
TL;DR
This paper introduces 'virtually expanding' dynamical systems, a broad class including expanding and some partially hyperbolic maps, and demonstrates the quasi-compactness of the Perron-Frobenius operator on associated Sobolev spaces.
Contribution
The paper defines the class of virtually expanding maps and proves the quasi-compactness of the Perron-Frobenius operator for this class.
Findings
Virtually expanding maps include all expanding and some partially hyperbolic volume-expanding maps.
Perron-Frobenius operator is quasi-compact on Sobolev spaces for these maps.
Provides a framework for analyzing statistical properties of a broad class of dynamical systems.
Abstract
We introduce a class of discrete dynamical systems that we call \emph{virtually expanding}. This is an open subset of self-covering maps on a closed manifold which contains all expanding maps and some partially hyperbolic volume-expanding maps. We show that the Perron-Frobenius operator is quasi-compact on a Sobolev space of positive order for such class of dynamical systems.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Quantum chaos and dynamical systems · Geometric and Algebraic Topology
