arXiv subjects
Publications and source records attributed to Mariusz Urbański.
We prove that in any metric space $(X,d)$ the singular integral operators {equation*} T^k_{μ,\ve}(f)(x)=\int_{X\setminus B(x,\varepsilon)}k(x,y)f(y)dμ(y).{equation*} converge weakly in some dense subspaces of $L^2(μ)$ under minimal regularity assumptions for the measures and the kernels.
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.
The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is bounded and the map $f$ restricted to its closure is expanding, the property refered to as sub-expanding. We first show the existence, uniqueness, conservativity and ergodicity of a conformal measure $m$ with minimal exponent $h$; furthermore, we show weak metrical exactness of this measure. Then we prove the existence of a $\sg$--finite invariant measure $μ$ absolutely continuous with respect to $m$. Our main result states that $μ$ is finite if and only if the order $ρ$ of the function $f$ satisfies the condition $h>3\fracρ{ρ+1}$. When finite, this measure is shown to be metrically exact. We also establish a version of Bowen's formula showing that the exponent $h$ equals the Hausdorff dimension of the Julia set of $f$.