it with subsystems where the contraction ratios are bounded by the number of mappings at each level. It is easier to estimate the dimension of a non-autonomous infinite iterated function system in Section 8. The following conclusion is inspired by [39].

Lemma 3.8**.**

Given $t_{0}>0$ and a non-autonomous finite iterated function system $\Phi$ satisfying (3.25). Let $\{\alpha_{n}\}_{n=1}^{\infty}$ be a real sequence strictly increasing to $\infty$. Then there exists a subsystem $\Phi^{\prime}$ of $\Phi$ such that $\overline{P}^{\prime}(t)=\overline{P}(t),\underline{P}^{\prime}(t)=\underline{P}(t)$ for all $t\geq t_{0}$ and for all integers $n>0$

[TABLE]

Proof.

Let $I^{\prime}_{n}$ be a set consisting of all indices $j\in I_{n}$ such that

[TABLE]

Then for $t\geq t_{0}$,

[TABLE]

Given $\delta>0$, since $\lim_{n\to\infty}\alpha_{n}=\infty$, for sufficiently large $n$, we have that

[TABLE]

Since $\#I_{n}<\infty$ for each $n\in\mathbb{N}_{+}$, it follows that $\sum_{k\in I_{n}}\overline{c}_{n,k}^{t}<\infty$ for all $n$. By Lemma 3.5, we have $\overline{P}^{\prime}(t)=\overline{P}(t)$ and $\underline{P}^{\prime}(t)=\underline{P}(t)$ for all $t\geq t_{0}$. ∎

4. Pressure functions of non-autonomous conformal iterated function systems

In this section, we define a pressure function of non-autonomous conformal iterated function systems, which is a powerful tool to study fractal dimensions.

Lemma 4.1**.**

Given a $NCIFS$, for each integer $k$ and $u\in I_{k}$,

[TABLE]

where $c$ is given by (1.5).

Proof.

Since $\overline{\mbox{int}(J)}=J$, for every $x\in J$ there exists $\{x_{n}\}_{n=1}^{\infty}\subset J$ such that $\lim_{n\to\infty}x_{n}=x$. Note that each $\varphi_{u}$ is a $C^{1}$ conformal diffeomorphism of $V$ into $V$. Thus, we have

[TABLE]

By (i) in definition, it follows that

[TABLE]

and $\|D\varphi_{u}\|\leq c$. ∎

Lemma 4.2**.**

Given a non-autonomous finite conformal iterated function system $\Phi$. For every $\theta\in(0,1]$, both upper and lower pressure functions are strictly monotonous in $t$, that is for $t_{1}<t_{2}$ if $\overline{P}(t_{1},\theta)$ and $\overline{P}(t_{2},\theta)$ are finite, we have $\overline{P}(t_{1},\theta)>\overline{P}(t_{2},\theta)$, where $\overline{P}(t,\theta)$ is given in (2.14).

Proof.

Given $\theta\in(0,1]$ and $t_{1}<t_{2}$ such that $\overline{P}(t_{1},\theta)$ and $\overline{P}(t_{2},\theta)$ are finite. By (1.5), we have

[TABLE]

where $c$ is given by (1.5), and it follows that

[TABLE]

Then we have $\overline{P}(t_{2},\theta)<\overline{P}(t_{1},\theta)$, since $0<c<1$. ∎

Corollary 4.3**.**

Let $\Phi$ be a non-autonomous finite or infinite conformal iterated function system. Both upper and lower pressure functions are strictly monotonous, that is for $t_{1}<t_{2}$ if $\overline{P}(t_{1})$ and $\overline{P}(t_{2})$ are finite, we have $\overline{P}(t_{1})>\overline{P}(t_{2})$, where $\overline{P}(t)$ is given in (2.16).

The following conclusions are the consequences of bounded distortion (see (II’) in Subsection 1.4), which are frequently used in our proofs.

Lemma 4.4**.**

Let $\Phi$ be a non-autonomous finite or infinite conformal iterated function system. For all integers $0<m<n<\infty$, every $\mathbf{u}\in\Sigma^{m}$ and $\mathbf{u}\in\Sigma_{m+1}^{n}$, we have that

[TABLE]

Proof.

Since $D\varphi_{\mathbf{u}\mathbf{v}}(x)=D\varphi_{\mathbf{u}}(\varphi_{\mathbf{v}}(x))D\varphi_{\mathbf{v}}(x)$, it follows that

[TABLE]

By Bounded distortion, for every $x\in J$, we have that

[TABLE]

and the conclusion holds. ∎

Lemma 4.5** (Quasi-differential Mean Value Theorem).**

Let $\Omega\subseteq\mathbb{R}^{d}$ be an open convex set, and let $f:\Omega\to\mathbb{R}^{d}$ be a differentiable mapping. Then for all distinct points $x,y\in\Omega$, there exist a point $\xi$ on the line segment connecting $x$ and $y$ such that

[TABLE]

The principle of bounded distortion makes precise the idea of a set being ’approximately nonautonomous similar’, in that any sufficiently small neighbourhood may be mapped onto a large part of the set by a transformation that is not unduly distorting.

Lemma 4.6**.**

Given NCIFS $\boldsymbol{\Phi}$, for all $\mathbf{u}\in\Sigma^{*}$, we have that for all $x,y\in J$,

[TABLE]

and moreover,

[TABLE]

Proof.

Recall that $J$ is compact and $V$ is an open connected set containing $J$. The collection $\mathcal{F}=\{B(x,r_{x})\subset V:x\in J\}$ of balls is a cover of $J$. Thus, there exists a finite subcover $\{B(x_{i},r_{i})\}_{i=1}^{k}\subset\mathcal{F}$ of $J$. Let $\delta$ be the Lebesgue number of $\{B(x_{i},r_{i})\}_{i=1}^{k}$; see [40]. Since $V$ is a connected set of $\mathbb{R}^{d}$, it is also path-connected. For every $1\leq i\leq k-1$, there exists a path connecting $x_{i}$ and $x_{i+1}$, and we choose a finite number of balls $\{B(z_{j},r_{j})\}_{j=1}^{K_{i}}$ with $z_{1}=x_{i}$ and $z_{K_{i}}=x_{i+1}$ satisfying $B(z_{j},r_{j})\cap B(z_{j+1},r_{j+1})\not=\emptyset$. We denote the new collection of these balls by $\{B(z_{j},r_{j})\}_{j=1}^{l}$. Note that for each $j$, $B(z_{j},r_{j})\cap B(z_{j+1},r_{j+1})\not=\emptyset$.

Fix $\mathbf{u}\in\Sigma^{*}$. Arbitrarily choose $x,y\in J$. There exist integers $m$ and $n$ such that $x\in B(z_{m},r_{m}),y\in B(z_{n},r_{n})$.

We first show

[TABLE]

If $0<|x-y|<\delta$, then by Lemma 4.5, it holds. Otherwise, for $|x-y|\geq\delta$, by Lemma 4.5, we have

[TABLE]

Hence (4.31) holds.

Next, we show

[TABLE]

Suppose $\varphi_{\mathbf{u}}(x)\in B(z_{p},r_{p})$ and $\varphi_{\mathbf{u}}(y)\in B(z_{q},r_{q})$ where $1\leq p\leq q\leq l$. If $p=q$, let $L$ be the line segment connecting $\varphi_{\mathbf{u}}(x)$ and $\varphi_{\mathbf{u}}(y)$. If $p<q$, let $L$ be the polyline connecting the points in the following order: $\varphi_{\mathbf{u}}(x)$, $z_{p}$, $z_{p+1}$, $\dots$, $z_{q}$, $\varphi_{\mathbf{u}}(y)$. We may parameterize $L$ by a continuous map $L:[0,1]\to\mathbb{R}^{d}$ with $L(0)=\varphi_{\mathbf{u}}(x)$ and $L(1)=\varphi_{\mathbf{u}}(y)$. For $s\in[0,1]$, define $L_{s}=\{L(t):0\leq t\leq s\}$ and let $|L_{s}|$ denote the length of $L_{s}$. Note that $|L|\leq 2\sum_{j=1}^{l}r_{j}$

Let $t_{0}=\sup\{t\in[0,1]:L_{t}\subset\varphi_{\mathbf{u}}(V)\}$. Then

[TABLE]

Since $\varphi_{\mathbf{u}}(x),\varphi_{\mathbf{u}}(y)\in J$, if $|\varphi_{\mathbf{u}}(x)-\varphi_{\mathbf{u}}(y)|\leq\delta$, then $|\varphi_{\mathbf{u}}(x)-\varphi_{\mathbf{u}}(y)|=|L|$, and if $|\varphi_{\mathbf{u}}(x)-\varphi_{\mathbf{u}}(y)|>\delta$, then

[TABLE]

Thus

[TABLE]

Since $\Psi_{\mathbf{u}}(x)=\varphi_{\mathbf{u}}(x)+\omega_{\mathbf{u}}$, it follows that

[TABLE]

and we have $\|D\varphi_{\mathbf{u}}\||x-y|\asymp|\varphi_{\mathbf{u}}(x)-\varphi_{\mathbf{u}}(y)|$. ∎

Lemma 4.7**.**

Let $\Phi$ be a non-autonomous finite or infinite conformal iterated function system. For all $\mathbf{u}\in\Sigma^{*}$ and $A\subset V$ we have

[TABLE]

Proof.

Since $\varphi_{\mathbf{u}}$ is a $C^{1}$ conformal diffeomorphism, it is clear that

[TABLE]

By bounded distortion, we have

[TABLE]

and the conclusion holds. ∎

5. intermediate dimension of non-autonomous finite conformal set

In this section, we study intermediate dimensions of the non-autonomous finite conformal set $E$ generated by the NFCIFS $\Phi$. Recall

[TABLE]

For $F\subset\mathbb{R}^{d}$ such that $E\cap F\neq\emptyset$, we write

[TABLE]

Let

[TABLE]

For each integer $k_{F}^{-}\leq k\leq k_{F}^{+}$, we write

[TABLE]

Lemma 5.1**.**

Let $\Phi$ be a non-autonomous finite conformal iterated function system satisfying OSC. Then there exists a constant $C_{2}$ such that for every $F\subset\mathbb{R}^{d}$ with $E\cap F\neq\emptyset$, we have

[TABLE]

Proof.

Given a set $F\subset\mathbb{R}^{d}$ such that $E\cap F\neq\emptyset$. Let $\delta=|F|$. For every $\mathbf{u}\in\mathcal{M}(\delta)$, we write $\mathbf{u}=u_{1}u_{2}\ldots u_{k}$. By Lemma 4.6, we have $\delta\leq\|D\varphi_{\mathbf{u}^{*}}\|.$ Recall $\underline{c}_{k}=\min_{1\leq j\leq\#I_{k}}\{\|D\varphi_{k,j}\|\}$, and by Lemma 4.4, it is clear that for all $x\in J$

[TABLE]

Arbitrarily choose $x\in F$, and we have $J_{\mathbf{u}}\subset B(x,2\delta)$ for every $J_{\mathbf{u}}\in A(F)$. By Lemma 4.6 and Lemma 4.7, it immediately follows that

[TABLE]

Setting $C_{2}=\frac{(2C_{1}C^{2})^{d}\mathcal{L}^{d}(B(0,1))}{\mathcal{L}^{d}((\mbox{int}J))}$, we obtain $\sum_{k=k_{F}^{-}}^{k_{F}^{+}}\underline{c}_{k}^{d}\#D(F,k)\leq C_{2}$. ∎

For each cut set $\mathcal{M}$ satisfying $\delta^{\frac{1}{\theta}}<\|D\varphi_{\mathbf{u}^{*}}\|$ and $\|D\varphi_{\mathbf{u}}\|\leq\delta$ for all $\mathbf{u}\in\mathcal{M}$. Let $\mathcal{M}^{\prime}=\{\mathbf{u}\in\mathcal{M}:|J_{\mathbf{u}}|\leq\delta\}$ and $\mathcal{M}^{\prime\prime}=\{\mathbf{u}\in\mathcal{M}:|J_{\mathbf{u}}|>\delta\}$.

We define $\mathcal{F}_{\mathcal{M}^{\prime}}=\{U_{\mathbf{u}}:\mathbf{u}\in\mathcal{M}^{\prime}\}$ by setting

[TABLE]

For $\mathbf{u}\in\mathcal{M}^{\prime\prime}$, we define a cover $\mathcal{F}_{\mathcal{\mathbf{u}}}=\{U_{\mathbf{u}_{i}}\}_{i=1}^{k_{\mathbf{u}}}$ of $J_{\mathbf{u}}$ satisfying $|U_{\mathbf{u}_{i}}|=\delta$.

Let

[TABLE]

Proof of Theorem 2.1.

We only give the proof for the lower intermediate dimension since the proof for the upper intermediate dimension is similar.

First, we prove $\underline{\mbox{\rm dim}}_{\rm\theta}\,E\geq s_{\theta}$. Arbitrarily choose $\alpha<s_{\theta}$. Recall that $s_{\theta}=\sup\{t:\underline{P}(t,\theta)>0\}$, and we have $\underline{P}(\alpha,\theta)>0$. There exists $\Delta_{1}>0$ such that for all $0<\delta<\Delta_{1}$, we have that

[TABLE]

Since $\lim_{k\to+\infty}\frac{\log\underline{c}_{k}}{\log M_{k}}=0$, for each $0<\eta<1$, there exists $K_{0}>0$ such that for $k>K_{0}$, we have

[TABLE]

Moreover there exists $\Delta_{2}$ such that for all $0<\delta<\Delta_{2}$, we have $|\mathbf{u}|>K_{0}$ for all $\mathbf{u}\in\Sigma^{*}$ satisfying $\|D\varphi_{\mathbf{u}}\|\leq\delta$.

Given a cover $\mathcal{F}$ of $E$ such that ${\delta^{\frac{1}{\theta}}}\leq|U|\leq\delta$ for each $U\in\mathcal{F}$. By Lemma 5.1, there exists a constant $C_{2}$ such that for every $U\in\mathcal{F}$ with $E\cap U\neq\emptyset$, we have

[TABLE]

where $k_{U}^{-}$ is given by (5.35). By (5.34), for $\mathbf{u}\in A(U)$, it follows that

[TABLE]

Combining this with (5.38) and (5.39), we have for $k_{U}^{-}>K_{0}$

[TABLE]

where $C^{\prime}$ is a constant independent of $\mathcal{F}$.

Let $\mathcal{F_{1}}=\{\mathbf{u}:\mathbf{u}\in A(U),U\in\mathcal{F}\}$. It is clear that $\mathcal{F_{1}}$ is a cut set satisfying $\delta^{\frac{1}{\theta}}<\|D\varphi_{\mathbf{u}^{*}}\|$ and $\|D\varphi_{\mathbf{u}}\|\leq\delta$. Moreover, we choose a finite cut set

[TABLE]

such that $\{\mathbf{u}_{i}\}_{i=1}^{n}\subset\mathcal{M}(\delta,\theta).$ By (5.37), it follows that

[TABLE]

and it implies that

[TABLE]

Setting $\epsilon_{0}=\frac{1}{C^{\prime}}$ and $\Delta_{0}=\min\{\Delta_{1},\Delta_{2}\}$, for every $0<\delta<\Delta_{0}$ and every cover $\mathcal{F}$ satisfying ${\delta^{\frac{1}{\theta}}}\leq|U|\leq\delta$ for all $U\in\mathcal{F}$, we have that

[TABLE]

It implies that $\underline{\mbox{\rm dim}}_{\rm\theta}\,E\geq\alpha-\eta$. Since $\eta>0$ and $\alpha<s^{\theta}$ are arbitrarily chosen, we obtain that $\underline{\mbox{\rm dim}}_{\rm\theta}\,E\geq s_{\theta}.$

Next, we prove $\underline{\mbox{\rm dim}}_{\rm\theta}\,E\leq s_{\theta}$. Arbitrarily choose $\beta>\gamma>s_{\theta}$. Since $\gamma>s_{\theta}$, that is $\underline{P}(\gamma,\theta)<0$, there exists a sequence $\{\delta_{k}\}_{k=1}^{\infty}$ convergent to [math] such that

[TABLE]

For each $k>0$, there exists a cut set $\mathcal{M}_{k}$ such that $\delta_{k}^{\frac{1}{\theta}}<\|D\varphi_{\mathbf{u}^{*}}\|$ and $\|D\varphi_{\mathbf{u}}\|\leq\delta_{k}$ for all $\mathbf{u}\in\mathcal{M}_{k}$ satisfying

[TABLE]

By (1.11), there exists an integer $K_{1}>0$, such that for all $k>K_{1}$,

[TABLE]

By Lemma 4.4 and Lemma 4.6, we have that

[TABLE]

Since $|J_{\mathbf{u}}|\leq C_{1}\delta$ there exists a constant $C^{\prime\prime}$ such that For each integer $k>0$ and each $\mathbf{u}\in\mathcal{M}_{k}^{\prime\prime}$,

[TABLE]

Since $\beta>\gamma>s_{\theta}$, by (5.40) and (5.41), we obtain that

[TABLE]

It follows that $\underline{\mbox{\rm dim}}_{\rm\theta}\,E\leq\beta$. Since $\beta\geq s^{\theta}$ is arbitrarily chosen, we obtain that $\underline{\mbox{\rm dim}}_{\rm\theta}\,E\leq s^{\theta}$. ∎

6. box dimension and packing dimension of non-autonomous finite conformal set

Let $E$ be the non-autonomous finite conformal set of the NFCIFS $\Phi$. According to the definition of the intermediate dimension, the box dimension is obtained by $s^{\theta}$ at $\theta=1$, and we may show the relationship between the box dimension and the jump point of the upper pressure function.

Lemma 6.1**.**

Let $E$ be the non-autonomous finite conformal set of the NFCIFS $\Phi$ satisfying (1.11) and OSC. Then the box dimensions of $E$ is given by

[TABLE]

where $s^{*}$ is given by (2.17).

Proof.

For $\theta=1$, by theorem 2.1, we have $\overline{\mbox{\rm dim}}_{\rm B}\,E=s^{1}$. Recall

[TABLE]

Then we may show $s^{1}\leq t^{*}\leq s^{*}\leq s^{1}$. First we show $s^{*}\geq t^{*}$. For each non-integral $t>s^{*}$, we have $\overline{P}(t)<0$. Hence there exists a $K_{0}>0$ such that for each $k>K_{0}$,

[TABLE]

and it follows that

[TABLE]

Then we have $t>t^{*}$, and $s^{*}\geq t^{*}$.

For every $t>t^{*}$, we have $\sum_{k=1}^{\infty}\sum_{\mathbf{u}\in\Sigma^{k}}\|D\varphi_{\mathbf{u}}\|^{t}<\infty,$ and it follows that

[TABLE]

which means $\overline{P}(t,1)\leq 0$. Then we have $t>s^{1}$, and $t^{*}\geq s^{1}$.

For each $t^{\prime}>t>s^{1}$, we have $\overline{P}(t,1)<0$, and there exists a $\Delta>0$ such that for each $0<\delta<\Delta$,

[TABLE]

Choosing $\rho$ such that $c<\rho<1$ where $c$ is the uniform contraction constant given by (1.5), there exists an integer $k_{2}$ such that $\rho^{k_{2}+1}<\Delta\leq\rho^{k_{2}}$. For each integer $k>0$, we write

[TABLE]

Since $c<\rho<1$, it is clear that $\Sigma^{*}=\bigcup_{k=0}^{\infty}\mathcal{M}(\rho^{k})$, and it follows that

[TABLE]

Hence, by (2.16), it is clear that $\overline{P}(t^{\prime})\leq 0$, and by (2.17), we have that $t^{\prime}>s^{*}$ and $s^{1}\geq s^{*}$. Therefore $s^{*}=s^{1}=\overline{\mbox{\rm dim}}_{\rm B}\,E.$ ∎

Recall that $J_{\mathbf{u}}=\Psi_{\mathbf{u}}(J)$ for each $\mathbf{u}\in\Sigma^{*}$. We write

[TABLE]

and it is clear that $E^{\mathbf{u}}\subset E$. Note that $E^{\mathbf{u}}$ may be regarded as the attractor of a certain non-autonomous iterated function system $\Phi^{\mathbf{u}}$, and we denote $\overline{P}^{\mathbf{u}}(t)$ as its corresponding upper topological pressures. In the following conclusion, we actually show that

[TABLE]

Proof of Theorem 2.2.

For each open set $O\subset\mathbb{R}^{d}$ such that $O\cap E\not=\emptyset$, we choose $x\in O\cap E$, and by Lemma 3.2, there exists $\mathbf{u}\in\Sigma^{\infty}$ such that $x=\pi_{\Phi}(\mathbf{u}).$ Since $\lim_{k\to\infty}|J_{\mathbf{u}|_{k}}|=0$, there exists an integer $k>0$ such that $J_{\mathbf{u}|_{k}}\subset O$, and we write $\mathbf{u}_{k}=\mathbf{u}|_{k}\in\Sigma^{k}$. For each $\mathbf{v}\in\Sigma^{k}$, we have

[TABLE]

Since $C^{-1}\|D\varphi_{\mathbf{v}}\|\|D\varphi_{\mathbf{u}}\|\leq\|D\varphi_{\mathbf{vu}}\|\leq\|D\varphi_{\mathbf{v}}\|\|D\varphi_{\mathbf{u}}\|,$ we have

[TABLE]

Let

[TABLE]

By Lemma 6.1, it follows that $\overline{\mbox{\rm dim}}_{\rm B}\,E^{\mathbf{w}}=s^{*}_{\mathbf{w}}=s^{*}_{\mathbf{v}}=\overline{\mbox{\rm dim}}_{\rm B}\,E^{\mathbf{v}}$ for each $\mathbf{v},\mathbf{w}\in\Sigma^{k}$, and by finite stability of box dimensions, we have

[TABLE]

for all $\mathbf{v},\mathbf{w}\in\Sigma^{k}$. Thus we have $\overline{\mbox{\rm dim}}_{\rm B}\,E^{\mathbf{u}_{k}}=\overline{\mbox{\rm dim}}_{\rm B}\,E,$ and this implies that $\overline{\mbox{\rm dim}}_{\rm B}\,(E\cap O)=\overline{\mbox{\rm dim}}_{\rm B}\,E$ for every open set $O\cap E\not=\emptyset$. By [13, Corollary 3.10], it follows that $\mbox{\rm dim}_{\rm P}\,E=\overline{\mbox{\rm dim}}_{\rm B}\,E=s^{*}$. ∎

By Theorem 2.2, we may relax the condition to (1.13) for the lower bound of the upper box dimensions.

Proposition 6.2**.**

Let $E$ be the non-autonomous finite conformal set satisfying OSC and

[TABLE]

Then the box and packing dimension of $E$ bounded below $s^{*}$, that is

[TABLE]

Proof.

Fix $t_{0}<s_{*}$, and let $\alpha_{k}=k$. By Lemma 3.8, there exists a subsystem $\Phi^{\prime}$ of $\Phi$ satisfying $\overline{c}_{k}=\overline{c}_{k}^{\prime}$, $\overline{P}^{\prime}(t)=\overline{P}(t)$ for all $t\geq t_{0}$ and for all $n$

[TABLE]

Let $E^{\prime}$ be the non-autonomous finite conformal set of $\Phi^{\prime}$.

Since

[TABLE]

it follows that

[TABLE]

Since $\|D\varphi_{\mathbf{u}}\|\leq c^{k}$ for each $\mathbf{u}\in\Sigma^{k}$ where $c$ is uniform contraction constant given by (1.5), we have

[TABLE]

Note that $M_{k}^{\prime}\leq M_{k}<1$, and it follows that

[TABLE]

By Theorem 2.2, $\mbox{\rm dim}_{\rm P}\,E^{\prime}={s^{*}}^{\prime}$. Note that $\underline{P}^{\prime}(t)=\underline{P}(t)$ for all $t\geq t_{0}$, that is ${s^{*}}^{\prime}=s^{*}$. Then we have $\overline{\mbox{\rm dim}}_{\rm B}\,E\geq\mbox{\rm dim}_{\rm P}\,E\geq\mbox{\rm dim}_{\rm P}\,E^{\prime}=s^{*}$. ∎

7. Hausdorff dimension of non-autonomous finite conformal set

In this section, we first study the Hausdorff dimension of non-autonomous finite conformal sets under $\mathcal{L}^{d}(\partial J)=0$ and (1.11). Then we further relax the latter condition to

[TABLE]

where $M_{k},\underline{c}_{k}$ and $\overline{c}_{k}$ are given by (1.10).

Lemma 3.4 shows that $s_{*}$ given by (3.21) is always an upper bound to the Hausdorff dimensions of non-autonomous finite conformal sets. Next, we show that $s_{*}$ actually gives the Hausdorff dimensions of non-autonomous finite conformal sets under the condition $\mathcal{L}^{d}(\partial J)=0$ and $\lim_{k\to+\infty}\frac{\log\underline{c}_{k}}{\log M_{k}}=0.$

Proof of Theorem 2.3.

By Lemma 3.4, it is sufficient to prove $\mbox{\rm dim}_{\rm H}\,E\geq s_{*}$.

For each $t^{\prime}<t<s_{*}$, we have $\underline{P}(t)>0$, and there exists $K_{1}$ such that

[TABLE]

for all $k>K_{1}$. Since $\mbox{int}(J_{\mathbf{u}})\cap\mbox{int}(J_{\mathbf{v}})=\emptyset$ and $\mathcal{L}^{d}(\partial J)=0$, we have that $\mathcal{L}^{d}(J_{\mathbf{u}}\cap J_{\mathbf{v}})=0$ for all $\mathbf{u},\mathbf{v}\in\Sigma^{*}$ with $|\mathbf{u}|=|\mathbf{v}|$. Since $\lim_{k\to+\infty}\frac{\log\underline{c}_{k}}{\log M_{k}}=0,$ there exists an integer $K_{0}>0$ such that for all $k>K_{0}$,

[TABLE]

Next, we construct a sequence of probability measures $\{\mu_{n}\}_{n=1}^{\infty}$ on $J$. For each $n>0$, Let $\mu_{n}$ be a uniformly distributed measure given by

[TABLE]

for all $J_{\mathbf{u}},\mathbf{u}\in\Sigma^{n}$. For every $0<m<n$ and every $\mathbf{u}\in\Sigma^{m}$, we have that

[TABLE]

By Lemma 4.4, it follows that

[TABLE]

For every $B(x,r)\subset\mathbb{R}^{d}$, by (5.34), we have

[TABLE]

By (5.35) and (5.36), letting $n>\max\{K_{1},K_{0},k^{+}_{B(x,r)}\}$, it implies that

[TABLE]

Combining (7.42), (7.43) with Lemma 4.6, it follows that

[TABLE]

where $C_{3}=(2C_{1})^{t^{\prime}}C^{2t}e^{\frac{-\underline{P}(t)}{2}}$ is a constant. By Lemma 5.1, we have

[TABLE]

Since $\{\mu_{n}\}$ is bounded on $J$, there exists a subsequence $\{\mu_{n_{k}}\}$ converging weakly to a measure $\mu$; see [8]. Hence, taking limit on both sides of (7.46), we obtain that

[TABLE]

By the mass distribution principle [13], this implies that $\mbox{\rm dim}_{\rm H}\,E\geq t^{\prime}$ for all $t^{\prime}<s_{*}$, and we have $\mbox{\rm dim}_{\rm H}\,E\geq s_{*}$, which completes the proof. ∎

Next, we show that $s_{*}$ still gives the Hausdoff dimension under (1.13) by applying Theorem 2.3 and approximation of subsystems.

Proof of Corollary 2.4.

By Lemma 3.4, if $s_{*}=0$, the conclusion holds. If $s_{*}>0$, by the similar argument to Proposition 6.2, there exists a subsystem $\Phi^{\prime}$ of $\Phi$ such that $\underline{P}^{\prime}(t)=\underline{P}(t)$ for all $t\geq t_{0}$, and $E^{\prime}$ is the non-autonomous finite conformal set of $\Phi^{\prime}$ satisfying $\lim_{k\to\infty}\frac{\log\underline{c}^{\prime}_{k}}{\log M^{\prime}_{k}}=0.$ By Theorem 2.3, $\mbox{\rm dim}_{\rm H}\,E^{\prime}=s_{*}^{\prime}$. Note that $\underline{P}^{\prime}(t)=\underline{P}(t)$ for all $t\geq t_{0}$, that is $s_{*}^{\prime}=s_{*}$. Since $\mbox{\rm dim}_{\rm H}\,E^{\prime}\leq\mbox{\rm dim}_{\rm H}\,E\leq s_{*}$, we have $\mbox{\rm dim}_{\rm H}\,E=s_{*}$. ∎

Given a set $J\subset\mathbb{R}^{d}$, we say $J$ satisfies cone condition if there exists $\alpha,l>0$ such that for every $x\in\partial J\subset\mathbb{R}^{d}$, there exists an open cone $\mbox{Con}(x,u_{x},\alpha,l)\subset\mbox{int}(J)$ with vertex $x$, direction vector $u_{x}$, central angle of Lebesgue measure $\alpha$ and altitude $l$. Given a non-autonomous finite or infinite conformal iterated function system $\Phi$, we say $\Phi$ satisfies cone condition if the initial set $J$ in the definition of $\Phi$ satisfies cone condition. See [35] for details. It is a standard fact that the cone condition implies $\mathcal{L}^{d}(\partial J)=0$, and we include a proof for the convenience of readers.

Lemma 7.1**.**

Given $\Phi$ satisfying cone condition and OSC. For every $x\in J$, $r>0$, Let $\{\mathbf{u}_{\lambda}\}_{\lambda\in\Lambda}\subset\Sigma^{*}$ satisfy that $|J_{\mathbf{u}_{\lambda}}|>r$ and $J_{\mathbf{u}_{\lambda}}\cap B(x,r)\not=\emptyset$ for $\lambda\in\Lambda$, and that $\Psi_{\mathbf{u}_{\lambda_{1}}}(\mbox{int}(J))\cap\Psi_{\mathbf{u}_{\lambda_{2}}}(\mbox{int}(J))=\emptyset$ for $\lambda_{1}\not=\lambda_{2}$. Then

[TABLE]

where $M_{1}$ is a constant independent of $x$ and $r$.

Proof.

By [35, (2.10)], there exists a constant $D>1$ and $0<\beta\leq\alpha$ such that for all $x\in J$ and for all $\mathbf{u}\in\Sigma^{*}$, by (1.6), it follows that

[TABLE]

Since $|J_{\mathbf{u}_{\lambda}}|>r$, it is clear that $\{\mathbf{u}_{\lambda}\}_{\lambda\in\Lambda}$ is finite, i.e. $\#\Lambda<\infty$. For each $\lambda\in\Lambda$, since $B(x,r)\cap\Psi_{\mathbf{u}_{\lambda}}(J)\not=\emptyset$, by the bounded distortion, there exists $\mbox{Con}_{\lambda}\subset\Psi_{\mathbf{u}_{\lambda}}(\mbox{int}(J))$ with central angle of Lebesgue measure $\beta$ and altitude $(C_{1}D)^{-1}r$ such that $\mbox{Con}_{\lambda}\subset B(x,(1+(C_{1}D)^{-1}r)$, and $\mbox{Con}_{\lambda}\cap\mbox{Con}_{\lambda^{\prime}}=\emptyset$ for all $\lambda\neq\lambda^{\prime}$. It follows that

[TABLE]

Let $M_{1}=\frac{(1+(C_{1}D)^{-1})^{d}}{\beta((C_{1}D)^{-1})^{d}}$, and the conclusion holds. ∎

Lemma 7.2**.**

Let $J\subset\mathbb{R}^{d}$ be a compact set satisfying cone condition. Then we have $\mathcal{L}^{d}(\partial J)=0$.

Proof.

Let $Q=\{x_{1},\ldots,x_{d-1},0)\in\mathbb{R}^{d}:x_{i}\in\mathbb{Q},1\leq i\leq d-1\}$. For each $x\in\partial J\subset\mathbb{R}^{d}$, by cone condition, there exists an open cone $\mbox{Con}(x,u_{x},\alpha,l)\subset\mbox{int}(J)$ with vertex $x$, direction vector $u_{x}$, central angle of Lebesgue measure $\alpha$ and altitude $l$. Since $\mathbb{Q}$ is dense in $\mathbb{R}$, there exists another open cone $\mbox{Con}(x,u_{x}^{\prime},\frac{\alpha}{2},\frac{l}{2})\subset\mbox{int}(J)$ such that the line passing through the point $x$ with the direction of $u_{x}^{\prime}$ intersects $Q$. For each $x\in\partial J$, let $l_{x}$ be the straight line passing through the point $x$ with the direction of $u_{x}^{\prime}$.

Let $k=[\frac{2|J|}{l}]+1$. Fix $q\in Q$. For each $1\leq i\leq k$, let

[TABLE]

For $x\in A_{q,+}^{i}$, let

[TABLE]

where $\mbox{Con}(q,u_{x}^{\prime},|x-q|+\frac{l}{2})$ and $\mbox{Con}(q,-u_{x}^{\prime},|x-q|-\frac{l}{2})$ have the same base as $\mbox{Con}(x,u_{x}^{\prime},\frac{\alpha}{2},\frac{l}{2})$.

Since $Q$ is countable, we rewrite it as $Q=\{q_{j}\}_{j=1}^{\infty}$. For each $j\in\mathbb{N}$, let $B_{q_{j},+}^{i}=\bigcup_{x\in A_{q_{j},+}^{i}}B_{q_{j},+}^{i}(x).$ For $0<\epsilon<\frac{l}{2|J|}$, we have

[TABLE]

and it follows that

[TABLE]

Similarly, we have that

[TABLE]

Since

[TABLE]

it implies that

[TABLE]

and combining it with (7.47) and (7.48), we obtain that

[TABLE]

where $C_{4}$ is a constant. The conclusion follows by the arbitrariness of $\epsilon$. ∎

In [39], Rempe-Gillen and Urbański showed if a non-autonomous finite conformal set satisfies cone condition and $\lim_{k\to\infty}\frac{\log\#I_{k}}{k}=0$, then its Hausdorff dimension is equal to $s_{*}$. We slightly generalise their conclusion into the following form.

Corollary 7.3**.**

Given NFCIFS $\Phi$ satisfying cone condition and OSC. Suppose $m_{\Phi}=\inf\{\underline{P}(t):\underline{P}(t)>0\}>0$. If

[TABLE]

then $\mbox{\rm dim}_{\rm H}\,E=s_{*},$ where $s_{*}$ is given by (2.17).

Proof.

By Lemma 3.4, it is sufficient to prove $\mbox{\rm dim}_{\rm H}\,E\geq s_{*}$. For each $t<s_{*}$, we have $\underline{P}(t)>0$. By (7.49), there exist two constants $M\in(0,1)$ and $K>0$ such that for all $k>K$,

[TABLE]

Since $\Phi$ satisfies the cone condition, by Lemma 7.2, we construct the measure $\mu_{n}$ by (7.44). Given $x\in J$ and $r>0$, by (5.36), for each $\mathbf{u}\in D(B(x,r),k)$, it is clear that $|J_{\mathbf{u}^{*}}|>2r$ and $J_{\mathbf{u}^{*}}\cap B(x,r)\not=\emptyset$. Since $\Phi$ satisfies cone condition, by Lemma 7.1, we have

[TABLE]

Recall (7.45), that is

[TABLE]

By Lemma 4.6, we have $\|D\varphi_{\mathbf{u}}\|\leq C_{1}|J_{\mathbf{u}}|\leq C_{1}r$. Combining it with (7.50), we have that

[TABLE]

where $C_{5}$ is a constant independent of $r$ and $k$. Choose $n>\max\{K,k^{+}_{B(x,r)}\}$. By (5.34), it follows that

[TABLE]

Since $\{\mu_{n}\}$ is bounded on $J$, there exists $\{\mu_{n_{k}}\}$ weakly convergent to a measure $\mu$ (see [8]), and it follows that

[TABLE]

By the mass distribution principle in [13], it follows that $\mbox{\rm dim}_{\rm H}\,E\geq t^{\prime}$ for all $t<s_{*}$, and we have $\mbox{\rm dim}_{\rm H}\,E\geq s_{*}$. ∎

8. Dimensions of the non-autonomous infinite conformal set

In this section, we study the Hausdorff dimensions of non-autonomous infinite conformal sets by applying the strategy in Subsection 3.3.

Let $\Phi$ be a non-autonomous infinite conformal iterated function system. Recall that $M_{n}=\max_{\mathbf{u}\in\Sigma^{n}}\{\|D\varphi_{\mathbf{u}}\|\}$ and

[TABLE]

We assume that $\{\|D\varphi_{n,k}\|\}_{k=1}^{\infty}$ is monotonically decreasing, and we have

[TABLE]

Proof of Theorem 2.5.

By Proposition 3.6, there exists a finite subsystem $\Phi^{\prime}$ satisfying $\lim_{n\to+\infty}\frac{\log\#(I^{\prime}_{n})}{\log M^{\prime}_{n}}=0$ and $\underline{P}^{\prime}(t)=\underline{P}(t)$ for each $t\in(0,d)$. Let $E^{\prime}$ be the non-autonomous finite conformal set of $\Phi^{\prime}$. Since $\overline{c}_{n}=\overline{c}_{n}^{\prime}$ and $M_{n}\geq M_{n}^{\prime}$, we have $\lim_{n\to\infty}\frac{\log\underline{c}^{\prime}_{n}}{\log M^{\prime}_{n}}=0.$ By Proposition 6.2, we have

[TABLE]

Furthermore, since $\mathcal{L}^{d}(\partial J)=0$, by Lemma 3.4 and Corollary 2.4, we have $\mbox{\rm dim}_{\rm H}\,E=\mbox{\rm dim}_{\rm H}\,E^{\prime}=s_{*}$. ∎

Given the non-autonomous infinite conformal system $\Phi$ satisfying the cone condition. Rempe-Gillen and Urbański [39] showed that if the following hold for all $t\in(0,d)$ and all $\epsilon>0$:

• (1).

The sum $\sum_{k\in I_{n}}\|D\varphi_{n,k}\|^{t}$ are either infinite for all $n$ or finite for all $n$.

• (2).

If $\sum_{k\in I_{n}}\|D\varphi_{n,k}\|^{t}<\infty$, then $\lim_{n\to\infty}\frac{\sum_{k\leq e^{\epsilon n}}\|D\varphi_{n,k}\|^{t}}{\sum_{k\in I_{n}}\|D\varphi_{n,k}\|^{t}}=1.$

• (3).

If $\sum_{k\in I_{n}}\|D\varphi_{n,k}\|^{t}=\infty$, then $\lim_{n\to\infty}\sum_{k\leq e^{\epsilon n}}\|D\varphi_{n,k}\|^{t}=\infty.$

Then the Hausdorff dimension of the non-autonomous conformal set is equal to $s_{*}$.

Compared with their conclusions, since $\lim_{k\to\infty}\frac{k}{-\log M_{k}}\leq\frac{1}{-\log c}$, we reduce the requirements for the convergence rate of $\sum_{k\in I_{n}}\|D\varphi_{n,k}\|^{t}$, but we need the condition $\lim_{k\to+\infty}\frac{\log\overline{c}_{k}}{\log M_{k}}=0$ holds in Theorem 2.5.

The following conclusion is a consequence of Corollary 7.3.

Corollary 8.1**.**

Let $E$ be the non-autonomous infinite conformal set of the NICIFS $\Phi$ satisfying OSC and cone condition with $m_{\Phi}=\inf\{\underline{P}(t):\underline{P}(t)>0\}>0$. Suppose there exists $0<M<1$ such that for all $t\in(0,d)$, it satisfies

• (1).

The sum $\sum_{k\in I_{n}}\|D\varphi_{n,k}\|^{t}$ are either infinite for all $n$ or finite for all $n$.

• (2).

If $\sum_{k\in I_{n}}\|D\varphi_{n,k}\|^{t}<\infty$, then

[TABLE]

• (3).

If $\sum_{j\in I_{n}}\|D\varphi_{n,k}\|^{t}=\infty$, then $\lim_{n\to\infty}\sum_{k\leq{e^{Mm_{\Phi}n}}}\|D\varphi_{n,k}\|^{t}=\infty.$

Then we have $\mbox{\rm dim}_{\rm H}\,E=s_{*}$, where $s_{*}$ is given by (2.17).

Proof.

Since $E$ satisfies OSC, by Theorem 3.4, it follows that $\mbox{\rm dim}_{\rm H}\,E\leq s_{*}$.

Let $\beta_{n}=e^{-2Mm_{\Phi}n}$. By Corollary 3.7, there exists a finite subsystem $\Phi^{\prime}$ such that

[TABLE]

Since $\Phi^{\prime}$ satisfies OSC and cone condition, by Corollary 7.3, we have $\mbox{\rm dim}_{\rm H}\,E^{\prime}=s_{*}^{\prime}$, where $s_{*}^{\prime}$ is given by (2.17) with respect to the subsystem $\Phi^{\prime}$. By Corollary 3.7, we obtain $\mbox{\rm dim}_{\rm H}\,E\geq\mbox{\rm dim}_{\rm H}\,E^{\prime}=s_{*}^{\prime}=s_{*}$, and the conclusion holds. ∎