On the decay of correlation for piecewise monotonic mappings. II
type: Item
instance of: scholarly article
title: On the decay of correlation for piecewise monotonic mappings. II
zbMATH Open document ID: 0619.58029
author: Makoto Mori
published in: Tokyo Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: [For part I see ibid. 8, 389-414 (1985; Zbl 0597.58014).] The author considers a class of mappings F of the unit interval \(I=[0,1]\) into itself which are piecewise linear with nonzero slope on every subinterval of a (finite or countable) partition of I with the additional property that every subinterval (except one fixed) has an image enclosing (0,1). After introducing such technical notions as word (a finite sequence of names of subintervals), admissible word, type of a word, generating functions and Fredholm eigenvalue of F a so-called renewal equation for admissible words is derived. This is the main tool for proving the two theorems of the paper: If the infimum \(\xi\) of the lower Lyapunov numbers is positive and the second Fredholm eigenvalue \(\eta\) of F is less than one then F has a unique invariant distribution \(\mu\) being absolutely continuous with respect to Lebesgue measure. Moreover, the dynamical system ([0,1],F,\(\mu)\) is mixing and the correlations for any pair of functions decay asymptotically by a power law with rate \(\eta^ n\). If \(\xi\) is negative then there exists an attracting periodic orbit.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 26A18
zbMATH DE Number: 4005199
zbMATH Keywords: iterated maps
zbMATH Keywords: piecewise linear maps
zbMATH Keywords: mixing systems
zbMATH Keywords: decay of correlations
zbMATH Keywords: Fredholm eigenvalue
zbMATH Keywords: invariant distribution
zbMATH Keywords: attracting periodic orbit
reviewed by: Gottfried Jetschke
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270150982
OpenAlex ID: W2057564494
DOI: 10.3836/TJM/1270150982
Recommended article: On the decay of correlation for piecewise monotonic mappings. I
Recommended article: On the intermittency of a piecewise linear map (Takahashi model)
Recommended article: Q3784930
Recommended article: Q4208292
Recommended article: Properties of invariant measures for piecewise expanding one-dimensional transformations with summable oscillations of derivative
Recommended article: Higher order mixing property of piecewise linear transformations
Recommended article: ON THE EVENTUAL PERIODICITY OF PIECEWISE LINEAR CHAOTIC MAPS
Recommended article: ORDER AND CHAOS FOR A CLASS OF PIECEWISE LINEAR MAPS
Recommended article: Intermittency generated by attracting and weakly repelling fixed points
Recommended article: On the ergodicity of a model of drilling
description: scientific article; zbMATH DE number 4005199