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We develop a versatile framework which allows us to rigorously estimate the Hausdorff dimension of maximal conformal graph directed Markov systems in $\mathbb{R}^n$ for $n \geq 2$. Our method is based on piecewise linear approximations of the eigenfunctions of the Perron-Frobenius operator via a finite element framework for discretization and iterative mesh schemes. One key element in our approach is obtaining bounds for the derivatives of these eigenfunctions, which, besides being essential for the implementation of our method, are of independent interest.

In this paper, we study orbit counting problems for inner functions using geodesic and horocyclic flows on Riemann surface laminations. For a one component inner function of finite Lyapunov exponent with $F(0) = 0$, other than $z \to z^d$, we show that the number of pre-images of a point $z \in \mathbb{D} \setminus \{ 0\}$ that lie in a ball of hyperbolic radius $R$ centered at the origin satisfies $$ \mathcal{N}(z, R) \, \sim \, \frac{1}{2} \log \frac{1}{|z|} \cdot \frac{1}{\int_{\partial \mathbb{D}} \log |F'| dm}, \quad \text{as }R \to \infty. $$ For a general inner function of finite Lyapunov exponent, we show that the above formula holds up to a Ces\`aro average. Our main insight is that iteration along almost every inverse orbit is asymptotically linear. We also prove analogues of these results for parabolic inner functions of infinite height., Comment: 75 pages, 2 figures

In this paper, we use thermodynamic formalism to study the dynamics of inner functions $F$ acting on the unit disk. If the Denjoy-Wolff point of $F$ is in the open unit disk, then without loss of generality, we can assume that $F(0) = 0$ so that 0 is an attracting fixed point of $F$ and the Lebesgue measure on the unit circle is invariant under $F$. Utilizing the connection between composition operators, Aleksandrov-Clark measures and Perron-Frobenius operators, we develop a rudimentary thermodynamic formalism which allows us to prove the Central Limit Theorem and the Law of Iterated Logarithm for Sobolev multipliers and H\"older continuous observables. Under the more restrictive, but natural hypothesis that $F$ is a one component inner function, we develop a more complete thermodynamic formalism which is sufficient for orbit counting, assuming only the $(1+\varepsilon)$ integrability of $\log|F'|$. As one component inner functions admit countable Markov partitions of the unit circle, we may work in the abstract symbolic setting of countable alphabet subshifts of finite type. Due to the very weak hypotheses on the potential, we need to pay close attention to the regularity of the complex Perron-Frobenius operators $\mathcal L_s$ with $\text{Re }s > 1$ near the boundary. Finally, we discuss inner functions with a Denjoy-Wolff point on the unit circle. We assume a parabolic type behavior of $F$ around this point and we introduce the class of parabolic one component inner functions. By making use of the first return map, we deduce various stochastic laws and orbit counting results from the aforementioned abstract symbolic results., Comment: 84 pages, 1 figure

Polynomials and entire functions whose hyperbolic dimension is strictly smaller than the Hausdorff dimension of their Julia set are known to exist but in all these examples the latter dimension is maximal, i.e. equal to two. In this paper we show that there exist hyperbolic entire functions $f$ having Hausdorff dimension of the Julia set $\HD (\J _f)<2$ and hyperbolic dimension $\HypDim(f)<\HD(\J_f)$., Comment: 15 pages

By employing the recurrence method worked out in `Estimating the Hausdorff measure by recurrence', we provide effective lower estimates of the proper--dimensional Hausdorff measure of minimal sets of circle homeomorphisms that are not conjugate to any rotation.

We study new relations between countable iterated function systems (IFS) with overlaps, Smale endomorphisms and random systems with complete connections. We prove that stationary measures for countable conformal IFS with overlaps and placedependent probabilities, are exact dimensional; moreover we determine their Hausdorff dimension. Next, we construct a family of fractals in the limit set of a countable IFS with overlaps S, and study the dimension for certain measures supported on these subfractals. In particular, we obtain families of measures on these subfractals which are related to the geometry of the system.

We show that any dynamics on any planar set $S$ discrete in some domain $D$ can be realized by the postcritical dynamics of a function holomorphic in $D$, up to a small perturbation. A key step in the proof, and a result of independent interest, is that any planar domain $D$ can be equilaterally triangulated with triangles whose diameters $\rightarrow0$ (at any prescribed rate) near $\partial D$.

We consider the subclass of class ${\mathcal B} $ consisting of meromorphic functions $f:{\mathbb C}\to\hat{\mathbb C}$ for which infinity is not an asymptotic value and whose all poles have orders uniformly bounded from above. This class was introduced in \cite{BwKo2012} and the Hausdorff dimension ${\rm HD}({{\mathcal I} (f)})$ of the set ${{\mathcal I} (f)}$ of all points escaping to infinity under forward iteration of $f$ was estimated therein. In this paper we provide a closed formula for the exact value of ${\rm HD}({{\mathcal I} (f)})$ identifying it with the critical exponent of the natural series introduced in \cite{BwKo2012}. This exponent is very easy to calculate for many concrete functions. In particular, we construct a function from this class which is of infinite order and for which ${\rm HD}({{\mathcal I} (f)})=0$., Comment: 27 pages

This text provides an overview of the (geometric) thermodynamic formalism for transcendental meromorphic and entire functions with particular emphasis on geometric/fractal aspects such as Bowen's Formula expressing the hyperbolic dimension as a unique zero of a pressure function and the behavior of the latter when the transcendental functions vary in an analytic family., Comment: Revised Version, 36 pages

The ultimate goal of our book is to present a unified approach to the dynamics, ergodic theory, and geometry of elliptic functions from $\C$ to $\oc$. We consider elliptic functions as a most regular class of transcendental meromorphic functions. Poles form an essential feature of such functions but the set of critical values is finite and an elliptic function is "the same" on its of its fundamental regions. In a sense this is the class of transcendental meromorphic functions which resembles rational functions most. On the other hand, the differences are huge. We will touch on them in the course of this introduction. In order to comprehensively cover the dynamics and geometry of elliptic functions we make large preparations. This is done in the first two parts of the book: Part 1, "Ergodic Theory and Measures" and Part 2,"Geometry and Conformal Measures". We intend our book to be as self contained as possible and we use essentially all major results of Part~1 and Part~2 in Part~3 and Part~4 dealing with elliptic functions. This book can be thus treated as a fairly comprehensive account of dynamics, ergodic theory, and fractal geometry of elliptic functions but also as a reference book (with proofs) for many results of geometric measure theory, finite and infinite abstract ergodic theory, Young's towers, measure--theoretic Kolmogorov--Sinai entropy, thermodynamic formalism, geometric function theory (in particular Koebe's Distortion Theorems and Riemann--Hurwitz Formulas), various kinds of conformal measures, conformal graph Directed Markov systems and iterated function systems, classical general theory of elliptic functions, and topological dynamics of transcendental meromorphic functions.

We consider small perturbations of a conformal iterated function system (CIFS) produced by either adding or removing some generators with small derivative from the original. We establish a formula, utilizing transfer operators arising from the thermodynamic formalism \`a la Sinai--Ruelle--Bowen, which may be solved to express the Hausdorff dimension of the perturbed limit set in series form: either exactly, or as an asymptotic expansion. Significant applications include strengthening Hensley's asymptotic formula from 1992, which improved on earlier bounds due to Jarn\'ik and Kurzweil, for the Hausdorff dimension of the set of real numbers whose continued fraction expansion partial quotients are all $\leq N$; as well as its counterpart for reals whose partial quotients are all $\geq N$ due to Good from 1941.

We study skew-product dynamics for a large class of finitely-generated semi--hyperbolic semigroups of rational maps acting on the Riemann sphere, which generalizes both the theory of iteration of a single rational map of a single complex variable complex/holomorphic dynamics) and the theory of countable alphabet conformal iterated function systems (CIFSs). We construct the thermodynamic formalism for such dynamical systems and geometric potentials by developing the notion of nice families that extend to the case of our highly disconnected skew product phase space the powerful notion of nice sets due to Rivera--Letelier and Przytycki, and the allied earlier notion of $K(V)$ sets due to Denker and the last named author. We leverage out techniques to prove the existence and uniqueness of equilibrium states for a wide class of H\"older potentials, and concomitant statistical laws: central limit theorem, law of iterated logarithm, and exponential decay of correlations. We devote lots of space and effort to control (non-recurrent) critical points which is a notoriously challenging task even for a single rational function; more generators add qualitatively new challenges. Beyond dynamics, but still with dynamical methods, we advance the study of finer fractal geometrical properties of the intricate Julia sets associated to such systems and, in particular, via equilibrium states, we perform a multifractal analysis of Lyapunov exponents. We use the Nice Open Set Condition (NOSC) introduced by the last two authors, and apply our new techniques to settle a long-standing problem in the theory of rational semigroups by proving that for our class of semigroups the Hausdorff dimension of each fiber Julia set is strictly smaller than the Hausdorff dimension of the global Julia set of the semigroup., Comment: 168 pages, 6 figures

In this paper we study various aspects of porosities for conformal fractals. We first explore porosity in the general context of infinite graph directed Markov systems (GDMS), and we show that, under some natural assumptions, their limit sets are porous in large (in the sense of category and dimension) subsets, and they are mean porous almost everywhere. On the other hand, we prove that if the limit set of a GDMS is not porous then it is not porous almost everywhere. We also revisit porosity for finite graph directed Markov systems, and we provide checkable criteria which guarantee that limit sets have holes of relative size at every scale in a prescribed direction. We then narrow our focus to systems associated to complex continued fractions with arbitrary alphabet and we provide a novel characterization of porosity for their limit sets. Moreover, we introduce the notions of upper density and upper box dimension for subsets of Gaussian integers and we explore their connections to porosity. As applications we show that limit sets of complex continued fractions system whose alphabet is co-finite, or even a co-finite subset of the Gaussian primes, are not porous almost everywhere, while they are mean porous almost everywhere. We finally turn our attention to complex dynamics and we delve into porosity for Julia sets of meromorphic functions. We show that if the Julia set of a tame meromorphic function is not the whole complex plane then it is porous at a dense set of its points and it is almost everywhere mean porous with respect to natural ergodic measures. On the other hand, if the Julia set is not porous then it is not porous almost everywhere. In particular, if the function is elliptic we show that its Julia set is not porous at a dense set of its points.

We consider a class of dynamical systems, which we call weakly coarse expanding, which is a generalization to the postcritically infinite case of expanding Thurston maps as discussed by Bonk-Meyer and is closely related to coarse expanding conformal systems as defined by Haissinsky-Pilgrim. We prove existence and uniqueness of equilibrium states for a wide class of potentials, as well as statistical laws such as a central limit theorem, law of iterated logarithm, exponential decay of correlations and a large deviation principle. Further, if the system is defined on the 2-sphere, we prove all such results even in presence of periodic (repelling) branch points., Comment: 30 pages, revised version

We study Smale skew product endomorphisms (introduced in [27]) now over countable graph directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply for instance to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We give also applications in ergodic number theory, for example to the continued fraction expansion, and the backward fractions expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions $\mathcal T_\beta$ of $\beta$-maps $T_\beta$, for arbitrary $\beta > 1$., Comment: Published in Ergodic Theory and Dynamical Systems, vol 41, 2021, 2508-2541. arXiv admin note: substantial text overlap with arXiv:1705.05880

We extend the parametric geometry of numbers (initiated by Schmidt and Summerer, and deepened by Roy) to Diophantine approximation for systems of $m$ linear forms in $n$ variables, and establish a new connection to the metric theory via a variational principle that computes fractal dimensions of a variety of sets of number-theoretic interest. The proof relies on two novel ingredients: a variant of Schmidt's game capable of computing the Hausdorff and packing dimensions of any set, and the notion of templates, which generalize Roy's rigid systems. In particular, we compute the Hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, resolving a conjecture of Kadyrov, Kleinbock, Lindenstrauss and Margulis, as well as a question of Bugeaud, Cheung and Chevallier. As a corollary of Dani's correspondence principle, the divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs has equal Hausdorff and packing dimensions. Other applications include quantitative strengthenings of theorems due to Cheung and Moshchevitin, which originally resolved conjectures due to Starkov and Schmidt respectively; as well as dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions, completing partial results due to Baker, Bugeaud, Cheung, Chevallier, Dodson, Laurent and Rynne., Comment: now with a uniform version of the main variational principle

We prove a new result in the area of hitting time statistics. Currently, there is a lot of papers showing that the first entry times into cylinders or balls are often faster than the Birkhoff's Ergodic Theorem would suggest. We provide an opposite counterpart to these results by proving that the hitting times into shrinking balls are also often much larger than these theorems would suggest, by showing that for many dynamical systems $$ \displaystyle \limsup_{r\to 0} \tau_{B(y,r)}(x)\mu(B(y,r))=+\infty, $$ for an appropriately large, at least of full measure, set of points $y$ and $x$. We first do this for all transitive open distance expanding maps and Gibbs/equilibrium states of H\"older continuous potentials; in particular for all irreducible subshifts of finite type with a finite alphabet. Then we prove such result for all finitely irreducible subshifts of finite type with a countable alphabet and Gibbs/equilibrium states for H\"older continuous summable potentials. Next, we show that the \emph{limsup} result holds for all graph directed Markov systems (far going natural generalizations of iterated function systems) and projections of aforementioned Gibbs states on their limit sets. By utilizing the first return map techniques, we then prove the \emph{limsup} result for all tame topological Collect--Eckmann multimodal maps of an interval, all tame topological Collect--Eckmann rational functions of the Riemann sphere, and all dynamically semi--regular transcendental meromorphic functions from $\mathbb{C}$ to $\widehat{\mathbb{C}}$., Comment: arXiv admin note: text overlap with arXiv:1609.03610

The quantization dimension function for an $F$-conformal measure $m_F$ generated by an infinite conformal iterated function system satisfying the strong open set condition and by a summable H\"{o}lder family of functions is expressed by a simple formula involving the temperature function of the system. The temperature function is commonly used to perform the multifractal analysis, in our context of the measure $m_F$. The result in this paper extends a similar result of Lindsay and Mauldin established for finite conformal iterated function systems [Nonlinearity 15 (2002)].

We consider random multimodal $C^3$ maps with negative Schwarzian derivative, defined on a finite union of closed intervals in $[0,1]$, onto the interval $[0,1]$ with the base space $\Omega$ and a base invertible ergodic map $\theta:\Omega\to\Omega$ preserving a probability measure $m$ on $\Omega$. We denote the corresponding skew product map by $T$ and call it a critically finite random map of an interval. We prove that there exists a subset $AA(T)$ of $[0,1]$ with the following properties: (1) For each $t\in AA(T)$ a $t$-conformal random measure $\nu_t$ exists. We denote by $\lambda_{t,\nu_t,\omega}$ the corresponding generalized eigenvalues of the corresponding dual operators $\mathcal{L}_{t,\omega}^*$, $\omega\in\Omega$. (2) Given $t\ge 0$ any two $t$-conformal random measures are equivalent. (3) The expected topological pressure of the parameter $t$: $$\mathcal{E}P(t):=\int_{\Omega}\log\lambda_{t,\nu,\omega}dm(\omega) $$ is independent of the choice of a $t$-conformal random measure $\nu$. (4) The function $$ AA(T)\ni t\longmapsto \mathcal{E}P(t)\in\mathbb R $$ is monotone decreasing and Lipschitz continuous. (5) With $b_T$ being defined as the supremum of such parameters $t\in AA(T)$ that $\mathcal{E}P(t)\ge 0$, it holds that $$ \mathcal{E}P(b_T)=0 \ \ \ {\rm and} \ \ \ [0,b_T]\subset \text{Int}(AA(T)). $$ (6) $\text{HD}(\mathcal{J}_\omega(T))=b_T$ for $m$-a.e $\omega\in\Omega$, where $\mathcal{J}_\omega(T)$, $\omega\in\Omega$, form the random closed set generated by the skew product map $T$. (7) $b_T=1$ if and only if $\bigcup_{\Delta\in \mathcal{G}}\Delta=[0,1]$, and then $\mathcal{J}_\omega(T)=[0,1]$ for all $\omega\in\Omega$.

In this paper we study the dimension spectrum of continued fractions with coefficients restricted to infinite subsets of natural numbers. We prove that if $E$ is any arithmetic progression, the set of primes, or the set of squares $\{n^2\}_{n \in \mathbb{N}}$, then the continued fractions whose digits lie in $E$ have full dimension spectrum, which we denote by $DS(\mathcal{CF}_E)$. Moreover we prove that if $E$ is an infinite set of consecutive powers then the dimension spectrum $DS(\mathcal{CF}_E)$ always contains a non trivial interval. We also show that there exists some $E \subset \mathbb{N}$ and two non-trivial intervals $I_1, I_2$, such that $DS(\mathcal{CF}_E) \cap I_1=I_1$ and $DS(\mathcal{CF}_E) \cap I_2$ is a Cantor set. On the way we employ the computational approach of Falk and Nussbaum in order to obtain rigorous effective estimates for the Hausdorff dimension of continued fractions whose entries are restricted to infinite sets.

We consider random iteration of exponential entire functions, i.e. of the form ${\mathbb C}\ni z\mapsto f_\lambda(z):=\lambda e^z\in\mathbb C$, $\lambda\in{\mathbb C}\setminus \{0\}$. Assuming that $\lambda$ is in a bounded closed interval $[A,B]$ with $A>1/e$, we deal with random iteration of the maps $f_\lambda$ governed by an invertible measurable map $\theta:\Omega\to\Omega$ preserving a probability ergodic measure $m$ on $\Omega$, where $\Omega$ is a measurable space. The link from $\Omega$ to exponential maps is then given by an arbitrary measurable function $\eta:\Omega\longmapsto [A,B]$. We in fact work on the cylinder space $Q:={\mathbb C}/\sim$, where $\sim$ is the natural equivalence relation: $z\sim w$ if and only if $w-z$ is an integral multiple of $2\pi i$. We prove that then for every $t>1$ there exists a unique random conformal measure $\nu^{(t)}$ for the random conformal dynamical system on $Q$. We further prove that this measure is supported on the, appropriately defined, radial Julia set. Next, we show that there exists a unique random probability invariant measure $\mu^{(t)}$ absolutely continuous with respect to $\mu^{(t)}$. In fact $\mu^{(t)}$ is equivalent with $\nu^{(t)}$. Then we turn to geometry. We define an expected topological pressure $\mathcal E P(t)\in{\mathbb R}$ and show that its only zero $h$ coincides with the Hausdorff dimension of $m$--almost every fiber radial Julia set $J_r(\omega)\subset Q$, $\omega\in\Omega$. We show that $h\in (1,2)$ and that the omega--limit set of Lebesgue almost every point in $Q$ is contained in the real line $\mathbb R$. Finally, we entirely transfer our results to the original random dynamical system on $\mathbb C$. As our preliminary result, we show that all fiber Julia sets coincide with the entire complex plane $\mathbb C$.

In this paper we study the dimension spectrum of general conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We perform a comprehensive study of the dimension spectrum addressing questions regarding its size and topological structure. As a corollary we obtain that the dimension spectrum of infinite conformal iterated function systems is compact and perfect. On the way we revisit the role of the parameter $\theta$ in graph directed Markov systems and we show that new phenomena arise. We also establish topological pressure estimates for subsystems in the abstract setting of symbolic dynamics with countable alphabets. These estimates play a crucial role in our proofs regarding the dimension spectrum, and they allow us to study Hausdorff dimension asymptotics for subsystems. Finally we narrow our focus to the dimension spectrum of conformal iterated function systems and we prove, among other things, that the iterated function system resulting from the complex continued fractions algorithm has full dimension spectrum. We thus give a positive answer to the Texan conjecture for complex continued fractions.

We provide the full theory of thermodynamic formalism for a very general collection of entire functions in class $\mathcal B$. This class overlaps with the collection of all entire functions for which thermodynamic formalism has been so far established and contains many new functions. The key point is that we introduce an integral means spectrum for logarithmic tracts which takes care of the fractal behavior of the boundary of the tract near infinity. It turns out that this spectrum behaves well as soon as the tracts have some sufficiently nice geometry which, for example, is the case for quasicircle, John or H\"older tracts. In this case we get a good control of the corresponding transfer operators, leading to full thermodynamic formalism along with its applications such as exponential decay of correlations, central limit theorem and a Bowen's formula for the Hausdorff dimension of radial Julia sets. Our approach applies in particular to every hyperbolic function from any Eremenko-Lyubich analytic family of Speiser class $\mathcal S$ provided this family contains at least one function with H\"older tracts. The latter is, for example, the case if the family contains a Poincar\'e linearizer.

We introduce and study skew product Smale endomorphisms over finitely irreducible topological Markov shifts with countable alphabets. We prove that almost all conditional measures of equilibrium states of summable and locally Holder continuous potentials are dimensionally exact, and that their dimension is equal to the ratio of the (global) entropy and the Lyapunov exponent. We also prove for them a formula of Bowen type for the Hausdorff dimension of all fibers. We develop a version of thermodynamic formalism for finitely irreducible two-sided topological Markov shifts with countable alphabets. We describe then the thermodynamic formalism for Smale skew products over countable-to-1 endomorphisms, and give several applications to measures on natural extensions of endomorphisms. We show that the exact dimensionality of conditional measures on fibers, implies the global exact dimensionality of the measure, in certain cases. We then study equilibrium states for skew products over endomorphisms generated by graph directed Markov systems, in particular for skew products over expanding Markov-Renyi(EMR) maps, and we settle the question of the exact dimensionality of such measures. In particular, this applies to skew products over the continued fractions transformation, and over parabolic maps. We prove next two results related to Diophantine approximation, which make the renowned Doeblin-Lenstra Conjecture more general and more precise, for a different class of measures than in the classical case. In the end, we prove exact dimensionality and find a computable formula for the dimension of equilibrium measures, for induced maps of natural extensions $\mathcal T_\beta$ of beta-maps, for arbitrary \beta > 1., Comment: This was accepted and will appear in Ergodic Theory and Dynamical Systems

In this article we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the former. We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic orbits ordered hy a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features of the distribution of their weights. These have direct applications to a variety of examples, including the case of Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions, Mannenville-Pomeau maps, Schottky groups, Fuchsian groups, and many more. A fairly complete collection of asymptotic counting results for them is presented. Our new approach is founded on spectral properties of complexified Ruelle--Perron--Frobenius operators and Tauberian theorems as used in classical problems of prime number theory.

We establish a new connection between metric Diophantine approximation and the parametric geometry of numbers by proving a variational principle facilitating the computation of the Hausdorff and packing dimensions of many sets of interest in Diophantine approximation. In particular, we show that the Hausdorff and packing dimensions of the set of singular $m\times n$ matrices are both equal to $mn \big(1-\frac1{m+n}\big)$, thus proving a conjecture of Kadyrov, Kleinbock, Lindenstrauss, and Margulis (preprint 2014) as well as answering a question of Bugeaud, Cheung, and Chevallier (preprint 2016). We introduce the notion of a $template$, which generalizes the notion of a $rigid$ $system$ (Roy, 2015) to the setting of matrix approximation. Our main theorem takes the following form: for any class of templates $\mathcal F$ closed under finite perturbations, the Hausdorff and packing dimensions of the set of matrices whose successive minima functions are members of $\mathcal F$ (up to finite perturbation) can be written as the suprema over $\mathcal F$ of certain natural functions on the space of templates. Besides implying KKLM's conjecture, this theorem has many other applications including computing the Hausdorff and packing dimensions of the set of points witnessing a conjecture of Starkov (2000), and of the set of points witnessing a conjecture of Schmidt (1983)., Comment: Announcement