Mariusz Urbański

Ergodic theory for Markov fibred systems and parabolic rational maps

A paraboIic rational map of the Riemann sphere admits a non-atomic h-conformal measure on its Julia set where h = the Hausdorff dimension of the Julia set and satisfies 1/2 < h < 2. With respect to this measure the rational map is conservative, exact and there is an equivalent a-finite invariant measure. Finiteness of the measure is characterised. Central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one. A theory of Markov fibred systems is developed, and parabolic rational maps are considered within this framework

On the existence of conformal measures

A general notion of conformal measure is introduced and some basic properties are studied. Sufficient conditions for the existence of these measures are obtained, using a general construction principle. The geometric properties of conformal measures relate equilibrium states and Hausdorff measures. This is shown for invariant subsets of S under expanding maps.

Ergodic theory of equilibrium states for rational maps

Let T be a rational map of degree d>or=2 of the Riemann sphere C=C union ( infinity ). The authors develop the theory of equilibrium states for the class of Holder continuous functions f for which the pressure is larger than sup f. They show that there exist a unique conformal measure (reference measure) and a unique equilibrium state, which is equivalent to the conformal measure with a positive continuous density. The associated Perron-Frobenius operator acting on the space of continuous functions is almost periodic and they show that the system is exact with respect to the equilibrium measure.

Gibbs states on the symbolic space over an infinite alphabet

We consider subshifts of finite type on the symbolic space generated by incidence matrices over a countably infinite alphabet. We extend the definition of topological pressure to this context and, as our main result, we construct a new class of Gibbs states of Hölder continuous potentials on these symbol spaces. We establish some basic stochastic properties of these Gibbs states: exponential decay of correlations, central limit theorem and an a.s. invariance principle. This is accomplished via detailed studies of the associated Perron-Frobenius operator and its conjugate operator.

Conformal iterated function systems with applications to the geometry of continued fractions

In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior. We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmetic density properties of the set of allowed entries.

On the transfer operator for rational functions on the Riemann sphere

Let T be a rational function of degree 2 on the Riemann sphere. Denote L the transfer operator of a HH older-continuous function on its Julia set J = J(T) satisfying P(T;) > sup z2J (z). We study the behavior of fL n : n 1g for HH older-continuous functions and show that the sequence is (uniformly) norm-bounded in the space of HH older-continuous functions for suuciently small exponent. As a consequence we obtain that the density of the equilibrium measure for with respect to the exppP(T;) ? ]-conformal measure is HH older-continuous. We also prove that the rate of convergence of L n to this density in sup-norm is O ? exp(? p n). >From this we deduce the central limit theorem for .

Thermodynamic Formalism and Multifractal Analysis of Conformal Infinite Iterated Function Systems

We develop the thermodynamic formalism for equilibrium states of strongly Hölder families of functions. These equilibrium states are supported on the limit set generated by iterating a system of infinitely many contractions. The theory of these systems was laid out in an earlier paper of the last two authors. The first five sections of this paper except Section 3 are devoted to developing the thermodynamic formalism for equilibrium states of Hölder families of functions. The first three sections provide us with the tools needed to carry out the multifractal analysis for the equilibrium states mentioned above assuming that the limit set is generated by conformal contractions. The theory of infinite systems of conformal contractions is laid out in [13]. The multifractal analysis is then given in Section 7. In Section 8 we apply this theory to some examples from continued fraction systems and Apollonian packing.

Measures and dimensions in conformal dynamics

This survey collects basic results concerning fractal and ergodic properties of Julia sets of rational functions of the Riemann sphere. Frequently these results are compared with their counterparts in the theory of Kleinian groups, and this enlarges the famous Sullivan dictionary. The topics concerning Hausdorff and packing measures and dimensions are given most attention. Then, conformal measures are constructed and their relations with Hausdorff and packing measures are discussed throughout the entire article. Also invariant measures absolutely continuous with respect to conformal measures are touched on. While the survey begins with facts concerning all rational functions, much time is devoted toward presenting the well-developed theory of hyperbolic and parabolic maps, and in Section 3 the class NCP is dealt with. This class consists of such rational functions f that all critical points of f which are contained in the Julia set of f are non-recurrent. The NCP class comprises in particular hyperbolic, parabolic and subhyperbolic maps. Our last section collects some recent results about other subclasses of rational functions, e.g. Collet-Eckmann maps and Fibonacci maps. At the end of this article two appendices are included which are only loosely related to Sections 1-4. They contain a short description of tame mappings and the theory of equilibrium states and Perron-Frobenius operators associated with Hölder continuous potentials. 1. Dimensions of Julia sets A first issue we will be dealing with in this article is to describe various fractals of Julia sets as captured by the Hausdorff, packing and box dimensions. Afterwards, we address the question of when the corresponding Hausdorff and packing measures are positive and finite. Given a subset A of a metric space (X, d) a countable family {B(xi, ri)}i=1 of open balls centered at points of A is said to be a packing of A if and only if for any pair i 6= j d(xi, xj) ≥ ri + rj . The supremum sup{ri : i ≥ 1} is called the radius of the packing {B(xi, ri)}i=1. Given in addition a positive radius r > 0, denote by N(A, r) the minimal number of open balls with radius r needed to cover A and by P (A, r) the maximal number of open balls with radius r forming a packing of A. In order to get an idea of what the box dimension is, imagine a two-dimensional smooth surface A in the Euclidean space R. It is reasonable to expect the minimal number N(A, r) and the maximal number P (A, r) to be some multiples of r−2. The Received by the editors December 22, 1999, and, in revised form, January 8, 2003. 2000 Mathematics Subject Classification. Primary 35F35, 37D35; Secondary 37F15, 37D20, 37D25, 37D45, 37A40, 37A05. Research partially supported by NSF Grant DMS 9801583. c ©2003 American Mathematical Society 281 282 MARIUSZ URBAŃSKI coefficient is not important to understand dimensionality of A, but the exponent is crucial, and it is captured by the following limits: lim r→0 logN(A, r) − log r = lim r→0 logP (A, r) − log r . In general these limits fail to exist, and the following two quantities are defined: BD(A) = lim inf r→0 logN(A, r) − log r = lim inf r→0 logP (A, r) − log r

Absolutely Continuous Invariant Measures for Expansive Rational Maps with Rationally Indifferent Periodic Points.

Expansive rational maps T: C -> C which are not expanding with respect to the spherical metric are those which have rationally indifferent periodic points. For an atomless f-conformal measure m of such a rational map we prove the existence of a unique (up to a multiplicative constant) σ-fmite, Γ-invariant measure μ absolutely continuous with respect to m. We also give a necessary and sufficient condition for the measure μ to be finite. 1980 Mathematics Subject Classification (1985 Revision): 28DOS, 58F11, 30C99. §

Invariant measures for parabolic ifs with overlaps and random continued fractions

We study parabolic iterated function systems (IFS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no “overlaps.” We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.