2.1 Rarely switching systems

The first main results of this paper describe the Hausdorff and upper box-counting dimensions in the case where the sequence $\overline {c}$ does not switch too often between values. Namely, let $\kappa _n$ be the number of switches up to the time n:

$$ \begin{align*} \kappa_n:=\# \{ m<n \mid c_m \neq c_{m+1} \}. \end{align*} $$

We will say that the sequence $\overline {c}$ switches rarely if $\kappa _n=o(n)$ .

Also, given the sequence $\overline {c}$ , we consider the number of times $N_{n,j}$ when each index j occurs in the first n letters of the sequence, and the corresponding frequencies:

$$ \begin{align*} \mathrm{Freq}_{n,j}:=\frac{{N_{n,j}}}{n} \quad {\text{where} \ N_{n,j}:=\#\{m\le n\mid c_m=j\}}; \end{align*} $$

we say that the sequence $\overline {c}$ satisfies the frequencies condition if

(2.1) $$ \begin{align} \text{for all } j=1,\ldots,k{,} \quad \limsup_{n\to\infty} \, \mathrm{Freq}_{n,j}=1, \end{align} $$

where k is the size of the set of systems $\mathcal {F}$ .

Example 2.1. The sequences $\overline {c}$ that satisfy both rarely switching and frequencies assumptions exist. For instance, the sequence

$$ \begin{align*} \overline{c}= \underbrace{00\cdots 0}_{l_1\text{ times}}\, \underbrace{11\cdots 1}_{l_2\text{ times}}\, \underbrace{00\cdots 0}_{l_3\text{ times}}\, \underbrace{11\cdots 1}_{l_4\text{ times}}\cdots \end{align*} $$

satisfies both assumptions once ${l_{j+1}}/{l_j} \to \infty $ as $j\to \infty $ .

Theorem 2.2. Let $\mathcal {F}=\{F_1,\ldots ,F_k\}$ be a system of $C^{1+\tau }$ expanding cookie-cutter maps, where $\tau>0$ . Assume that the sequence $\overline {c}=(c_n)$ is rarely switching, $\kappa _n=o(n)$ , and satisfies the frequencies condition (2.1). Then, the Hausdorff and upper box-counting dimensions of the associated non-stationary cookie-cutter-like set $J_{\overline {c}}=J(\mathcal {F};\overline {c})$ are given by

(2.2) $$ \begin{align} \dim_H J_{\overline{c}} = \min_{j=1,\ldots,k} \dim J(F_j), \end{align} $$

(2.3) $$ \begin{align} \mathop{\overline{\dim}_B}J_{\overline{c}}= \max_{j=1,\ldots,k} \dim J(F_j). \end{align} $$

A particular case of cookie-cutter maps are ‘piecewise-affine’ maps F, such that the associated IFS consists of affine contracting maps $f_i$ . Cookie-cutter-like sets, associated to families of such maps, and their dimensions have already been previously studied, see [Reference Hua, Rao, Wen and WuHRWW2000, Reference Feng, Rao and WuFRW1997]. Consideration of such sets allows to illustrate also Theorem 2.2, providing an intuition for its conclusions.

Example 2.3. For $\unicode{x3bb} \in (0,\tfrac 12)$ , let $\Phi _{\unicode{x3bb} }$ be a piecewise-affine cookie-cutter, defined on ${I=[0,1]}$ by

$$ \begin{align*} \Phi_{\unicode{x3bb}}(x)=\begin{cases} \dfrac{1}{\unicode{x3bb}} x, & x\in [0,\unicode{x3bb}],\\ 1-\dfrac{1}{\unicode{x3bb}} (1-x),& x\in [1-\unicode{x3bb},1]; \end{cases} \end{align*} $$

see Figure 3(a). Such a map defines a Cantor set $J(\Phi _{\unicode{x3bb} })$ of dimension $\dim _H J(\Phi _{\unicode{x3bb} }) = {\log 2}/{\log \unicode{x3bb} ^{-1}}$ .

Figure 3

(a) Piecewise-affine cookie cutter $\Phi _{\unicode{x3bb} }$ . (b) Graphs of the dimensions for $F_{0}^{\mathbf {a}},F_{1}^{\mathbf {a}}$ from Example 2.8.

Now, take $\mathcal {F}=\{\Phi _{\unicode{x3bb} _1},\ldots ,\Phi _{\unicode{x3bb} _k}\}$ and any sequence $\overline {c}=(c_n)$ of elements of $\{1,\ldots ,k\}$ . Then, the sets $K_{c_1\cdots c_n}$ , defined by (1.3), consist of $2^n$ intervals of length $L_n:=\prod _{m=1}^n \unicode{x3bb} _{c_m}$ each. On the scale $L_n$ in the definition of the box counting dimension, one thus has

(2.4) $$ \begin{align} s_n&:=\frac{\log 2^n}{\log (L_n)^{-1}} = \frac{n \log 2}{ \sum_{j=1}^k N_{n,j} \log \unicode{x3bb}_j^{-1}}\nonumber\\& = \bigg( \sum_{j=1}^k \frac{N_{n,j}}{n} \frac{\log \unicode{x3bb}_j^{-1}}{\log 2} \bigg)^{-1} = \bigg( \sum_{j=1}^k \frac{\mathrm{Freq}_{n,j}}{\dim_H J(\Phi_{\unicode{x3bb}_j})} \bigg)^{-1} \end{align} $$

and the lower and the upper box counting dimensions are the lower and the upper limits of the sequence $s_n$ , given by (2.4). Inside the parentheses in the right-hand side of (2.4), one has a linear combination of values ${1}/{\dim _H J(\Phi _{\unicode{x3bb} _j})}$ with positive coefficients $\mathrm {Freq}_{n,j}$ , whose sum is equal to $1$ for any n. This value thus lies between the minimum and maximum of these inverse dimensions, and assuming that the sequence $\overline {c}$ satisfies the frequencies condition (2.1), accumulates to both these maximum and minimum values. Passing to the upper and lower limit in (2.4) thus gives

$$ \begin{align*} \mathop{\overline{\dim}_B} J(\mathcal{F},\overline{c}) = \max_{j=1,\ldots,k} \dim J(\Phi_{\unicode{x3bb}_j}); \quad \mathop{\underline{\dim}_B} K= \min_{j=1,\ldots,k} \dim J(\Phi_{\unicode{x3bb}_j}), \end{align*} $$

the first value being exactly one of the conclusions of Theorem 2.2. The lower box counting dimension is also equal to $\dim _H J(\Phi _{\unicode{x3bb} _j})$ , but this requires some more arguments to show (one has to check that there is no more efficient way to find a cover of the Cantor set); see [Reference Feng, Rao and WuFRW1997, Theorem 2] for the formal proof.

Remark 2.4. In the particular case where all the maps forming each of the cookie-cutters $F_j$ are affine, the rarely switching condition $\kappa _n=o(n)$ can be omitted. The reason for that is that, in this case, the log-partition functions become exactly additive, see Remark 5.3. Hence, the conclusion of the key Proposition 5.1 holds without the rare switching assumption; Proposition 5.1 is the only place in the proofs of the main results where this assumption is used.

However, the rare switching assumption cannot be omitted in general, as we will see from Example 6.2.

Finally, note that the applicability of Theorem 2.2 can be sometimes extended by grouping letters into groups. For instance, the sequence

$$ \begin{align*} \overline{c} = (00)^{l_1} (11)^{l_2} (01)^{l_3} \ldots (00)^{l_{3m+1}} (11)^{l_{3m+2}} (01)^{l_{3m+3}} \ldots \end{align*} $$

(where the powers represent the number of repetitions and the products represent the concatenation of words), used in Example 6.2, does not satisfy the rarely switching assumption, but it does satisfy it once we group letters into pairs. Finally, as we will see at the end of §5 (see Proposition 5.4 therein), the frequencies condition can be omitted, at the cost of conclusions (2.2) and (2.3) becoming more complicated, involving lower and upper limits, respectively.

2.2 Implications for parameter-dependent systems

Consider now families of cookie-cutter maps/IFSs that depend on a parameter $\mathbf {a}$ :

$$ \begin{align*} \mathcal{F}^{\mathbf{a}}=\{F^{\mathbf{a}}_1,\ldots,F^{\mathbf{a}}_k\}. \end{align*} $$

From Ruelle [Reference RuelleR1982], we know that the dimension of cookie-cutter sets of analytic cookie-cutter maps, depending analytically on a parameter, depend analytically on the same parameter.

However, even though such an analytic dependence takes place for individual (stationary) cookie-cutter sets, we see that it might not take place for the associated (non-stationary) cookie-cutter-like sets. Namely, applying Theorem 2.2 individually at each parameter $\mathbf {a}$ , we obtain the following corollary.

Corollary 2.6. Assume that a family $\mathcal {F}^{\mathbf {a}}$ of parameter-dependent cookie-cutter sets satisfies the assumptions of Theorem 2.2 for each parameter $\mathbf {a}$ , and that the sequence $\overline {c}=(c_n)$ satisfies the rarely switching condition $\kappa _n=o(n)$ , as well as the frequencies condition (2.1). Then, for the cookie-cutter-like sets $J^{\mathbf {a}}_{\overline {c}}:=J(\mathcal {F}^{\mathbf {a}};\overline {c})$ , one has

(2.5) $$ \begin{align} \dim_H J^{\mathbf{a}}_{\overline{c}}= \min_{j=1,\ldots,k}\dim J(F^{\mathbf{a}}_j), \end{align} $$

(2.6) $$ \begin{align} \mathop{\overline{\dim}_B} J^{\mathbf{a}}_{\overline{c}} = \max_{j=1,\ldots,k}\dim J(F^{\mathbf{a}}_j). \end{align} $$

Now, the minimum and maximum of two analytic functions lose differentiability at any point where their graphs have a transverse intersection (see Figure 4). We thus get an example of non-differentiable dependence of the Hausdorff and upper-box counting dimensions on the parameter.

Figure 4

Intersecting graphs of the dimensions $\dim _H(J(F_j^{\mathbf {a}}))$ , and the corresponding non-differentiable graphs of $\dim _H J^{\mathbf {a}}_{\overline {c}}$ and $\mathop {\overline {\dim }_B} J^{\mathbf {a}}_{\overline {c}}$ .

Figure 4 Long description

Top: two transversally intersecting graphs of the dimensions of two different parameter-dependent cookie-cutters. Bottom: graphs of Hausdorff (left) and upper box-counting (right) dimensions for the corresponding cookie-cutter-like sets. These dimensions are equal to the minimum and maximum of the initial graphs, both having a break-type singularity at their intersection.

In the same way as for Theorem 2.2, it can be illustrated by consideration of piecewise-affine maps. Namely, one has the following example.

Example 2.8. For $\mathbf {a}\in [\tfrac 14,\tfrac 25]$ , take $\unicode{x3bb} _0(\mathbf {a})=\mathbf {a}$ and $\unicode{x3bb} _1(\mathbf {a})=\tfrac 13$ , and take $\mathcal {F}^{\mathbf {a}}=\{F_{0}^{\mathbf {a}},F_1^{\mathbf {a}}\}=\{\Phi _{\unicode{x3bb} _0(\mathbf {a})},\Phi _{\unicode{x3bb} _1(\mathbf {a})}\},$ where $\Phi _{\unicode{x3bb} }$ is the map from Example 2.3. Take the sequence $\overline {c}=(c_n)$ , given by Example 2.1, for instance, for the lengths $l_j=j!$ , so that the frequencies condition holds. Then, one has

$$ \begin{align*} \dim_H J^{\mathbf{a}}_{\overline{c}}= \min (\log_{1/\mathbf{a}} 2, \log_{3} 2), \quad \mathop{\overline{\dim}_B} = \max (\log_{1/\mathbf{a}} 2, \log_{3} 2), \end{align*} $$

in particular, both functions $\dim _H J$ and $\mathop {\overline {\dim }_B} J$ have a break-type singularity at $\mathbf {a}=\tfrac 13$ , where the graphs of $\dim J(F_{i}^{\mathbf {a}})=\dim J(\Phi _{\unicode{x3bb} _i(\mathbf {a})})$ for $i=1,2$ intersect; see Figure 3(b).

4.1 Cookie-cutter sets

For a single cookie-cutter map F, for which the corresponding maps $f_i$ are affine,

$$ \begin{align*} f_j(x) = a_j x + b_j, \quad j=1,\ldots,q, \end{align*} $$

the dimension of the corresponding cookie-cutter set $J(F)$ is given by the so-called Moran formula (see [Reference MoranMo1946, Theorem II]): one has $\dim _H J(F) = s$ , where s is the solution to the equation

(4.1) $$ \begin{align} \sum_{j=1}^q |a_j|^s =1. \end{align} $$

This formula can be seen as an equality between the s-dimensional volume of the set $J(F)$ and the sum of volumes of its parts $f_j(J(F))$ .

When the maps $f_i$ are no longer affine, but only contracting $C^{1+\tau }$ -diffeomorphisms, the dimension of the set $J(F)$ is given by the Bowen formula. Namely, for every n and every word

$$ \begin{align*} w=w_1\cdots w_n\in \Omega_n:= \{1,\ldots,q\}^n, \end{align*} $$

one considers the interval

$$ \begin{align*} I_w:=f_{w_1}\circ \cdots \circ f_{w_n}(I). \end{align*} $$

The partition function of n iterates is then defined as

$$ \begin{align*} Z_{n,F}(s) := \sum_{w\in\Omega_n} |I_w|^s, \end{align*} $$

and the associated n-iterates pressure function as

$$ \begin{align*} P_{n,F}(s):=\frac{1}{n} \log Z_{n{,F}}(s). \end{align*} $$

This sequence of functions converges to a continuous decreasing stationary pressure function,

$$ \begin{align*} P_F(s):=\lim_{n\to\infty} P_{n,F}(s). \end{align*} $$

The Bowen formula then states (see [Reference BowenB1980, Lemma 10]) that the dimension s (both Hausdorff and box-counting) of the cookie-cutter set $J(F)$ is given by the solution of the equation $P_F(s)=0$ .

Note that for the case of a cookie-cutter map F with affine maps $f_i$ , one has ${Z_{n,F}=Z_{1,F}^n}$ and, thus, the pressure function $P_F(s)$ is simply the logarithm of the left-hand side of Moran’s formula (4.1).

4.2 Thermodynamic formalism for cookie-cutter-like sets

Assume that we are given a finite set $\mathcal {F}= \{F_1,\ldots ,F_k\}$ of $C^{1+\tau }$ -cookie-cutter maps and a sequence $\overline {c}=(c_n)_{n\in \mathbb {N}}$ , taking values in $\{1,\ldots ,k\}$ . Consider the cookie-cutter-like set $J(\mathcal {F};\overline {c})$ , defined by (1.2); its Hausdorff and box-counting dimensions are then determined (see [Reference Ma, Rao and WenMRW2001, Reference Mauldin and UrbańskiMU2003, Reference Rempe-Gillen and UrbańskiRU2016]) via a generalization of the Bowen formula.

Namely, let $(f_{j,1},\ldots , f_{j,q_j})$ be the IFS corresponding to the cookie-cutter map $F_j$ . For any sequence $c_1,\ldots , c_n\in \{1,\ldots ,k\}$ define

$$ \begin{align*} \Omega_{c_1\cdots c_n} := \{w=w_1\cdots w_n \mid \text{for all } j=1,\ldots,n \ w_j \in \{1,\ldots,q_{c_j}\} \} \end{align*} $$

and let

$$ \begin{align*} I_{w_1\cdots w_n}^{c_1\cdots c_n}:= f_{c_1,w_1}\circ \cdots \circ f_{c_n,w_n}(I), \quad w_1\cdots w_n\in \Omega_{c_1\cdots c_n}, \end{align*} $$

be the intervals that are connected components of the associated set $K_{c_1\cdots c_n}$ (see (1.3)).

One then defines [Reference Ma, Rao and WenMRW2001, Reference Mauldin and UrbańskiMU2003, Reference Rempe-Gillen and UrbańskiRU2016] non-stationary versions of the (n-iterations) partition and pressure functions by

(4.2) $$ \begin{align} Z_{c_1\cdots c_n}(s)& :=\sum_{w_1\cdots w_n\in \Omega_{c_1\cdots c_n}} |I_{w_1\cdots w_n}^{c_1\cdots c_n}|^s, \nonumber\\P_{c_1\cdots c_n}(s)& :=\frac{1}{n} \log Z_{c_1\cdots c_n}(s). \end{align} $$

Note that in the non-stationary setting, the sequence of the pressure functions $P_{c_1\cdots c_n}$ a priori does not converge, and passing to the upper or lower limit would lead to the loss of some information. Thus, we are working with the finite-iterates non-stationary pressure functions.

In the same way as for stationary cookie-cutters, as there is a finite number of contracting diffeomorphisms $f_{i,j}$ , there exist constants $1<\unicode{x3bb} <L$ , such that for all $(i,j)$ , one has

$$ \begin{align*} \text{for all } x\in I, \quad L^{-1}<Df_{i,j}(x)<\unicode{x3bb}^{-1}. \end{align*} $$

This implies the uniform exponential bounds for the intervals

$$ \begin{align*} L^{-n} < |I_{w_1\cdots w_n}^{c_1\cdots c_n}| < \unicode{x3bb}^{-n}; \end{align*} $$

hence, each summand in the definition (4.2) of the partition function satisfies the estimates

(4.3) $$ \begin{align} \text{for all } r>0, \text{ for all } s, \quad |I_{w_1\cdots w_n}^{c_1\cdots c_n}|^{s} L^{-nr} < |I_{w_1\cdots w_n}^{c_1\cdots c_n}|^{s+r} < |I_{w_1\cdots w_n}^{c_1\cdots c_n}|^{s} \unicode{x3bb}^{-nr}; \end{align} $$

adding them up and taking the logarithm, one gets the uniform decrease and continuity estimates

(4.4) $$ \begin{align} P_{{c_1,\ldots,c_n}}(s)-(\log L)\cdot r <P_{{c_1,\ldots,c_n}}(s+r)< P_{{c_1,\ldots,c_n}}(s)-(\log \unicode{x3bb})\cdot r. \end{align} $$

In particular, there exists a unique zero $s_{c_1\cdots c_n}$ of the function $P_{c_1\cdots c_n}(s)$ . It is easy to see that it belongs to $[0,1]$ : it suffices to substitute $s=0$ and $s=1$ in (4.2) to get

$$ \begin{align*} {P_{c_1\cdots c_n}(0)=\log Z_{c_1\cdots c_n}(0)>0, \quad P_{c_1\cdots c_n}(1)=\log Z_{c_1\cdots c_n}(1)<0.} \end{align*} $$

Now, let

(4.5) $$ \begin{align} s_*:=\liminf_{n\to\infty} s_{c_1\cdots c_n} \quad \text{and} \quad s^*:=\limsup_{n\to\infty} s_{c_1\cdots c_n}. \end{align} $$

Then, the dimensions of the cookie-cutter-like set $J_{\overline {c}}=J(\mathcal {F};(c_n))$ are given by the following theorem.

Theorem 4.1. [Reference Ma, Rao and WenMRW2001, Theorem 1.3]

In the notation above,

(4.6) $$ \begin{align} \dim_H J_{\overline{c}}=s_* \quad \text{and} \quad \mathop{\overline{\dim}_B} J_{\overline{c}}=s^*. \end{align} $$

5.1 Controlling additivity: main proposition

Given a sequence $\overline {c}=(c_n)$ , for every n, let us consider a linear combination of the stationary pressure functions $P_{F_j}(s)$ , associated to individual maps $F_j$ : denote

(5.1) $$ \begin{align} \widetilde{P}_{c_1\cdots c_n}:=\sum_{j=1}^k \mathrm{Freq}_{n,j} \cdot P_{F_j}(s). \end{align} $$

The following key proposition states that in ‘rare switching’ mode, the non-stationary pressure function for a large number n of iterations becomes increasingly better approximated by this linear combination.

Proposition 5.1. Assume that the sequence $(c_n)$ satisfies the rare switching condition $\kappa _n=o(n)$ . Then, the difference

(5.2) $$ \begin{align} P_{c_1\cdots c_n}(s)- {\widetilde{P}_{c_1\cdots c_n}(s)} \end{align} $$

converges to zero as $n\to \infty $ and, for any compact interval $[a,b]\subset \mathbb {R}$ , this convergence is uniform in $s\in [a,b]$ .

This proposition follows from the quasi-additivity of the log-partition functions.

Lemma 5.2. There exists a constant $C_{\varkappa }$ such that for any n, any $s\in \mathbb {R}$ , and any $c_{1},\ldots , c_{n+m}$ , one has

(5.3) $$ \begin{align} |\!\log Z_{c_1\cdots c_{n+m}}(s)-(\log Z_{c_1\cdots c_n}(s)+\log Z_{c_{n+1}\cdots c_{n+m}}(s))|\le C_{\varkappa} \cdot |s|. \end{align} $$

The proof of Lemma 5.2 is based on the control of distortion; we postpone it until §5.3, where we introduce the distortion control technique. Finally, we note that as we will see from this proof, one can take the constant $C_{\varkappa }$ to be the constant from Lemma 5.5.

Proof of Proposition 5.1.

Given any sequence $c_1,\ldots ,c_n$ , cut it into consecutive groups of identical letters: let

$$ \begin{align*} c_1\cdots c_n = (j_1)^{l_1} \cdots (j_{\kappa_n+1})^{l_{\kappa_n+1}}, \end{align*} $$

where $j_1,\ldots ,j_{\kappa _n+1}\in \{1,\ldots ,k\}$ are symbols and powers signify number of repetitions. Applying consecutively Lemma 5.2, we get

(5.4) $$ \begin{align} \bigg|\!\log Z_{c_1\cdots c_n}(s) - \sum_i \log Z_{(j_i)^{l_i}}(s) \bigg| \le \kappa_n \cdot C_{\varkappa}|s|. \end{align} $$

Recall that $N_{n,j}$ is the number of occurrences of the letter j among the first n letters of $\overline {c}$ :

$$ \begin{align*} {N_{n,j}}=\#\{m\le n \mid c_m=j\} = \sum_{i: \, j_i=j} l_i. \end{align*} $$

Let us group the identical letters together:

(5.5) $$ \begin{align} \bigg|\!\sum_i \log Z_{(j_i)^{l_i}}(s) - \sum_{j=1}^k \log Z_{(j)^{{N_{n,j}}}}(s) \bigg| \le \kappa_n \cdot C_{\varkappa}|s|. \end{align} $$

Finally, note that for every j, l, ${n'}$ , one has

$$ \begin{align*} | {n'}\log Z_{(j)^l} (s) - \log Z_{(j)^{{n'}l}} (s) | \le {n'}\cdot C_{\varkappa}|s|; \end{align*} $$

dividing by ${n'}$ and passing to the limit as ${n'}\to \infty $ imply that

$$ \begin{align*} |\!\log Z_{(j)^l} (s) - l \cdot P_{F_j}(s) | \le C_{\varkappa}|s|. \end{align*} $$

Joining this with (5.4) and (5.5), we see that

(5.6) $$ \begin{align} \bigg|\!\log Z_{c_1\cdots c_n}(s) - \sum_{j=1}^k {N_{n,j}} P_{F_j}(s) \bigg| \le (2\kappa_n+k) \cdot C_{\varkappa}|s|. \end{align} $$

Dividing by n and using the rarely switching assumption $\kappa _n=o(n)$ completes the proof.

5.2 Estimating the dimension

Proof of Theorem 2.2.

We are going to use the result from [Reference Ma, Rao and WenMRW2001], Theorem 4.1, cited earlier. Namely, the Hausdorff and upper box-counting dimensions are respectively the lower and upper limits of the zeros $s_{c_1\cdots c_n}$ of the pressure functions $P_{c_1\cdots c_n}(s)$ (see (4.6)).

Now, note that due to the uniform lower bound (4.4) for the slope of $P_{c_1\cdots c_n}(s)$ , for every $s', s"$ , we have

$$ \begin{align*} |P_{c_1\cdots c_n}(s') - P_{c_1\cdots c_n}(s")| \ge \log \unicode{x3bb} \cdot |s' - s"| \end{align*} $$

and, hence,

(5.7) $$ \begin{align} |s_{c_1\cdots c_n} - s'| \le \frac{1}{\log \unicode{x3bb}} |P_{c_1\cdots c_n}(s')|. \end{align} $$

Now, for every n, let $s^{\prime }_{c_1\cdots c_n}$ be the root of the linear combination $\widetilde {P}_{c_1\cdots c_n}$ of the stationary pressure functions, given by (5.1):

$$ \begin{align*} \widetilde{P}_{c_1\cdots c_n}(s^{\prime}_{c_1\cdots c_n})=0. \end{align*} $$

Then, Proposition 5.1 implies that the value $P_{c_1\cdots c_n}(s^{\prime }_{c_1\cdots c_n})$ tends to zero; hence (due to the estimate (5.7)), the same applies to the difference $|s_{c_1\cdots c_n}-s^{\prime }_{c_1\cdots c_n}|$ ; see Figure 5.

Figure 5

The graphs of non-stationary pressure function $P_{c_1\cdots c_n}$ and its linear combination approximation $\widetilde {P}_{c_1\cdots c_n}$ . The difference $|s_{c_1\cdots c_n}-s^{\prime }_{c_1\cdots c_n}|$ (shown in bold) is estimated via the value $P_{c_1\cdots c_n}(s^{\prime }_{c_1\cdots c_n})$ and the slope estimate (5.7).

Figure 5 Long description

The graphs of two decreasing functions, the nonstationary pressure function and its linear combination approximation, as well as the zeros s c 1 dots c n and s' c 1 dots c n are marked. Both zeros belong to the interval between the minimum and the maximum of the dimensions dim J(F j) of cookie-cutter-sets J(F j). The difference between the roots of these functions is compared to the value of the pressure function P c 1 dots c n at the root s' c 1 dots c n of the approximation.

Applying Theorem 4.1 immediately leads to

(5.8) $$ \begin{align} \dim_H J_{\overline{c}} = \liminf_{n\to\infty} s^{\prime}_{c_1\cdots c_n}, \quad \mathop{\overline{\dim}_B} J_{\overline{c}} = \limsup_{n\to\infty} s^{\prime}_{c_1\cdots c_n}. \end{align} $$

Now, on one hand, due to the monotonicity of each of the pressure functions $P_{F_j}(s)$ , for every n, the root $s^{\prime }_{c_1\cdots c_n}$ lies between the minimum and the maximum of the roots of these functions, which are exactly the minimum and the maximum of the dimensions of the sets $J(F_j)$ . Hence,

(5.9) $$ \begin{align} \liminf_{n\to\infty} s^{\prime}_{c_1\cdots c_n} \ge \min_j \dim J(F_j), \quad \limsup_{n\to\infty} s^{\prime}_{c_1\cdots c_n} \le \max_j \dim J(F_j). \end{align} $$

On the other hand, denote $C_P:=\max _{s\in [0,1]} \max _j |P_{F_j}(s)|$ . If for some $n,j_0,\varepsilon $ we have $\mathrm {Freq}_{n,j_0}\ge 1-\varepsilon $ , then for $s'=\dim J(F_{j_0})$ , one has

$$ \begin{align*} {|\widetilde{P}_{c_1\cdots c_n}({s'})| =}\, \bigg|\!\sum_{j=1}^k \mathrm{Freq}_{n,j} \cdot P_{F_j}({s'})\bigg| \le \varepsilon\cdot \max_{j} |P_{F_j}(s')| {\, \le C_P \varepsilon.} \end{align*} $$

Now, inequality (4.4) (after passing to the limit and taking an average) implies the same slope estimate for $\widetilde {P}_{c_1\cdots c_n}$ , and thus,

$$ \begin{align*} |{s^{\prime}_{c_1\cdots c_n}} - s'| { \le \frac{1}{\log \unicode{x3bb}} |\widetilde{P}_{c_1\cdots c_n}({s'})|} \le \frac{{C_P}}{\log \unicode{x3bb}} \, \varepsilon. \end{align*} $$

The frequencies condition then immediately implies that the inequalities in (5.9) are actually equalities. Together with (5.8), this concludes the proof of Theorem 2.2.

We conclude this section by noticing that Theorem 2.2 can be stated in a way that does not require the frequency conditions. Namely, consider the simplex

$$ \begin{align*} \Delta:=\bigg\{(\alpha_1,\ldots,\alpha_k)\mid \text{for all } j\ \alpha_j\ge 0, \ \sum_j \alpha_j=1\bigg\} \end{align*} $$

and a map $\mathop {\mathrm {Root}}:\Delta \to [0,1]$ that associates to $(\alpha _1,\ldots ,\alpha _k)$ the root of the corresponding linear combination of pressure functions:

$$ \begin{align*} \mathop{\mathrm{Root}}(\alpha_1,\ldots,\alpha_k)=s: \quad \sum_j \alpha_j P_{F_j}(s)=0. \end{align*} $$

This map is continuous due to the same slope arguments as before, and we have

$$ \begin{align*} s^{\prime}_{c_1\cdots c_n} = \mathop{\mathrm{Root}}(\mathrm{Freq}_{n,1},\ldots,\mathrm{Freq}_{n,k}). \end{align*} $$

Hence, the following statement holds.

Proposition 5.4. Assume that all the assumptions of Theorem 2.2 hold, except possibly for the frequency assumptions. Then, the dimensions are given by the formulas

$$ \begin{align*} \dim_H J_{\overline{c}} & = \liminf_{n\to\infty} \, \mathop{\mathrm{Root}}(\mathrm{Freq}_{n,1},\ldots,\mathrm{Freq}_{n,k}),\\ \mathop{\overline{\dim}_B} J_{\overline{c}} & = \limsup_{n\to\infty} \, \mathop{\mathrm{Root}}(\mathrm{Freq}_{n,1},\ldots,\mathrm{Freq}_{n,k}). \end{align*} $$

In particular, for given cookie-cutter maps $F_1,\ldots , F_k$ , assuming that the sequence $(c_n)$ is rarely switching, the corresponding dimensions depend only on the set of accumulation points of the vector of frequencies in the simplex $\Delta $ .

5.3 Distortion control and quasi-additivity lemma

The proof of the quasi-additivity in Lemma 5.2 relies upon the distortion control techniques; let us briefly recall the necessary statements. Given a $C^1$ -diffeomorphism $f:I\to I'$ , one defines its distortion on I as

$$ \begin{align*} \varkappa(f;I):=\mathop{\mathrm{osc}}_{x\in I} \, \log |Df(x)|. \end{align*} $$

The distortion is composition-subadditive:

$$ \begin{align*} \varkappa(f_1\circ f_2;I) \le \varkappa(f_2;I) + \varkappa(f_1;f_2(I)). \end{align*} $$

At the same time, for a diffeomorphism f of class $C^{1+\tau }$ , its distortion on a subinterval $I'\subset I$ can be estimated as

$$ \begin{align*} \varkappa(f;I') \le C_f |I'|^{\tau} \end{align*} $$

for some constant $C_f$ . A standard corollary to these two facts is that the distortion of compositions of $C^{1+\tau }$ -contracting diffeomorphisms stays uniformly bounded. Namely, consider the compositions that occur in the non-stationary IFSs describing the cookie-cutter-like sets, given by

$$ \begin{align*} f_{w_1\cdots w_{n}}^{c_1\cdots c_{n}}:=f_{c_1,w_1}\circ \cdots \circ f_{c_n,w_n}. \end{align*} $$

We then have the following estimate.

Lemma 5.5. There exists a constant $C_{\varkappa }$ , such that for any $c_1,\ldots ,c_n$ and any word ${w\in \Omega _{c_1\cdots c_n}}$ , one has

(5.10) $$ \begin{align} \varkappa(f_{w_1\cdots w_{n}}^{c_1\cdots c_{n}}; I) \le C_{\varkappa}. \end{align} $$

Proof. Indeed, denoting $C_{\mathcal {F}}:=\max _{i,j} C_{f_{i,j}}$ , we see that due to the subadditivity of the distortion, the left-hand side of (5.10) does not exceed the sum of the distortions of the individual maps on the corresponding intervals. This sum can be estimated as

$$ \begin{align*} \sum_{m=0}^{n-1} \max_{i,j} (C_{f_{j,i}}) \cdot |I_{w_1\cdots w_m}|^{\tau} \le C_{\mathcal{F}} \sum_{m=0}^{\infty} \unicode{x3bb}^{-\tau m}= \frac{C_{\mathcal{F}}}{1-\unicode{x3bb}^{-\tau}}=:C_{\varkappa}, \end{align*} $$

where we have used $|I_{w_1\cdots w_m}|\le \unicode{x3bb} ^{-m}$ .

We are now ready to complete the estimates for the proof of Lemma 5.2.

Proof of Lemma 5.2.

We have

(5.11) $$ \begin{align} I_{w_1\cdots w_{n+m}}^{c_1\cdots c_{n+m}}=f_{w_1\cdots w_n}^{c_1\cdots c_{n}} (I_{w_{n+1}\cdots w_{n+m}}^{c_{n+1}\cdots c_{n+m}}), \quad I_{w_1\cdots w_{n}}^{c_1\cdots c_{n}} = f_{w_1\cdots w_n}^{c_1\cdots c_{n}} (I). \end{align} $$

Due to the mean value theorem, for a diffeomorphism $f:I'\to I"$ , one has $|I"|=|Df(x)| \cdot |I'|$ for some $x\in I'$ . Applying it to both images in (5.11), we get

$$ \begin{align*} |I_{w_1\cdots w_{n+m}}^{c_1\cdots c_{n+m}}| & = |Df_{w_1\cdots w_n}^{c_1\cdots c_{n}} (x)| \cdot |I_{w_{n+1}\cdots w_{n+m}}^{c_{n+1}\cdots c_{n+m}}|,\\ |I_{w_1\cdots w_{n}}^{c_1\cdots c_{n}}| & = |Df_{w_1\cdots w_n}^{c_1\cdots c_{n}} (y)| \cdot |I| \end{align*} $$

for some $x,y\in I$ . Now, the quotient of derivatives of $f_{w_1\cdots w_n}^{c_1\cdots c_{n}}$ at any two points $x,y\in I$ does not exceed $e^{C_{\varkappa }}$ due to Lemma 5.5, and as $|I|=1$ , we have

$$ \begin{align*} e^{-C_{\varkappa}} < \frac{|I_{w_1\cdots w_{n+m}}^{c_1\cdots c_{n+m}}|}{|I_{w_1\cdots w_{n}}^{c_1\cdots c_{n}}|\cdot |I_{w_{n+1}\cdots w_{n+m}}^{c_{n+1}\cdots c_{n+m}}|} < e^{C_{\varkappa}}. \end{align*} $$

Taking sth powers, multiplying by the denominator, and summing over all the words w, we get

$$ \begin{align*} e^{-sC_{\varkappa}} Z_{c_1\cdots c_n}(s) \cdot Z_{c_{n+1}\cdots c_{n+m}}(s) \le Z_{c_1\cdots c_{n+m}}(s) \le e^{sC_{\varkappa}} Z_{c_1\cdots c_n}(s) \cdot Z_{c_{n+1}\cdots c_{n+m}}(s) \end{align*} $$

for $s\ge 0$ , and the reversed inequalities for $s<0$ . An application of the logarithm completes the proof.