Escape rates and Perron-Frobenius operators: open and closed dynamical systems
type: Item
instance of: scholarly article
title: Escape rates and Perron-Frobenius operators: open and closed dynamical systems
zbMATH Open document ID: 1213.37012
author: Gary Froyland
author: Ognjen Stancevic
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2010-09-22T00:00:00Z
review text: They study the Perron-Frobenius operator \({\mathcal P}\) of dynamical systems and certain open dynamical systems. We prove that the presence of a large positive eigenvalue of \({\mathcal P}\) guarantees the existence of a 2-partition of the phase space for which the escape rates of the open systems defined on the two partition sets are both slower than log. The open systems with slow escape rates are easily identified from the Perron-Frobenius operators of the closed systems. Numerical results are presented for expanding maps of the unit interval. We also apply our technique to shifts of finite type to show that if the adjacency matrix for the shift has a large positive second eigenvalue, then the shift may be decomposed into two disjoint subshifts, both of which have high topological entropies.
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 37M25
Mathematics Subject Classification ID: 37C40
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 37B40
zbMATH DE Number: 5790142
zbMATH Keywords: Perron-Frobenius operator
zbMATH Keywords: open dynamical system
zbMATH Keywords: escape rate
zbMATH Keywords: almost-invariant set
zbMATH Keywords: topological entropy
reviewed by: Ahmed Hegazi
MaRDI profile type: Publication
full work available at URL: dcdsb.2010.14.457
OpenAlex ID: W1986893967
Wikidata QID: Q59139584
DOI: 10.3934/DCDSB.2010.14.457
description: scientific article; zbMATH DE number 5790142