arXiv subjects

Pawel Gora

Publications and source records attributed to Pawel Gora.

5 recordsLinked to original sources

arXiv2025

Quasi-compactness of Frobenius-Perron Operator for Piecewise Convex Maps with Countable Branches

In this paper, we prove the quasi-compactness of the Frobenius-Perron operator for a piecewise convex map $τ$ with a countably infinite number of branches on the interval $I=[0,1]$. We establish that for high enough $n$ iterates of $τ$, $τ^n$ are piecewise expanding. Using the Lasota-Yorke Inequality derived from references \cite{hofbauer1982} and \cite{keller1985}, adapted to meet the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem, we demonstrate the existence of absolutely continuous invariant measure (ACIM) $μ$ for $τ$, the exactness of the dynamical system $(I, τ,μ)$ and the quasi-compactness of Frobenius-Perron operator $P_τ$ induced by $τ$. The last fact implies a multitude of strong ergodic properties of $τ$.

math.DS↗

arXiv2019

Absolutely Continuous Invariant measures for non-autonomous dynamical systems

We consider the non autonomous dynamical system $\{τ_{n}\},$ where $τ_{n}$ is a continuous map $X\rightarrow X,$ and $X$ is a compact metric space. We assume that $\{τ_{n}\}$ converges uniformly to $τ.$ The inheritance of chaotic properties as well as topological entropy by $τ$ from the sequence $\{τ_{n}\}$ has been studied in \cite{Can1, Can2, Li,Ste,Zhu}. In \cite{You} the generalization of SRB\ measures to non-autonomous systems has been considered. In this paper we study absolutely continuous invariant measures (acim) for non autonomous systems. After generalizing the Krylov-Bogoliubov Theorem \cite{KB} and Straube's Theorem \cite{Str} to the non autonomous setting, we prove that under certain conditions the limit map $τ$ of a non autonomous sequence of maps $\{τ_n\}$ with acims has an acim.

math.DS↗

arXiv2011

Eventually Expanding Maps

In this paper we show that the piecewise linear map f(x) = px for x in [0,1/p], and sx-s/p for x in (1/p,1], p > 1, 0 < s < 1 which has an expanding, onto branch and a contracting branch is eventually piecewise expanding and exact.

math.DS↗

arXiv2009

SRB Measures For Certain Markov Processes

We study Markov processes generated by iterated function systems (IFS). The constituent maps of the IFS are monotonic transformations of the interval. We first obtain an upper bound on the number of SRB (Sinai-Ruelle-Bowen) measures for the IFS. Then, when all the constituent maps have common fixed points at 0 and 1, theorems are given to analyze properties of the ergodic invariant measures $δ_0$ and $δ_1$. In particular, sufficient conditions for $δ_0$ and/or $δ_1$ to be, or not to be, SRB measures are given. We apply some of our results to asset market games.

math.DS↗

arXiv2006

Basis Markov Partitions and Transition Matrices for Stochastic Systems

We analyze dynamical systems subjected to an additive noise and their deterministic limit. In this work, we will introduce a notion by which a stochastic system has something like a Markov partition for deterministic systems. For a chosen class of the noise profiles the Frobenius-Perron operator associated to the noisy system is exactly represented by a stochastic transition matrix of a finite size K. This feature allows us to introduce for these stochastic systems a basis--Markov partition, defined herein, irrespectively of whether the deterministic system possesses a Markov partition or not. We show that in the deterministic limit, corresponding to K --> infinity, the sequence of invariant measures of the noisy systems tends, in the weak sense, to the invariant measure of the deterministic system. Thus by introducing a small additive noise one may approximate transition matrices and invariant measures of deterministic dynamical systems.

nlin.CD↗