Bartlomiej Skorulski

Distance expanding random mappings, thermodynamical formalism, Gibbs measures and fractal geometry

1 Introduction.- 2 Expanding Random Maps.- 3 The RPF-theorem.- 4 Measurability, Pressure and Gibbs Condition.- 5 Fractal Structure of Conformal Expanding Random Repellers.- 6 Multifractal Analysis.- 7 Expanding in the Mean.- 8 Classical Expanding Random Systems.- 9 Real Analyticity of Pressure.

Finer fractal geometry for analytic families of conformal dynamical systems

We prove several results establishing real analyticity of Hausdorff dimensions of limit sets of analytic families of conformal graph directed Markov systems. With this tool and with iterated function systems resulting from the existence of nice sets in the sense of Rivera-Letelier, we prove that the canonical Hausdorff measure restricted to the radial Julia set of a tame meromorphic function (can be rational) is σ-finite and that the Hausdorff dimension of the radial Julia sets for fairly general families of meromorphic functions (can be rational) is real analytic.

The Law of Iterated Logarithm and Equilibrium Measures Versus Hausdorff Measures for Dynamically Semi-regular Meromorphic Functions

The Law of Iterated Logarithm for dynamically semi-regular meromorphic mappings and loosely tame observables is established. The equilibrium states of tame potentials are compared with an appropriate one-parameter family of generalized Hausdorff measures. The singularity/absolute continuity dichotomy is established. Both results utilize the concept of nice sets and the theory of infinite conformal iterated function systems.

Dynamical rigidity of transcendental meromorphic functions

We prove the form of dynamical rigidity of transcendental meromorphic functions which asserts that if two tame transcendental meromorphic functions restricted to their Julia sets are topologically conjugate via a locally bi-Lipschitz homeomorphism, then they, treated as functions defined on the entire complex plane , are topologically conjugate via an affine map, i.e. a map from to of the form z???az?+?b. As an intermediate step we show that no tame transcendental meromorphic function is essentially affine.

The RPF-Theorem

We now establish a version of Ruelle–Perron–Frobenius (RPF) Theorem along with a mixing property. Notice that this quite substantial fact is proved without any measurable structure on the space \(\mathcal{J}\). In particular, we do not address measurability issues of λ x and q x . In order to obtain this measurability we will need and we will impose a natural measurable structure on the space \(\mathcal{J}\). This will be done in the next chapter.

Fractal Structure of Conformal Expanding Random Repellers

We now deal with conformal expanding random maps. We prove an appropriate version of Bowen’s Formula, which asserts that the Hausdorff dimension of almost every fiber \({\mathcal{J}}_{x}\), denoted throughout the paper by \(\mathrm{HD}\), is equal to a unique zero of the function \(t\mapsto \mathcal{E}\!P(t)\).

Measurability, Pressure and Gibbs Condition

In this monograph we develop the thermodynamical formalism for measurable expanding random mappings. This theory is then applied in the context of conformal expanding random mappings where we deal with the fractal geometry of fibers.

Expanding in the Mean

In this chapter we show that the main achievements of this manuscript, including thermodynamical formalism, Bowen’s formula and multifractal analysis, also hold for a class of random maps satisfying an allegedly weaker expanding condition

Classical Expanding Random Systems

\int \nolimits \nolimits \log {\gamma }_{x}\mathit{dm}(x) > 0.