Dimension theory of Non-Autonomous iterated function systems

Let $\{\Psi_{i}\}_{1\leq i\leq M}$ be a finite set of affine contractions on $\mathbb{R}^{d}$ with $M\geq 2$ and

(1.1)

$$\Psi_{i}(x)=T_{i}(x)+a_{i}\qquad(1\leq i\leq M),$$

where $a_{i}$ is a translation vector, and $T_{i}$ is a linear transformation on $\mathbb{R}^{d}$. The set $\{\Psi_{i}\}_{1\leq i\leq M}$ is called a self-affine iterated function system. By the well-known theorem of Hutchinson, see [4, 19], the IFS has a unique attractor, that is a unique non-empty compact $E\subset\mathbb{R}^{d}$ such that

(1.2)

$$E=\bigcup_{1\leq i\leq M}\Psi_{i}(E),$$

which is called a self-affine set. If the affine transformations $\Psi_{i}$ are similarity mappings, we call $E$ a self- similar set, see [4] for details.

Formulae giving the Hausdorff and box-counting dimensions of self-similar sets satisfying the open set condition are well-known, see [4]. However, calculation of the dimensions of self-affine sets is more awkward, see [2, 9, 12, 15, 22, 25]. For each $k=1,2,\cdots$, we write $\mathcal{I}^{k}=\{i_{1}i_{2}\cdots i_{k}:1\leq i_{j}\leq M,\ j\leq k\}$ for the set of words of length $k$. Let $\phi^{s}$ be the singular value function defined by  (2.14). Falconer in [5] defined the criticla value $d(T_{1},\ldots,T_{M})$ which is the unique solution to

$$\lim_{k\to\infty}\left(\sum_{\mathbf{i}\in\mathcal{I}^{k}}\phi^{s}(T_{\mathbf{i}})\right)^{\frac{1}{k}}=1,$$

and it is often called affine dimension or Falconer dimension. Given $\|T_{i}\|<\frac{1}{2}$ for $i=1,2,\ldots M,$ it turns out that

(1.3)

$$\dim_{\rm H}F=\dim_{\rm B}F=\min\{d,d(T_{1},\ldots,T_{M})\},$$

for almost all $\mathbf{a}=(a_{1},\ldots,a_{M})\in\mathbb{R}^{Md}$(in the sense of $Md$-dimensional Lebesgue measure), we refer the readers to [5, 27] for details. Note that $d(T_{1},\ldots,T_{M})$ is always the upper bound for the box-counting dimension of self-affine sets, Falconer in [6] proved that the box-counting dimension and affine dimension coincide by applying projection condition with other assumptions. From then on, a considerable amount of literature has been published on the validation of formula (1.3) under various conditions. In particular, in [20], Jordan, Pollicott and Simon studied perturbed self-affine sets where they changed translations into independently identically distributed random variables, and they showed that for $\|T_{i}\|<1$, $i=1,2,\ldots M,$ the dimension formula (1.3) holds almost surely. In [10], Falconer and Kempton showed the dimension formula (1.3) holds for all translations in $R^{2}$ under various assumptions. We refer readers to [1, 6, 7, 8, 11, 13, 14, 16, 18, 24] for various related studies.

Moran sets were first studied by Moran in [23], and we recall the definition for the readers’ convenience.

Let $\{n_{k}\}_{k\geq 1}$ be a sequence of integers greater than or equal to $2$. For each $k=1,2,\cdots$, we write

(1.4)

$$\Sigma^{k}=\{u_{1}u_{2}\cdots u_{k}:1\leq u_{j}\leq n_{j},\ j\leq k\}$$

for the set of words of length $k$, with $\Sigma^{0}=\{\emptyset\}$ containing only the empty word $\emptyset$, and write

(1.5)

$$\Sigma^{*}=\bigcup_{k=0}^{\infty}\Sigma^{k}$$

for the set of all finite words.

Suppose that $J\subset\mathbb{R}^{d}$ is a compact set with $\mbox{int}(J)\neq\varnothing$ (we always write int($\cdot)$ for the interior of a set). Let $\{\Xi_{k}\}_{k\geq 1}$ be a sequence of positive real vectors where $\Xi_{k}=(c_{k,1},c_{k,2},\cdots,c_{k,n_{k}})$ for each $k\in\mathbb{N}$. We say the collection $\mathcal{J}=\{J_{\mathbf{u}}:\mathbf{u}\in\Sigma^{*}\}$ of closed subsets of $J$ fulfils the Moran structure if it satisfies the following Moran structure conditions (MSC):

• (1).

  For each $\mathbf{u}\in\Sigma^{*}$, $J_{\mathbf{u}}$ is geometrically similar to $J$, i.e., there exists a similarity $\Psi_{\mathbf{u}}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ such that $J_{\mathbf{u}}=\Psi_{\mathbf{u}}(J)$. We write $J_{\emptyset}=J$ for the empty word $\emptyset$.

• (2).

  For all $k\in\mathbb{N}$ and $\mathbf{u}\in\Sigma^{k-1}$, the elements $J_{\mathbf{u}1},J_{\mathbf{u}2},\cdots,J_{\mathbf{u}n_{k}}$ of $\mathcal{J}$ are the subsets of $J_{\mathbf{u}}$ with disjoint interiors, ie., $\mbox{int}(J_{\mathbf{u}i})\cap\mbox{int}(J_{\mathbf{u}i^{\prime}})=\varnothing$ for $i\neq i^{\prime}$. Moreover, for all $1\leq i\leq n_{k}$,

  $$\frac{|J_{\mathbf{u}i}|}{|J_{\mathbf{u}}|}=c_{k,i},$$

  where $|\cdot|$ denotes the diameter.

The non-empty compact set

$$E=E(\mathcal{J})=\bigcap\nolimits_{k=1}^{\infty}\bigcup\nolimits_{\mathbf{u}\in\Sigma^{k}}J_{\mathbf{u}}$$

is called a Moran set determined by $\mathcal{J}$. For each $\mathbf{u}\in\Sigma^{k}$, the element $J_{\mathbf{u}}$ is called a  $k$th-level basic set of $E$. For each integer $k>0,$ let $d_{k}$ be the unique real solution of the equation

(1.6)

$$\prod\nolimits_{i=1}^{k}\left(\sum\nolimits_{j=1}^{n_{i}}(c_{i,j})^{d_{k}}\right)=1.$$

Let $d_{\ast}$ and $d^{\ast}$ be the real numbers given respectively by

(1.7)

$$d_{\ast}=\liminf_{k\rightarrow\infty}d_{k},\quad d^{\ast}=\limsup_{k\rightarrow\infty}d_{k}.$$

It was shown in [17, 28, 29] that if $c_{*}=\inf\{c_{k,j}:k\in\mathbb{N},1\leq j\leq n_{k}\}>0,$ the following dimension formulae hold

$$\dim_{\rm H}E=d_{\ast},\quad\dim_{\rm P}E={\overline{\dim}}_{\rm B}E=d^{\ast}.$$

The dimension theory of Moran sets has been studied extensively, and we refer the readers to [17, 28, 29] for detail and references therein. Note that, in the definition of Moran sets, the position of $J_{\mathbf{u}i}$ in $J_{\mathbf{u}}$ is very flexible, and the contraction ratios may also vary at each level. Therefore the structures of Moran sets are more complex than self-similar sets, and in general, the inequality

$$\dim_{\rm H}E\leq\underline{\dim}_{\rm B}E\leq{\overline{\dim}}_{\rm B}E$$

holds strictly for Moran fractals. The general lower box dimension formula for Moran sets is still an open question. Except providing various examples, Moran sets are also useful tools for analysing properties of fractal sets in various studies, for example, see [28] and references therein for applications.

Note that similarities and separation assumption are required in the Moran structure, Inspired by the structure of iterated function systems, we generalize the Moran structure to non-autonomous structure where we replace similarities by contractions or affine mappings, and remove the separation assumption. The geometric properties of non-autonomous sets are more complex than Moran sets since mappings may have different contraction ratios in different directions. Therefore, non-autonomous sets provide not only interesting phenomena but also useful tools for fractal analysis. Different to the attractors of iterated function systems, non-autonomous sets do not have dynamical properties any more, and the tools in ergodic theory cannot be invoked. As a result, it is more difficult to explore their dimension formulae and other fractal properties.

In this paper, we investigate the dimension theory of non-autonomous attractors, and particularly, we are interested in a classs of attractors, named “non-autonomous affine sets or affine sets”. In section 2, we first give the definition of non-autonomous attractors and estimate the dimensions of such sets, then we study the non-autonomous affine sets and state the main conclusions in subsection 2.3. The dimension estimates for non-autonomous sets are provided in section 3. Upper and lower box dimensions of non-autonomous affine sets are explored in section 4. We discuss the upper bounds of Hausdorff dimensions of non-autonomous affine sets in section 5, and we give the Hausdorff dimension formula for a spectial class of non-autonomous affine sets with finitely many translations. In section 6, we study the Hausdorff dimension of non-autonomous affine sets with random translations, and we prove that with probability one, the Hausdorff dimensions of such sets equal to a critical value. Finally, in section 7, we compare critical values of non-autonomous affine sets, and we also provide some examples to illustrate our conclusions.

Let $\{n_{k}\}_{k=1}^{\infty}$ be a sequence of integers such that $n_{k}\geq 2$ for all $k\geq 1$. Let $\Sigma^{k}$ and $\Sigma^{*}$ be given by (1.4) and (1.5), respectively. Suppose that $J\subset\mathbb{R}^{d}$ is a compact set with non-empty interior. Let $\{\Xi_{k}\}_{k=1}^{\infty}$ be a sequence of collections of contractive mappings, that is

(2.8)

$$\Xi_{k}=\{S_{k,1},S_{k,2},\ldots,S_{k,n_{k}}\},$$

where each $S_{k,j}$ satisfies that $|S_{k,j}(x)-S_{k,j}(y)|\leq c_{k,j}|x-y|$ for some $0<c_{k,j}<1$. We say the collection $\mathcal{J}=\{J_{\mathbf{u}}:\mathbf{u}\in\Sigma^{*}\}$ of closed subsets of $J$ fulfils the non-autonomous structure with respect to $\{\Xi_{k}\}_{k=1}^{\infty}$ if it satisfies the following conditions:

• (1).

  For all integers $k>0$ and all $\mathbf{u}\in\Sigma^{k-1}$, the elements $J_{\mathbf{u}1},J_{\mathbf{u}2},\cdots,J_{\mathbf{u}n_{k}}$ of $\mathcal{J}$ are the subsets of $J_{\mathbf{u}}$. We write $J_{\emptyset}=J$ for the empty word $\emptyset$.

• (2).

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

  $$J_{\mathbf{u}}=\Psi_{\mathbf{u}}(J)=\Psi_{u_{1}}\circ\ldots\circ\Psi_{u_{j}}\ldots\circ\Psi_{u_{k}}(J),$$

  where $\Psi_{u_{j}}(x)=S_{j,u_{j}}x+\omega_{u_{1}\ldots u_{j}}$, for some $\omega_{u_{1}\ldots u_{j}}\in\mathbb{R}^{d}$, and $S_{j,u_{j}}\in\Xi_{j}$, $j=1,2,\ldots k$.

• (3).

  The maximum of the diameters of $J_{\mathbf{u}}$ tends to 0 as $|\mathbf{u}|$ tends to $\infty$, that is,

  $$\lim_{k\to\infty}\max_{\mathbf{u}\in\Sigma^{k}}|J_{\mathbf{u}}|=0.$$

We call $\{\Xi_{k}\}_{k=1}^{\infty}$ a non-autonomous iterated function systems(NIFS) determined by $\mathcal{J}$, and the non-empty compact set

(2.9)

$$E=E(\mathcal{J})=\bigcap\nolimits_{k=1}^{\infty}\bigcup\nolimits_{\mathbf{u}\in\Sigma^{k}}J_{\mathbf{u}}$$

is called a non-autonomous attractor determined by $\mathcal{J}$. For all $\mathbf{u}\in\Sigma^{k}$, the elements $J_{\mathbf{u}}$ are called  $k$th-level basic sets of $E$. If the non-autonomous attractor $E$ satisfies that for all integers $k\geq 1$ and $\mathbf{u}\in\Sigma^{k-1}$,

$$\mathrm{int}(J_{\mathbf{u}i})\cap\mathrm{int}(J_{\mathbf{u}i^{\prime}})=\emptyset\quad\textit{ for }i\neq i^{\prime}\in\{1,2,\ldots,n_{k}\},$$

we say $E$ satisfies Moran separation condition (MSC).

(2) In the definition, the transformation $\Psi_{\mathbf{u}}$ may be determined in the following way. For each $\mathbf{u}=u_{1}\in\Sigma^{1}$ and $J_{\mathbf{u}}\in\mathcal{J}$, there exists an mapping $\Psi_{\mathbf{u}}:J\to J_{\mathbf{u}}$ such that

$$J_{\mathbf{u}}=\Psi_{\mathbf{u}}(J)=S_{1,u_{1}}(J)+\omega_{u_{1}}\qquad\qquad\textit{ for some }\omega_{u_{1}}\in\mathbb{R}^{d}.$$

Suppose that for each $\mathbf{u}\in\Sigma^{k-1}$ and $J_{\mathbf{u}}\in\mathcal{J}$, the affine mapping $\Psi_{\mathbf{u}}:J\to J_{\mathbf{u}}$ is defined. For each $1\leq j\leq n_{k}$ and $J_{\mathbf{u}j}\in\mathcal{J}$, there exists an affine mapping $\Psi_{\mathbf{u}j}:J\to J_{\mathbf{u}j}$ such that

$$J_{\mathbf{u}j}=\Psi_{\mathbf{u}j}(J)=\Psi_{\mathbf{u}}\circ\Big{(}S_{k,j}(J)+\omega_{\mathbf{u}j}\Big{)}\qquad\textit{ for some }\omega_{\mathbf{u}j}\in\mathbb{R}^{d}.$$

(3) The assumption $\lim_{k\to\infty}\max_{\mathbf{u}\in\Sigma^{k}}|J_{\mathbf{u}}|=0$ in the definition is necessary, otherwise the set $E$ may have positive finite Lebesgue measure, and we do not consider such case in this paper, see Example  1 in section 7.

(4) Non-autonomous iterated function system is also a generalization of the iterated function system. However, for $\mathbf{u}\neq\mathbf{v}\in\Sigma^{k-1}$, the transformations $S_{k,j}(x)+\omega_{\mathbf{u}j}$ and $S_{k,j}(x)+\omega_{\mathbf{v}j}$ in $\Psi_{\mathbf{u}j}$ and $\Psi_{\mathbf{v}j}$ have the same contractive part $S_{k,j}\in\Xi_{k}$ but with different translations. Therefore the non-autonomous attractor $E$ may be different to the attractor of classic iterated function system even if the sequence $\{\Xi_{k}\}$ is identical, that is, $n_{k}=M$ and $\Xi_{k}=\{S_{1},\ldots,S_{M}\}$ for all $k>0$.

For general non-autonomous sets, it is difficult to find their fractal dimensions, but we are still able to provide some rough estimates if contraction ratios are known.

We next obtain a lower bound for Hausdorff and box-counting dimensions in the case where the basic sets of $E$ satisfy the following condition. We say the non-autonomous set $E$ satisfies gap separation condition(GSC) if there exists a constant $C$ such that for all $\mathbf{u}\in\Sigma^{*}$, $i\neq j$, we have that

$$\inf\{|x-y|:x\in J_{\mathbf{u}i},y\in J_{\mathbf{u}j}\}\geq C|J_{\mathbf{u}}|$$

Note that if the contractions are all similarities in the NIFS $\{\Xi_{k}\}_{k=1}^{\infty}$, then $c_{*}>0$ is frequently used to guarranttee the lower bounds in Theorem 2.2. For general NIFSs, $c_{*}>0$ is not sufficient for the theorem, see Example 3 for a counterexample.

To study the dimension properties of non-autonomous sets, the following notations are frequently used in our context. Let $\Sigma^{k}$ and $\Sigma^{*}$ be given by (1.4) and (1.5), respectively. Let $\Sigma^{\infty}=\{(u_{1}u_{2}\ldots u_{k}\ldots):1\leq u_{k}\leq n_{k}\}$ be the corresponding set of infinite words, where $\{n_{k}\geq 2\}_{k\geq 1}$ is the sequence of integers.

For $\mathbf{u}=u_{1}\cdots u_{k}\in\Sigma^{k}$, we write $\mathbf{u}^{-}=u_{1}\ldots u_{k-1}$ and write $|\mathbf{u}|=k$ for the length of $\mathbf{u}$. For each $\mathbf{u}=u_{1}u_{2}\cdots u_{k}\in\Sigma^{*}$, and $\mathbf{v}=v_{1}v_{2}\cdots\in\Sigma^{\infty}$, we say $\mathbf{u}$ is a curtailment of $\mathbf{v}$, denote by $\mathbf{u}\preceq\mathbf{v}$, if $\mathbf{u}=v_{1}\cdots v_{k}=\mathbf{v}|k$. We call the set $\mathcal{C}_{\mathbf{u}}=\{\mathbf{v}\in\Sigma^{\infty}:\mathbf{u}\preceq\mathbf{v}\}$ the cylinder of $\mathbf{u}$, where $\mathbf{u}\in\Sigma^{*}$. If $\mathbf{u}=\emptyset$, its cylinder is $\mathcal{C}_{\mathbf{u}}=\Sigma^{\infty}$.

For $\mathbf{u},\mathbf{v}\in\Sigma^{\infty}$, let $\mathbf{u}\wedge\mathbf{v}\in\Sigma^{*}$ denote the maximal common initial finite word of both $\mathbf{u}$ and $\mathbf{v}$. We topologise $\Sigma^{\infty}$ 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 $\Sigma^{\infty}$ a compact metric space. The cylinders $\mathcal{C}_{\mathbf{u}}=\{\mathbf{v}\in\Sigma^{\infty}:\mathbf{u}\preceq\mathbf{v}\}$ for $\mathbf{u}\in\Sigma^{*}$ form a base of open and closed neighbourhoods for $\Sigma^{\infty}$. We call a set of finite words $A\subset\Sigma^{*}$ a covering set for $\Sigma^{\infty}$ if $\Sigma^{\infty}\subset\bigcup_{\mathbf{u}\in A}\mathcal{C}_{\mathbf{u}}$.

For $\mathbf{v}=v_{1}\ldots v_{k}\in\Sigma^{k}$, we denote compositions of mappings by $S_{\mathbf{v}}=S_{1,v_{1}}\ldots S_{k,v_{k}}$. Let $\Pi:\Sigma^{\infty}\rightarrow\mathbb{R}^{d}$ be the projection given by

	$\displaystyle\Pi(\mathbf{u})$	$\displaystyle=$	$\displaystyle\bigcap_{k=0}^{\infty}(S_{1,u_{1}}+\omega_{u_{1}})(S_{2,u_{2}}+\omega_{u_{1}u_{2}})\cdots(S_{k,u_{k}}+\omega_{\mathbf{u}|k})(J)$
		$\displaystyle=$	$\displaystyle\lim_{k\to\infty}(S_{1,u_{1}}+\omega_{u_{1}})(S_{2,u_{2}}+\omega_{u_{1}u_{2}})\cdots(S_{k,u_{k}}+\omega_{\mathbf{u}|k})(x)$
		$\displaystyle=$	$\displaystyle\omega_{u_{1}}+S_{1,u_{1}}\omega_{u_{1}u_{2}}+\cdots+S_{\mathbf{u}|k}\omega_{\mathbf{u}|k+1}+\cdots.$

It is clear that the attractor $E$ is the image of $\Pi$, i.e. $E=\Pi(\Sigma^{\infty})$. Note that the projection $\Pi$ is surjective. To emphasize the dependence on translations, we sometimes write $\Pi^{\omega}(\mathbf{u})$ and $E^{\omega}$ instead of $\Pi(\mathbf{u})$ and $E$.

Let $\mu$ be a finite Borel regular measure on $\Sigma^{\infty}$. We define $\mu^{\omega}$, the projection of the measure $\mu$ onto $\mathbb{R}^{d}$, by

(2.11)

$$\mu^{\omega}(A)=\mu\{\mathbf{u}:\Pi^{\omega}(\mathbf{u})\in A\},$$

for $A\subseteq\mathbb{R}^{d}$, or equivalently by

(2.12)

$$\int f(x)d\mu^{\omega}(x)=\int f(\Pi^{\omega}(\mathbf{u}))d\mu(\mathbf{u}),$$

for every continuous $f:\mathbb{R}^{d}\to\mathbb{R}$. Then $\mu^{\omega}$ is a Borel measure supported by $E^{\omega}$.

Let $\{\Xi_{k}\}_{k=1}^{\infty}$ be a sequence of collections of contractive matrices, that is

(2.13)

$$\Xi_{k}=\{T_{k,1},T_{k,2},\ldots,T_{k,n_{k}}\},$$

where $T_{k,j}$ are $d\times d$ matrices with $\|T_{k,j}\|<1$ for $j=1,2,\ldots n_{k}$. The collection $\mathcal{J}=\{J_{\mathbf{u}}:\mathbf{u}\in\Sigma^{*}\}$ of closed subsets of $J$ fulfils the non-autonomous structure with respect to the sequence $\{\Xi_{k}\}_{k=1}^{\infty}$ of contractive matrices. We call $\{\Xi_{k}\}_{k=1}^{\infty}$ the non-autonomous affine iterated function system (NAIFS) determined by $\mathcal{J}$, and we call the attractor $E$ given by (2.9) a non-autonomous affine set(NAS) or affine set determined by $\mathcal{J}$.

Note that Moran sets may be regarded as a special case of non-autonomous affine sets satisfying Moran separation condition, where $T_{k,i}$ is the multiplication of contraction ratio $c_{k,i}$ and a rotation matrix $A_{k,i}$.

Let $T\in\mathcal{L}(\mathbb{R}^{d},\mathbb{R}^{d})$ be a contracting and non-singular linear mapping. The singular values $\alpha_{1}(T),\alpha_{2}(T),\ldots$, $\alpha_{d}(T)$ of $T$ are the lengths of the (mutually perpendicular) principle semi-axes of $T(B)$, where $B$ is the unit ball in $\mathbb{R}^{d}$. Equivalently they are the positive square roots of the eigenvalues of $T^{\ast}T$, where $T^{\ast}$ is the transpose of $T$. Conventionally, we write that $1>\alpha_{1}(T)\geq\ldots\geq\alpha_{d}(T)>0$.

For $0\leq s\leq d$, the singular value function of $T$ is defined by

(2.14)

$$\phi^{s}(T)=\alpha_{1}(T)\alpha_{2}(T)\ldots\alpha_{m-1}(T)\alpha_{m}^{s-m+1}(T),$$

where $m$ is the integer such that $m-1<s\leq m$. For technical convenience, we set $\phi^{s}(T)=(\alpha_{1}(T)\alpha_{2}(T)\ldots\alpha_{d}(T))^{s/d}=(\det T)^{s/d}$ for $s>d$. It is clear that $\phi^{s}(T)$ is continuous and strictly decreasing in $s$. The singular value function is submultiplicative, that is, for all $s\geq 0$,

(2.15)

$$\phi^{s}(TU)\leq\phi^{s}(T)\phi^{s}(U),$$

for all $T,U\in\mathcal{L}(\mathbb{R}^{d},\mathbb{R}^{d})$, see  [5] for details.

Let $\{\Xi_{k}\}_{k=1}^{\infty}$ be given by (2.13). We write

(2.16)		$\displaystyle\alpha_{+}$	$\displaystyle=$	$\displaystyle\sup\{\alpha_{1}(T_{k,i}):1\leq i\leq n_{k}\textit{, }k\in\mathbb{N}\},$
	$\displaystyle\alpha_{-}$	$\displaystyle=$	$\displaystyle\inf\{\alpha_{d}(T_{k,i}):1\leq i\leq n_{k}\textit{, }k\in\mathbb{N}\}.$

Immediately, for each $\mathbf{u}\in\Sigma^{k}$, the singular value function $\phi^{s}$ of $T_{\mathbf{u}}$ is bounded by

(2.17)

$$\alpha_{-}^{sk}\leq\phi^{s}(T_{\mathbf{u}})\leq\alpha_{+}^{sk}.$$

Note that for a self-affine set $E$, the sequence $\{\sum_{\mathbf{u}\in\mathcal{I}^{k}}\phi^{s}(T_{\mathbf{u}})\}_{k=1}^{\infty}$ is submultiplicative, and this implies that the function

$$p(s)=\lim_{k\to\infty}\left(\sum_{\mathbf{u}\in\mathcal{I}^{k}}\phi^{s}(T_{\mathbf{u}})\right)^{\frac{1}{k}}$$

is continuous and strictly decreasing in $s$. This property plays an important role in finding the affine dimensions of self-affine fractals. However, in general, the submultiplicative property does not hold for non-autonomous affine sets. The lack of submultiplicativity causes one of the main difficulties to determine the dimensions of non-autonomous affine sets.

Given $\{n_{k}\geq 2\}_{k=1}^{\infty}$. Let $\{\Xi_{k}\}_{k=1}^{\infty}$ be the NAIFS (2.13) and $E$ be the corresponding non-autonomous affine set defined by(2.9). From now on, we always assume that the matrices in $\Xi_{k}$ for all $k\in\mathbb{N}$ are nonsingular, and

(2.18)

$$0<\alpha_{-}\leq\alpha_{+}<1.$$

For each $s>0$ and $0<\epsilon<1$, let $m$ be the integer such that $m-1<s\leq m$ and define

$$\Sigma^{*}(s,\epsilon)=\{\mathbf{u}=u_{1}\ldots u_{k}\in\Sigma^{*}:\alpha_{m}(T_{\mathbf{u}})\leq\epsilon<\alpha_{m}(T_{\mathbf{u}^{-}})\}.$$

The set $\Sigma^{*}(s,\epsilon)$ is a cut-set or stopping in the sense that for every $\mathbf{u}\in\Sigma^{\infty}$, there is a unique integer $k$ such that $\mathbf{u}|k\in\Sigma^{*}(s,\epsilon)$. For each $\mathbf{u}\in\Sigma^{*}(s,\epsilon)$, by (2.15) and (2.16), we have that

$$\alpha_{-}\epsilon<\alpha_{m}(T_{\mathbf{u}})<\epsilon.$$

We define

(2.19)

$$s^{*}=\inf\Big{\{}s:\limsup_{\epsilon\to 0}\sum_{\mathbf{u}\in\Sigma^{*}(s,\epsilon)}\phi^{s}(T_{\mathbf{u}})<\infty\Big{\}}.$$

The critical value $s^{*}$ plays a key role in the box dimensions of non-autonomous affine sets. The following theorem shows that $s^{*}$ is an upper bound for the upper box dimension of $E$.

Suppose that the matrices in $\Xi_{k}$ for all $k>0$ are scalar matrices, that is $T_{k,j}=\mathrm{diag}\{\alpha_{k,j},\ldots,\alpha_{k,j}\}$ is a $d\times d$ scalar matrix for each $1\leq j\leq n_{k}$ and each $k>0$. Suppose that $E$ satisfies the MSC. Then $E$ is a Moran set, and $s^{*}$ gives the upper box dimension.

Next we show that under certain strong restrictions, the critical value $s^{*}$ gives the upper box dimension of $E$. We say the non-autonomous affine set $E$ satisfies the open projection condition(OPC) if there exists an open set $U$ such that $J\subset\overline{U}$ and for each $k>0$ and $\mathbf{u}\in\Sigma^{k-1}$,

$$U\supset\bigcup_{i=1,2,\ldots,n_{k}}(T_{k,i}+\omega_{\mathbf{u},i})(U),$$

with the union disjoint, and

$$\mathcal{L}^{d-1}\{\mathrm{proj}_{\Theta}U\}=\mathcal{L}^{d-1}\{\mathrm{proj}_{\Theta}\overline{U}\},$$

for all $(d-1)$-dimensional subspaces $\Theta$.

For self-affine fractals, the box-counting dimensions and Hausdorff dimensions are bounded by the same value $d(T_{1},\ldots,T_{M})$. Unfortunately, $s^{*}$ does not provide much information for the Hausdorff dimensions of non-autonomous affine sets, and we have to find a different candidate for Hausdorff dimensions.

Fix $s\geq 0$. By applying “Method II” in Rogers [26], we define a Hausdorff type measure on $\Sigma^{\infty}$ as follows. For each integer $k>0$, let

$$\mathcal{M}_{(k)}^{s}(G)=\inf\Big{\{}\sum_{\mathbf{u}}\phi^{s}(T_{\mathbf{u}}):G\subset\bigcup_{\mathbf{u}}\mathcal{C}_{\mathbf{u}},|\mathbf{u}|\geq k\Big{\}}.$$

We obtain a net measure of Hausdorff type by letting

(2.20)

$$\mathcal{M}^{s}(G)=\lim_{k\to\infty}\mathcal{M}_{(k)}^{s}(G),$$

for all $G\subset\Sigma^{\infty}$. Note that $\mathcal{M}^{s}$ is an outer measure which restricts to a measure on the Borel subsets of $\Sigma^{\infty}$.

We define

(2.21)

$$s_{A}=\inf\{s:\mathcal{M}^{s}(\Sigma^{\infty})=0\}=\sup\{s:\mathcal{M}^{s}(\Sigma^{\infty})=\infty\}.$$

The critical value $s_{A}$ is important in studying the Hausdorff dimensions for non-autonomous affine sets. The following theorem shows that $s_{A}$ is an upper bound for the Hausdorff dimension of $E$.

In section 7, we discuss the relations of $s^{*}$, $s_{A}$ and $d(T_{1},\ldots,T_{M})$, and we also give some examples to show that $s_{A}$ and $s^{*}$ are sharp bounds for Hausdorff dimensions and upper box dimensions of non-autonomous affine sets, respectively, see Example 4 in section 7.

Given the discontinuity of dimensions of $E^{\omega}$ in the translations $\omega$, we can only expect to show that $s_{A}$ is also a lower bound for Hausdorff dimensions for almost all constructions, in some sense.

First, we consider a special case where the translations of affine mappings in the non-autonomous structure are selected only from a finite set. Let $\Gamma=\{a_{1},\ldots,a_{\tau}\}$ be a finite collection of translations, where $a_{1},\ldots,a_{\tau}$ are regarded later as variables in $\mathbb{R}^{d}$. For each $\mathbf{u}=u_{1}\ldots u_{k}\in\Sigma^{*}$, we have that $J_{\mathbf{u}}=\Psi_{\mathbf{u}}(J)=\Psi_{u_{1}}\circ\ldots\circ\Psi_{u_{k}}(J)$. Suppose that the translation of $\Psi_{u_{j}}$ is an element of $\Gamma$, that is,

$$\Psi_{u_{j}}(x)=T_{j,u_{j}}x+\omega_{u_{1}\ldots u_{j}},\qquad\omega_{u_{1}\ldots u_{j}}\in\Gamma,$$

for $j=1,2,\ldots,k$. We write $\mathbf{a}=(a_{1},\ldots,a_{\tau})$ as a variable in $\mathbb{R}^{\tau d}$. To emphasize the dependence on these special translations in $\Gamma$, we denote the non-autonomous affine set by $E^{\mathbf{a}}$. The following conclusion shows that the Hausdorff dimension of $E^{\mathbf{a}}$ equals $s_{A}$ almost surely.

As you can see the set $E^{\mathbf{a}}$ is special and unnatural, the reason is that there is no obvious candidate which would take the place of the Lebesgue measure in infinite dimensional spaces. Inspired by [8, 20], we study the Hausdorff dimensions of non-autonomous affine sets in probabilistic language.

Let $\mathcal{D}$ be a bounded region in $\mathbb{R}^{d}$. For each $\mathbf{u}\in\Sigma^{*}$, let $\omega_{\mathbf{u}}\in\mathcal{D}$ be a random vector distributed according to some Borel probability measure $P_{\mathbf{u}}$ that is absolutely continuous with respect to $d$-dimensional Lebesgue measure. We assume that the $\omega_{\mathbf{u}}$ are independent identically distributed random vectors. Let $\mathbf{P}$ denote the product probability measure $\mathbf{P}=\prod_{\mathbf{u}\in\Sigma^{*}}P_{\mathbf{u}}$ on the family $\omega=\{\omega_{\mathbf{u}}:\mathbf{u}\in\Sigma^{*}\}.$ In this context, for each $\mathbf{u}=u_{1}\ldots u_{k}\in\Sigma^{*}$, we assume that the translation of $\Psi_{u_{j}}$ is an element of $\omega$, that is,

$$\Psi_{u_{j}}(x)=T_{j,u_{j}}x+\omega_{u_{1}\ldots u_{j}},\qquad\omega_{u_{1}\ldots u_{j}}\in\omega=\{\omega_{\mathbf{u}}:\mathbf{u}\in\Sigma^{*}\},$$

for $j=1,2,\ldots,k$. We also assume that the collection of $J_{\mathbf{u}}=\Psi_{\mathbf{u}}(J)=\Psi_{u_{1}}\circ\ldots\circ\Psi_{u_{k}}(J)$ fulfils the non-autonomous structure, and we call $E^{\omega}$ the non-autonomous affine set with random translations.

Next theorem states that, in this probabilistic setting, the Hausdorff dimension of $E^{\omega}$ equals $s_{A}$ with probability one.

Finally, we explore the lower box-counting dimension of non-autonomous affine sets. Under open projection condition, we show that $s_{A}$ is a lower bound for the lower box dimension.