Polynomial Fourier decay and a cocycle version of Dolgopyat’s method for self conformal measures

Let $\nu$ be a Borel probability measure on $\mathbb{R}$. For every $q\in\mathbb{R}$ the Fourier transform of $\nu$ at $q$ is defined by

$$\mathcal{F}_{q}(\nu):=\int\exp(2\pi iqx)d\nu(x).$$

The measure $\nu$ is called a Rajchman measure if $\mathcal{F}_{q}(\nu)=o(1)$ as $|q|\rightarrow\infty$. By the Riemann-Lebesgue Lemma, if $\nu$ is absolutely continuous then it is Rajchman. On the other hand, by Wiener’s Lemma if $\nu$ has an atom then it is not Rajchman. For measures that are both continuous (no atoms) and singular, determining whether or not $\nu$ is a Rajchman measure may be a challenging problem even for well structured measures. The Rajchman property has various geometric consequences on the measure $\nu$ and its support, e.g. regarding the uniqueness problem [36].

Further information about the rate of decay of $\mathcal{F}_{q}(\nu)$ has even stronger geometric consequences. For example, by a classical Theorem of Davenport-Erdős-LeVeque [20], if $\mathcal{F}_{q}(\nu)$ decays at a logarithmic rate then $\nu$-a.e. point is normal to all integer bases (see also [40]). Wide ranging geometric information can be derived if $\mathcal{F}_{q}(\nu)$ decays at a polynomial rate, that is, if there exists some $\alpha>0$ such that

$$\mathcal{F}_{q}(\nu)=O\left(\frac{1}{|q|^{\alpha}}\right).$$

For example, by a result of Shmerkin [44], it implies that the convolution of $\nu$ with any measure of dimension $1$ is absolutely continuous. We refer to Mattila’s recent book [37] for various further applications of Fourier decay in geometric measure theory and related fields.

The goal of this paper is to prove that every measure in a fundamental class of fractal measures has polynomial Fourier decay, as long as it satisfies some very mild non-linearity conditions; Furthermore, these conditions can be easily verified in concrete examples. To define this class of measures, let $\Phi=\{f_{1},...,f_{n}\}$ be a finite set of strict contractions of a compact interval $I\subseteq\mathbb{R}$ (an IFS - Iterated Function System), such that every $f_{i}$ is differentiable. We say that $\Phi$ is $C^{\alpha}$ smooth if every $f_{i}$ is at least $C^{\alpha}$ smooth for some $\alpha\geq 1$. It is well known that there exists a unique compact set $\emptyset\neq K=K_{\Phi}\subseteq I$ such that

$$K=\bigcup_{i=1}^{n}f_{i}(K).$$

(1)

The set $K$ is called the attractor of the IFS $\{f_{1},...,f_{n}\}$. We always assume that there exist $i,j$ such that the fixed point of $f_{i}$ does not equal the fixed point of $f_{j}$. This ensures that $K$ is infinite. We call $\Phi$ uniformly contracting if

$$0<\inf\{|f^{\prime}(x)|:\,f\in\Phi,x\in I\}\leq\sup\{|f^{\prime}(x)|:\,f\in\Phi,x\in I\}<1.$$

Next, writing $\mathcal{A}=\{1,...,n\}$, for every $\omega\in\mathcal{A}^{\mathbb{N}}$ and $m\in\mathbb{N}$ let

$$f_{\omega|_{m}}:=f_{\omega_{1}}\circ\circ\circ f_{\omega_{m}}.$$

Fix $x_{0}\in I$. Then we have a surjective coding map $\pi:\mathcal{A}^{\mathbb{N}}\rightarrow K$ given by

$$\omega\in\mathcal{A}^{\mathbb{N}}\mapsto x_{\omega}:=\lim_{m\rightarrow\infty}f_{\omega|_{m}}(x_{0}),$$

(2)

which is well defined because of uniform contraction (see e.g. [13, Section 2.1]).

Let $\textbf{p}=(p_{1},...,p_{n})$ be a strictly positive probability vector, that is, $p_{i}>0$ for all $i$ and $\sum_{i}p_{i}=1$, and let $\mathbb{P}=\mathbf{p}^{\mathbb{N}}$ be the corresponding Bernoulli measure on $\mathcal{A}^{\mathbb{N}}$. We call the measure $\nu=\nu_{\mathbf{p}}=\pi\mathbb{P}$ on $K$ the self conformal measure corresponding to $\mathbf{p}$, and note that our assumptions are known to imply that it is non-atomic. Equivalently, $\nu_{\mathbf{p}}$ is the unique Borel probability measure on $K$ such that

$$\nu=\sum_{i=1}^{n}p_{i}\cdot f_{i}\nu,\quad\text{ where }f_{i}\nu\text{ is the push-forward of }\nu\text{ via }f_{i}.$$

When all the maps in $\Phi$ are affine we call $\Phi$ a self-similar IFS and $\nu$ a self-similar measure.

Next, we say that a $C^{2}(\mathbb{R})$ IFS $\Psi$ is linear if $g^{\prime\prime}(x)=0$ for every $x\in K_{\Psi}$ and $g\in\Psi$. This notion was introduced in our previous work [3], though it implicitly appeared in the literature prior to that, notably in the work of Hochman-Shmerkin [28]. It is clear that if $\Psi$ is $C^{\omega}(\mathbb{R})$ and linear then it must be self-similar. While we believe such IFSs should exist, we are not aware of any known example of a linear $C^{r}(\mathbb{R})$ smooth IFS that is not self-similar for $r\geq 1$. A major source of examples of non-linear IFSs arise in smooth dynamics from certain homoclinic intersections, see e.g. [21, Section I].

We can now state the main result of this paper. We say that a $C^{r}$ IFS $\Phi$ is conjugate to an IFS $\Psi$ if there is a $C^{r}$ diffeomorphism $h$ such that $\Phi=\{h\circ g\circ h^{-1}\}_{g\in\Psi}$.

Moreover, our argument yields easily verifiable conditions when Theorem 1.1 may be applied (without directly checking whether the IFS is conjugate to linear); As we discuss later, it suffices for a $C^{2}(\mathbb{R})$ IFS to satisfy conditions (10) and (11) below for the conclusion of Theorem 1.1 to hold true.

A great deal of attention has been given to the case when IFS in question is real analytic. This is mainly because such IFSs arise naturally in number theory, e.g. as (finitely many) inverse branches of the Gauss map [29], as Furstenberg measures for some $\text{SL}(2,\mathbb{R})$ cocycles [52, 6], and are closely related to Patterson-Sullivan measures on limit sets of some Schottky groups [35, 39, 14], among others. Combining Theorem 1.1 with a new method from a paper of Algom et al. [2], we can derive the following Corollary regarding Fourier decay in this setting:

We emphasize that no separation conditions are imposed on the IFS in both Theorem 1.1 and Corollary 1.2. Thus, as we discuss below, these are essentially the first instances where polynomial decay is obtained for such measures without an underlying Markov partition, or a more specialized algebraic or dynamical setup. We remark that, simultaneously and independently of our work, Baker and Sahlsten [9] obtained similar results but with a different method. See Remark 1.3 for more details. Also, we note that Baker and Banaji [8] independently obtained a proof of Corollary 1.2, with a method that differs considerably from that of [2]. We refer to [2, 8] for more discussion and comparisons between the two techniques.

Relying on his own previous work [33] and on the work of Bourgain-Dyatlov [14], Li [34] proved polynomial Fourier decay for Furstenberg measures for $\text{SL}(2,\mathbb{R})$ cocycles under mild assumptions. Corollary 1.2 gives a new proof of these results when the measure is self-conformal (see [52, 6] for conditions that ensure this happens). We remark that, as we discuss in Section 1.2 below, the renewal theoretic parts of our argument are closely related to Li [33, 34]. Sahlsten-Stevens [43, Theorem 1.1] proved polynomial Fourier decay for a class of stationary measures that includes self-conformal measures, with respect to totally non-linear $C^{\omega}$ IFSs with strong separation (i.e. the union (1) is disjoint). Their methods also apply when the IFS is only $C^{2}(\mathbb{R})$ under the additional assumptions that $K_{\Phi}=[0,1]$ and some further separation conditions on $\Phi$. For self-conformal measures, Theorem 1.1 improves these results since we do not require any separation conditions on the IFS, or that the attractor is an interval. In fact, Corollary 1.2 removes virtually all the assumptions made in the corresponding result of Sahlsten-Stevens [43, Theorem 1.1], except for the existence of an analytic non-affine map in the IFS. Theorem 1.1 also improves our previous result [3, Theorem 1.1] by upgrading the rate of decay in the non-conjugate to linear setting from logarithmic to polynomial. Finally, we point out that when the IFS in question is conjugate to self-similar via a non-linear $C^{2}$ map, Kaufman [32] (for some Bernoulli convolutions) and later Mosquera-Shmerkin [38] (for homogeneous self-similar measures) proved polynomial Fourier decay for all self-conformal measures. The only concrete examples of polynomial Fourier decay in the fully self-similar setup were given by Dai-Feng-Wang [19] and Streck [50] for some homogeneous IFSs that enjoy nice number theoretic properties. It is known, though, that very few self-similar IFSs should fail this property; See Solomyak [46].

Finally, combining Corollary 1.2 with the recent works of Brémont [16] and Li-Sahlsten [36] we obtain the following complete characterization of the Rajchman property for $C^{\omega}(\mathbb{R})$ IFSs: Let $\Phi$ be a $C^{\omega}(\mathbb{R})$ IFS. If $\Phi$ admits a non-atomic self-conformal measure that is not Rajchman, then $\Phi=\{r_{1}\cdot x+t_{1},...,r_{n}\cdot x+t_{n}\}$ is self-similar and there exists a Pisot number $r^{-1}$ such that $r_{i}=r^{\ell_{i}}$ for some $\ell_{i}\in\mathbb{Z}_{+}$. Furthermore, $\Phi$ is affinely conjugated to an IFS that has all of its translations in $\mathbb{Q}(r)$. For more recent results on Fourier decay for self similar measures we refer to [46, 41, 51, 4, 38, 17, 19, 18, 50] and references therein.

We proceed to discuss the method of proof of Theorem 1.1. There is no loss of generality in assuming our IFS is $C^{2}([0,1])$. Fix a self-conformal measure $\nu=\nu_{\mathbf{p}}$, and assume the conditions as in Theorem 1.1 are met. We aim to show that $\nu$ has polynomial Fourier decay. Our proof consists of four steps:

The first and most involved step is arguing that the transfer operator corresponding to the derivative cocycle and $\mathbf{p}$ (Definition 2.3) satisfies a spectral gap-type estimate (Theorem 2.8). Both our result and method of proof are strongly related to the works of Dolgopyat [22, 24], Naud [39], and Stoyanov [49, 48] (see also [5, 7, 10]) . These papers utilize a proof technique that originates from the work of Dolgopyat [22], commonly known as Dolgopyat’s method. It is used to obtain spectral gap-type estimates by, roughly speaking, a reduction to an $L^{2}$ contraction estimate (in the spirit of Proposition 2.9), and then a proof of this estimate via the construction of so-called Dolgopyat operators (in the spirit of Lemma 2.12), that control cancellations of the transfer operator.

There are, among others, three critical properties of $\nu$ and $\Phi$ that are used in Naud’s version [39] of Dolgopyat’s method, which is the one we ultimately rely on here: That the union (1) corresponds to a Markov partition (i.e. for $i\neq j$, $f_{i}([0,1])$ and $f_{j}([0,1])$ may intersect only at their endpoints), that the measure $\nu$ satisfies the Federer property (see e.g. Theorem 2.4 Part (6)), and that $\Phi$ satisfies the uniform non-integrability condition (UNI in short - see Claim 2.2). In our setting, since we assume no separation conditions (in particular, the cylinder covering is not a Markov partition), we cannot assume $\nu$ has the Federer property. As for (UNI), in our previous work [3, proof of Claim 2.13] we showed that if $\Phi$ is not conjugate to linear then (UNI) does hold true (regardless of any separation assumptions). In fact, our non-linearity assumption is only used to obtain the (UNI) condition.

One of the main innovations of this paper is the introduction a cocycle version of Dolgopyat’s method, that we use get around these issues: First, we express $\nu$ as an integral over a certain family of random measures, and express the transfer operator as a corresponding convex combination of smaller ”pieces” of itself. This is based upon a technique that first appeared in [26], and was subsequently applied in several other papers. e.g. [30, 42, 47]. In our variant, that is closest to the construction in [1], these random measures typically satisfy a certain dynamical self-conformality relation with strong separation, and satisfy the Federer property. Critically, we are able to preserve the (UNI) condition into all the different pieces of the transfer operator corresponding to this decomposition. An important preliminary step is the construction of a special covering of our IFS by (possibly overlapping) sub-IFSs (Claim 2.1), followed by the construction of this disintegration in Theorem 2.4. We then proceed to state and prove our spectral gap-type estimate (Theorem 2.8) by first reducing to a certain randomized $L^{2}$ contraction estimate (Proposition 2.9), and then prove this estimate via the construction of corresponding randomized Dolgopyat operators (Lemma 2.12). All of this discussion takes place in Section 2.

In the second step of our argument, we deduce from Theorem 2.8 (our spectral gap estimate) a renewal theorem with an exponential error term. This is Proposition 3.2 in Section 3. Such strong renewal Theorems were first proved by Li [34] for random walks arising from certain random matrix products. Our proof technique is based on that of Li [33, 34], and Boyer [15], relating the error term in the renewal operator to the resolvent of the transfer operator, on which we have good control thanks to Theorem 2.8.

In the third step of our proof, we deduce from Proposition 3.2 (our renewal Theorem) an effective equidistribution result for certain random walks driven by the derivative cocycle. This is Theorem 4.1, that critically holds with an exponential error term. The exponent we obtain is related to the size of the strip where we have spectral gap (the $\epsilon$ from Theorem 2.8). Finally, in Section 5 we obtain the desired Fourier decay bound on $\mathcal{F}_{q}(\nu)$ by combining Theorem 4.1 with our previous proof scheme from [4, Section 4.2], that relies on delicate linerization arguments and estimation of certain oscillatory integrals as in [27].

As for the proof of Corollary 1.2, our main ingredient is that the following dichotomy holds for any given $C^{\omega}(\mathbb{R})$ IFS: Either it is not $C^{2}$ conjugate to linear, or it is $C^{\omega}$ conjugate to a self-similar IFS. This is Claim 6.1, that critically relies on the Poincaré-Siegel Theorem [31, Theorem 2.8.2]. Corollary 1.2 then follows from Theorem 1.1 (if the IFS is not conjugate to linear), or from a result of Algom et al. [2] (if the IFS is analytically conjugate to self-similar).

Fix a $C^{2}(\mathbb{R})$ IFS $\Phi=\{f_{1},...,f_{n}\}$ and write $\mathcal{A}=\{1,...,n\}$. We assume without the loss of generality that for every $1\leq a\leq n$ the map $f_{a}$ is a self map of $[0,1]$, and that

$$K=K_{\Phi}\text{ does not contain the points }0,1.$$

(3)

Indeed, this follows since we may assume the IFS acts on an open interval, and since Fourier decay estimates like in Theorem 1.1 are invariant under conjugation of the IFS by a non-singular affine map. Let

$$\tilde{\rho}:=\sup_{f\in\Phi}||f^{\prime}||_{\infty}\in(0,1),$$

and let

$$\rho_{\min}:=\min_{i\in\mathcal{A},x\in[0,1]}|f_{i}^{\prime}(x)|.$$

Let us recall the bounded distortion property [4, Lemma 2.1]: There is some $L=L(\Phi)$ such that for every $\eta\in\mathcal{A}^{*}$ we have

$$L^{-1}\leq\frac{\left|f_{\eta}^{\prime}(x)\right|}{\left|f_{\eta}^{\prime}(y)\right|}\leq L\text{ for all }x,y\in[0,1].$$

(4)

Next, note that

$$\sup_{x\in[0,1],\xi\in\mathcal{A}^{\mathbb{N}},n\in\mathbb{N}}\left|\frac{d}{dx}\left(\log f^{\prime}_{\xi|_{n}}\right)(x)\right|=\sup_{x\in[0,1],\xi\in\mathcal{A}^{\mathbb{N}},n\in\mathbb{N}}\left|\sum_{k=1}^{n}\frac{f_{\xi_{k}}^{\prime\prime}\circ f_{\sigma^{k}(\xi)|_{n-k}}(x)\cdot f_{\sigma^{k}(\xi)|_{n-k}}^{\prime}(x)}{f_{\xi_{k}}^{\prime}(x)\circ f_{\sigma^{k}(\xi)|_{n-k}}(x)}\right|$$

$$\leq\sup_{x\in[0,1],f\in\Phi}\left|\frac{f^{\prime\prime}(x)}{f^{\prime}(x)}\right|\cdot\frac{1}{1-\tilde{\rho}}.$$

(5)

So, there is a uniform constant $\tilde{C}:=2\cdot\sup_{x\in[0,1],f\in\Phi}\left|\frac{f^{\prime\prime}(x)}{f^{\prime}(x)}\right|$, such that, assuming as we may (see the next paragraph) that $\tilde{\rho}<\frac{1}{2}$,

$$\sup_{x\in[0,1],\xi\in\mathcal{A}^{\mathbb{N}},n\in\mathbb{N}}\left|\frac{d}{dx}\left(\log f^{\prime}_{\xi|_{n}}\right)(x)\right|\leq\tilde{C}.$$

(6)

For all $N$ let $\Phi^{N}$ be the corresponding induced IFS of generation $N$, that is,

$$\Phi^{N}=\{f_{\eta}:\,\eta\in\mathcal{A}^{N}\}.$$

Since $\Phi^{N}$ is finite, we may order its functions ($f_{1},f_{2}$ and so on) at our will. Note that the same constants $L$ from (4) and $\tilde{C}$ from (6) will work for the induced IFS $\Phi^{N}$ for all $N$.

Our first step is, assuming $\Phi$ is not conjugate to linear, to construct an induced IFS that has useful separation and non-linearity properties:

We first remark that the existence of an upper bound $m^{\prime}$ as in Part (2) holds for all $N$ and every $f_{i},f_{j}\in\Phi^{N}$; This is a standard fact, see e.g. [39, Section 4] or (6) for similar estimates. We thus focus on the lower bound, which is much harder to obtain, in our proof of Claim 2.1.

We require the following Claim:

.

We now define $N=n+k$ and our maps in $\Phi^{N}$ via

$$f_{1}:=f_{\eta_{1}}\circ f_{\xi},\quad f_{2}:=f_{\eta_{1}}\circ f_{\zeta},\quad f_{3}:=f_{\eta_{2}}\circ f_{\xi},\quad f_{4}:=f_{\eta_{2}}\circ f_{\zeta}.$$

Let us show that all parts of the Claim hold: First, for part (2), for all $x\in[0,1]$,

$$\left|\frac{d}{dx}\left(\log f_{1}^{\prime}-\log f_{2}^{\prime}\right)\left(x\right)\right|=\left|\frac{d}{dx}\left(\log\left(f_{\eta_{1}}\circ f_{\xi}\right)^{\prime}-\log\left(f_{\eta_{1}}\circ f_{\zeta}\right)^{\prime}\right)\left(x\right)\right|$$

$$=\left|\frac{d}{dx}\left(\log f_{\xi}^{\prime}-\log f_{\zeta}^{\prime}+\log\left(f_{\eta_{1}}^{\prime}\circ f_{\xi}\right)-\log\left(f_{\eta_{1}}^{\prime}\circ f_{\zeta}\right)\right)\left(x\right)\right|$$

$$\geq\left|\frac{d}{dx}\left(\log f_{\xi}^{\prime}-\log f_{\zeta}^{\prime}\right)\left(x\right)\right|-\left|\frac{d}{dx}\left(\log\left(f_{\eta_{1}}^{\prime}\circ f_{\xi}\right)-\log\left(f_{\eta_{1}}^{\prime}\circ f_{\zeta}\right)\right)\left(x\right)\right|.$$

Now, by Claim 2.2 since $n>N_{0}$,

$$\left|\frac{d}{dx}\left(\log f_{\xi}^{\prime}-\log f_{\zeta}^{\prime}\right)\left(x\right)\right|\geq c.$$

On the other hand, arguing similarly to (6),

$$\left|\frac{d}{dx}\log\left(f_{\eta_{1}}^{\prime}\circ f_{\xi}\right)(x)\right|\leq\tilde{C}\cdot\left|f_{\xi}^{\prime}(x)\right|\leq\tilde{C}\cdot\tilde{\rho}^{n}.$$

Since we may assume $n$ is also large depending on $\tilde{C},c$ (that are both known a-priori), we conclude that

$$\left|\frac{d}{dx}\left(\log f_{1}^{\prime}-\log f_{2}^{\prime}\right)\left(x\right)\right|\geq c-2\cdot\tilde{C}\cdot\tilde{\rho}^{n}>0.$$

The same calculation holds for $f_{3},f_{4}$. This shows Part (2) holds true for our functions with

$$m:=c-2\cdot\tilde{C}\cdot\tilde{\rho}^{n}.$$

Part (3) essentially follows from the same argument since, assuming $n$ is sufficiently large depending on $c,\tilde{C}$, we can get that also

$$c-2\cdot\tilde{C}\cdot\tilde{\rho}^{n}-2\cdot\tilde{C}\cdot\tilde{\rho}^{n+k}>0.$$

For Part (1), for every $x\in f_{1}([0,1])$ and $y\in f_{2}([0,1])$ there is some $z$ with

$$|f_{1}(x)-f_{2}(y)|=|f_{\eta_{1}}^{\prime}(z)|\cdot|f_{\xi}(x)-f_{\zeta}(y)|\geq\rho_{\min}^{k}\cdot\text{dist}\left(f_{\xi}([0,1]),\,f_{\zeta}([0,1])\right)$$

$$\geq\rho_{\min}^{k}\cdot 3\cdot\tilde{\rho}^{n}>0.$$

This shows that

$$\text{dist}\left(f_{1}([0,1]),\,f_{2}([0,1])\right),\quad\text{dist}\left(f_{3}([0,1]),\,f_{4}([0,1])\right)>\rho_{\min}^{k}\cdot 3\cdot\tilde{\rho}^{n}.$$

Next, by the choice of $\eta_{1},\eta_{2}$ and $f_{1},f_{2},f_{3},f_{4}$ it is clear that

$$\text{dist}\left(f_{1}([0,1]),\,f_{3}([0,1])\right),\,\text{dist}\left(f_{1}([0,1]),\,f_{4}([0,1])\right),\,\text{dist}\left(f_{2}([0,1]),\,f_{3}([0,1])\right)\,\text{dist}\left(f_{2}([0,1]),\,f_{4}([0,1])\right)$$

$$\geq\frac{\text{diam}(K)}{2}.$$

Then for all large $n^{\prime}>N_{0}$ and $\xi,\zeta\in\mathcal{A}^{n^{\prime}}$ as in Claim 2.2, for all large $n$ for all $x\in[0,1]$, we have

$$\left|\frac{d}{dx}\left(\log\left(f_{\omega_{1}|_{n}}\circ f_{\xi}\right)^{\prime}-\log\left(f_{\omega_{2}|_{n}}\circ f_{\zeta}\right)^{\prime}\right)\left(x\right)\right|$$

$$=\left|\frac{d}{dx}\left(\log f_{\omega_{1}|_{n}}^{\prime}\circ f_{\xi}-\log f_{\omega_{2}|_{n}}^{\prime}\circ f_{\zeta}\right)\left(x\right)-\frac{d}{dx}\left(\log f_{\zeta}^{\prime}-\log f_{\xi}^{\prime}\right)\left(x\right)\right|$$

$$\geq\left|\frac{d}{dx}\left(\log f_{\zeta}^{\prime}-\log f_{\xi}^{\prime}\right)\left(x\right)\right|-\left|\frac{d}{dx}\left(\log f_{\omega_{1}|_{n}}^{\prime}\circ f_{\xi}-\log f_{\omega_{2}|_{n}}^{\prime}\circ f_{\zeta}\right)\left(x\right)\right|$$

$$\geq c-2\tilde{C}\cdot\tilde{\rho}^{n^{\prime}}=c-o(1).$$

Thus, $\xi^{\prime\prime}=\omega_{1}|_{n}*\xi$ and $\zeta^{\prime\prime}=\omega_{2}|_{n}*\zeta$ are words that can be made arbitrarily long, that satisfy that for some $c^{\prime}>0$ and for all $x$

$$\left|\frac{d}{dx}\left(\log f_{\xi^{\prime\prime}}^{\prime}-\log f_{\zeta^{\prime\prime}}^{\prime}\right)\left(x\right)\right|>c^{\prime}.$$

Also, by (9),

$$\text{dist}\left(f_{\xi^{\prime\prime}}([0,1]),\,f_{\zeta^{\prime\prime}}([0,1])\right)>3(\tilde{\rho})^{n}>3(\tilde{\rho})^{n+n^{\prime}}.$$

Thus, we are back in Case 1 with these $\xi^{\prime\prime},\zeta^{\prime\prime}$ (that we recall can be made arbitrarily long) and $c^{\prime}$.

Thus, we have completely reduced Case 2 to Case 1, completing the proof of Claim 2.1. $\Box$

Fix a $C^{2}$ IFS $\Phi$ as in Section 2.1, and let us retain the other assumptions and notations from that Section. Let $\textbf{p}=(p_{1},...,p_{n})$ be a strictly positive probability vector on $\mathcal{A}$, and let $\mathbb{P}=\mathbf{p}^{\mathbb{N}}$ be the corresponding product measure on $\mathcal{A}^{\mathbb{N}}$. Let $\nu=\nu_{\mathbf{p}}$ be the corresponding self-conformal measure.

Note that for every $P$ the induced IFS $\Phi^{P}$ has the same attractor $K$ as $\Phi$. That is, $K_{\Phi^{P}}=K_{\Phi}$. Furthermore, $\nu$ is also a self-conformal measure with respect $\Phi^{P}$ and the induced probability vector $\mathbf{p}^{P}$ on $\mathcal{A}^{P}$.

Thus, by working with the induced IFS $\Phi^{N}$ as in Claim 2.1, we may assume without the loss of generality that our (not conjugate to linear) original IFS $\Phi$ already admits $f_{1},f_{2},f_{3},f_{4}\in\Phi$ that satisfy:

1. 1.

   For every $k\geq 1$ there exists $i\in\{1,3\}$ such that

   $$f_{i}([0,1])\cup f_{i+1}([0,1])\cup f_{k}([0,1]).$$

   (10)

   is a disjoint union. In particular, the union $f_{i}([0,1])\cup f_{i+1}([0,1])$ is disjoint for $i=1,3$.

2. 2.

   There exists $m^{\prime},m>0$ such that: For every $x\in[0,1]$, for both $i=1,3$,

   $$m\leq\left|\frac{d}{dx}\left(\log f_{i}^{\prime}-\log f_{i+1}^{\prime}\right)\left(x\right)\right|\leq m^{\prime},\text{ and }m-2\cdot\tilde{C}\cdot\sup_{f\in\Phi}||f^{\prime}||_{\infty}>0.$$

   (11)

Note that for the second part of (11) we use Claim 2.1 Part (3) and that, with the notation $\tilde{\rho}$ as in Section 2.1,

$$\sup_{f\in\Phi^{N}}||f^{\prime}||_{\infty}\leq\tilde{\rho}^{N}.$$

Let $G$ to be the free semigroup generated by the family $\{f_{a}:1\leq a\leq n\}$, which acts on $[0,1]$ by composing the corresponding $f_{a}$’s. We define the derivative cocycle $c:G\times[0,1]\rightarrow\mathbb{R}$ via

$$c(I,x)=-\log\left|f^{\prime}_{I}(x)\right|.$$

(12)

Slightly adjusting our notation from Section 2.1, let

$$\rho:=\sup_{f\in\Phi}||f^{\prime}||_{\infty}\in(0,1).$$

Also, we note that by uniform contraction

$$0<D:=\min\{-\log|f^{\prime}(x)|:f\in\Phi,x\in I\},\quad D^{\prime}:=\max\{-\log|f^{\prime}(x)|:f\in\Phi,x\in I\}<\infty.$$

(13)

A major part in our argument is played by the transfer operator:

Note that $\nu$ is the unique stationary measure corresponding to the measure $\sum_{a\in\mathcal{A}}p_{a}\cdot\delta_{\{f_{a}\}}$ on $G$. So, the detailed discussion about this operator as in [12, Section 11.5] applies in the setting we are considering.

We can now construct the random measures we discussed in Section 1.2. We begin by recalling the notion of a model (as in e.g. [45]): Let $I$ be a finite set of $C^{2}(\mathbb{R})$ iterated function systems $\{f_{1}^{(i)},...,f_{k_{i}}^{(i)}\},i\in I$. Let $\Omega=I^{\mathbb{N}}$. For $\omega\in\Omega$ and $n\in\mathbb{N}\cup\{\infty\}$ let

$$X_{n}^{(\omega)}:=\prod_{j=1}^{n}\{1,...,k_{\omega_{j}}\}.$$

Define a coding map $\Pi_{\omega}:X_{\infty}^{(\omega)}\rightarrow\mathbb{R}$ via

$$\Pi_{\omega}(u)=\lim_{n\rightarrow\infty}f^{(\omega_{1})}_{u_{1}}\circ\circ\circ f^{(\omega_{n})}_{u_{n}}(0).$$

Next, for each $i\in I$ let $\mathbf{p}_{i}=(p_{1}^{(i)},...,p_{k_{i}}^{(i)})$ be a probability vector with strictly positive entries. On each $X_{\infty}^{(\omega)}$ we define the product measure

$$\eta^{(\omega)}=\prod_{n=1}^{\infty}\mathbf{p}^{(\omega_{n})}.$$

We can now define

$$\mu_{\omega}:=\Pi_{\omega}\left(\eta^{(\omega)}\right).$$

Letting $\sigma:\Omega\rightarrow\Omega$ be the left shift, we have the following dynamical self-conformality relation:

$$\mu_{\omega}=\sum_{u\in X_{1}^{(\omega)}}p_{u}^{(\omega_{1})}f_{u}^{(\omega_{1})}\mu_{\sigma(\omega)}.$$

(14)

We also define

$$K_{\omega}:=\text{supp}(\mu_{\omega})=\Pi_{\omega}\left(X_{\infty}^{(\omega)}\right),$$

and note that here we have

$$K_{\omega}=\bigcup_{u\in X_{1}^{(\omega)}}f_{u}^{(\omega_{1})}K_{\sigma(\omega)}.$$

Next, for $\omega\in\Omega,N\in\mathbb{N}$, and $s\in\mathbb{C}$ we define an operator $P_{s,\omega,N}:C^{1}\left([0,1]\right)\rightarrow C^{1}\left([0,1]\right)$ by

$$P_{s,\omega,N}\left(g\right)(x):=\sum_{I\in X_{N}^{(\omega)}}\eta^{(\omega)}(I)e^{2\pi sc(I,x)}g\circ f_{I}(x),\quad\text{ where }\eta^{(\omega)}(I):=\prod_{n=1}^{|I|}\textbf{p}^{(\omega_{n})}_{I_{n}}.$$

(15)

Note that we only need to know $\omega|_{N}$ in order for $P_{s,\omega,N}$ to be well defined. Iterating (14) we have the following equivariance relations, whose proof is left to the reader: First, for every $\omega\in\Omega$, $N\in\mathbb{N}$, and $g\in C^{1}\left([0,1]\right)$,

$$\int g(x)\,d\mu_{\omega}(x)=\int P_{0,\omega,N}\left(g\right)(x)\,d\mu_{\sigma^{N}\omega}(x).$$

(16)

Furthermore, for all integers $0\leq\tilde{n}\leq n,N,$ we have that

$$P_{s,\omega,nN}\left(g\right)(x)=P_{s,\sigma^{\tilde{n}N}\omega,Nn-N\tilde{n}}\left(P_{s,\omega,\tilde{n}N}\left(g\right)\right)\left(x\right).$$

(17)