arXiv subjects

Bartlomiej Skorulski

Publications and source records attributed to Bartlomiej Skorulski.

3 recordsLinked to original sources

arXiv2012

Regularity and irregularity of fiber dimension of non-autonomous dynamical systems

This note concerns non-autonomous dynamics of rational functions and, more precisely, the fractal behavior of the Julia sets under perturbation of non-autonomous systems. We provide a necessary and sufficient condition for holomorphic stability which leads to Hölder continuity of dimensions of hyperbolic non-autonomous Julia sets with respect to the $l^\infty$-topology on the parameter space. On the other hand we show that, for some particular family, the Hausdorff and packing dimension functions are not differentiable at any point and that these dimensions are not equal on an open dense set of the parameter space still with respect to the $l^\infty$-topology.

math.DS↗

arXiv2010

Distance Expanding Random Mappings, Thermodynamic Formalism, Gibbs Measures, and Fractal Geometry

In this paper we define distance expanding random dynamical systems. We develop the appropriate thermodynamic formalism of such systems. We obtain in particular the existence and uniqueness of invariant Gibbs states, the appropriate pressure function and exponentially fast convergence of iterates of Perron-Frobenius operators resulting, in particular, in an exponential decay of correlations. We also obtain the formula for the derivative of the expected value of the pressure function. Next, we define and investigate in detail conformal random expanding repellers. Applying the developed machinery of thermodynamic formalism we prove a version of Bowen's formula which identifies the Hausdorff dimension $h$ of almost all fibers with the only zero of expected value of the pressure function. We then turn to more refined fractal properties by, firstly, showing that the multifractal formalism of the Gibbs states is valid and, secondly, that the $h$--Hausdorff measure vanishes while the corresponding packing measure is infinite provided the system is not quasi-deterministic.

math.DS↗

arXiv2009

Non-existence of absolutely continuous invariant probabilities for exponential maps

We show that for entire maps of the form $z \mapsto λ\exp(z)$ such that the orbit of zero is bounded and such that Lebesgue almost every point is transitive, no absolutely continuous invariant probability measure can exist. This answers a long-standing open problem.

math.DS↗