Multifractal analysis for conformal graph directed Markov systems
Mario Roy Mariusz Urbański
We derive the multifractal analysis of the conformal measure (or, equivalently, of the invariant measure) associated to a family of weights imposed upon a graph directed Markov system (GDMS) using balls as the filtration. Our analysis is done over a subset of the limit set, a subset which is often large. In particular, this subset is the entire limit set when the GDMS under scrutiny satisfies a boundary separation condition. Our analysis also applies to more general situations such as real and complex continued fractions.
keywords: Multifractal analysis real and complex continued fractions. graph directed Markov systems thermodynamic formalism conformal maps
Random graph directed Markov systems
Mario Roy Mariusz Urbański
We introduce and explore random conformal graph directed Mar-kov systems governed by measure-preserving ergodic dynamical systems. We first develop the symbolic thermodynamic formalism for random finitely primitive subshifts of finite type with a countable alphabet (by establishing tightness in a narrow topology). We then construct fibrewise conformal and invariant measures along with fibrewise topological pressure. This enables us to define the expected topological pressure $\mathcal EP(t)$ and to prove a variant of Bowen's formula which identifies the Hausdorff dimension of almost every limit set fiber with $\inf\{t:\mathcal EP(t)\leq0\}$, and is the unique zero of the expected pressure if the alphabet is finite or the system is regular. We introduce the class of essentially random systems and we show that in the realm of systems with finite alphabet their limit set fibers are never homeomorphic in a bi-Lipschitz fashion to the limit sets of deterministic systems; they thus make up a drastically new world. We also provide a large variety of examples, with exact computations of Hausdorff dimensions, and we study in detail the small random perturbations of an arbitrary elliptic function.
keywords: random dynamical systems elliptic functions. thermodynamic formalism Iterated function systems limit set countable shifts Hausdorff dimension
Bowen parameter and Hausdorff dimension for expanding rational semigroups
Hiroki Sumi Mariusz Urbański
We estimate the Bowen parameters and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map $\tilde{f}$ and the Lyapunov exponent of $\tilde{f}$ with respect to the maximal entropy measure for $\tilde{f}$. Moreover, we show that the equality holds if and only if the generators are simultaneously conjugate to the form $a_{j}z^{\pm d}$ by a M\"{o}bius transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than $2$.
keywords: Bowen parameter Hausdorff dimension random complex dynamics. Julia set rational semigroups Complex dynamical systems expanding semigroups
The dynamics and geometry of the Fatou functions
Janina Kotus Mariusz Urbański
We deal with the Fatou functions $f_\lambda(z)=z+e^{-z}+\lambda$, Re$\lambda\ge 1$. We consider the set $J_r(f_\lambda)$ consisting of those points of the Julia set of $f_\lambda$ whose real parts do not escape to infinity under positive iterates of $f_\lambda$. Our ultimate result is that the function $\lambda\mapsto$HD$(J_r(f_\lambda))$ is real-analytic. In order to prove it we develop the thermodynamic formalism of potentials of the form $-t$log$|F_\lambda'|$, where $F_\lambda$ is the projection of $f_\lambda$ to the infinite cylinder. It includes appropriately defined topological pressure, Perron-Frobenius operators, geometric and invariant generalized conformal measures (Gibbs states). We show that our Perron-Frobenius operators are quasicompact, that they embed into a family of operators depending holomorphically on an appropriate parameter and we obtain several other properties of these operators. We prove an appropriate version of Bowen's formula that the Hausdorff dimension of the set $J_r(f_\lambda)$ is equal to the unique zero of the pressure function. Since the formula for the topological pressure is independent of the set $J_r(f_\lambda)$, Bowen's formula also indicates that $J_r(f_\lambda)$ is the right set to deal with. What concerns geometry of the set $J_r(f_\lambda)$ we also prove that the HD$(J_r(f_\lambda))$-dimensional Hausdorff measure of the set $J_r(F_\lambda)$ is positive and finite whereas its HD$(J_r(f_\lambda))$-dimensional packing measure is locally infinite. This last property allows us to conclude that HD$(J_r(f_\lambda))<2$. We also study in detail the properties of quasiconformal conjugations between the maps $f_\lambda$. As a byproduct of our main course of reasoning we prove stochastic properties of the dynamical system generated by $F_\lambda$ and the invariant Gibbs states $\mu_t$ such as the Central Limit Theorem and the exponential decay of correlations.
keywords: Fatou functions Julia set conformal measures Hausdorff and packing measures Perron-Frobenius operator topological pressure real analyticity of Hausdorff dimension. Hausdorff dimension
Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups
Hiroki Sumi Mariusz Urbański
We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that for the canonically associated skew-product map there exists a unique $h$-conformal measure. Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely continuous invariant (under the skew-product map) measure. In fact these two measures are equivalent, and the invariant measure is metrically exact, hence ergodic.
keywords: Complex dynamical systems rational semigroups Julia set Hausdorff dimension skew product. semi-hyperbolic semigroups conformal measure
Holomorphic maps for which the unstable manifolds depend on prehistories
Eugen Mihailescu Mariusz Urbański
For points $x$ belonging to a basic set $\Lambda$ of an Axiom A holomorphic endomorphism of $\mathbb P^2$, one can construct the local stable manifold $W_{\varepsilon_0}^s(x)$ and the local unstable manifolds $W_{\varepsilon_0}^u(\hat x)$. A priori, $W_{\varepsilon_0}^u(\hat x)$ should depend on the entire prehistory $\hat x$ of $x$. However, all known examples have all their local unstable manifolds depending only on the base point $x$. Therefore a natural problem is to give actual examples where, for different prehistories of points in the basic sets of holomorphic Axiom A maps, we obtain different unstable manifolds. We solve this problem by considering the map $(z^4+\varepsilon w^2, w^4)$ and then also show that, by perturbing $(z^2+c, w^2)$, one can get also maps $f_\varepsilon$ which are injective on $\Lambda_\varepsilon$, their corresponding basic sets, hence the cardinality of the set $(f_\varepsilon|_{\Lambda_\varepsilon})^{-1}(x), x \in \Lambda_\varepsilon$, is not stable under perturbation.
keywords: prehistories injectivity on basic sets. perturbations of holomorphic maps Unstable manifolds
Mariusz Urbański
The conference "Dynamical Systems II, Denton 2009" held in the University of North Texas in Denton from May 17, 2009 through May 23, 2009 gathered approximately forty participants working on various subbranches of dynamical systems such as holomorphic and conformal dynamics, transcendental dynamics, random dynamical systems, thermodynamic formalism, and iterated function systems, including random behavior of deterministic systems, the theory of fractal sets, including dimension theory. Apart from stimulating, highly informative mathematical talks delivered by nearly all participants of the conference, its outcome resulted also in thirteen outstanding research articles presented in this volume. The subject of these papers varies from author to author reflecting their scientific interests. The articles were written by A. Badeńska (Warsaw University of Technology), D. Hensley (Texas A&M), P. Haissinsky (Université de Provence) , M. Kesseboehmer (University of Bremen), D. Mayer (TU Clausthal), E. Mihailescu (Romanian Academy), T. Mühlenbruch (FernUniversität in Hagen), F. Naud (Université d'Avignon), S. Munday (University of St Andrews), K. Pilgrim (Indiana University), Mario Roy (York University), H. H. Rugh, D. Simmons (University of North Texas), B. Stratmann (University of Bremen), F. Strömberg (TU Darmstadt), H. Sumi (University of Osaka), and M. Urbański (University of North Texas).

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Transversal families of hyperbolic skew-products
Eugen Mihailescu Mariusz Urbański
We study families of hyperbolic skew products with the transversality condition and in particular, the Hausdorff dimension of their fibers, by using thermodynamical formalism. The maps we consider can be non-invertible, and the study of their dynamics is influenced greatly by this fact.
    We introduce and employ probability measures (constructed from equilibrium measures on the natural extension), which are supported on the fibers of the skew product. A stronger condition, that of Uniform Transversality is then considered in order to obtain a general formula for Hausdorff dimension of fibers for all base points and almost all parameters.
    In the end we study a large class of examples of transversal hyperbolic families which locally depend linearly on the parameters, and also another class of examples related to complex dynamics.
keywords: skew products Hausdorff dimension hyperbolic maps transversality.
Continuity of Hausdorff measure for conformal dynamical systems
Tomasz Szarek Mariusz Urbański Anna Zdunik
Developing the pioneering work of Lars Olsen [14], we deal with the question of continuity of the numerical value of Hausdorff measures of some natural families of conformal dynamical systems endowed with an appropriate natural topology. In particular, we prove such continuity for hyperbolic polynomials from the Mandelbrot set, and more generally for the space of hyperbolic rational functions of a fixed degree. We go beyond hyperbolicity by proving continuity for maps including parabolic rational functions, for example that the parameter $1/4$ is such a continuity point for quadratic polynomials $z\mapsto z^2+c$ for $c\in [0,1/4]$. We prove the continuity of the numerical value of Hausdorff measures also for the spaces of conformal expanding repellers and parabolic ones, more generally for parabolic Walters conformal maps. We also prove some partial continuity results for all conformal Walters maps; these are in general of infinite degree. In order to do this, as one of our tools, we provide a detailed local analysis, uniform with respect to the parameter, of the behavior of conformal maps around parabolic fixed points in any dimension. We also establish continuity of numerical values of Hausdorff measures for some families of infinite $1$-dimensional iterated function systems.
keywords: Hausdorff measure conformal dynamical systems. Dimension theory

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