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Family of piecewise expanding maps having singular measure as a limit of ACIMs

Title:

Family of piecewise expanding maps having singular measure as a limit of ACIMs

LI, ZHENYANG, GÓRA, PAWEŁ, BOYARSKY, ABRAHAM, PROPPE, HARALD and ESLAMI, PEYMAN (2013) Family of piecewise expanding maps having singular measure as a limit of ACIMs. Ergodic Theory and Dynamical Systems, 33 (01). pp. 158-167. ISSN 0143-3857

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Official URL: http://dx.doi.org/10.1017/S0143385711000836

Abstract

Keller [Stochastic stability in some chaotic dynamical systems. Monatsh. Math.94(4) (1982), 313–333] introduced families of W-shaped maps that can have a great variety of behaviors. As a family approaches a limit W map, he observed behavior that was either described by a probability density function (PDF) or by a singular point measure. Based on this, Keller conjectured that instability of the absolutely continuous invariant measure (ACIM) can result only from the existence of small invariant neighborhoods of the fixed critical point of the limit map. In this paper, we show that the conjecture is not true. We construct a very simple family of W-maps with ACIMs supported on the whole interval, whose limiting dynamical behavior is captured by a singular measure. Key to the analysis is the use of a general formula for invariant densities of piecewise linear and expanding maps [P. Góra. Invariant densities for piecewise linear maps of interval. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1549–1583].

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Article
Refereed:Yes
Authors:LI, ZHENYANG and GÓRA, PAWEŁ and BOYARSKY, ABRAHAM and PROPPE, HARALD and ESLAMI, PEYMAN
Journal or Publication:Ergodic Theory and Dynamical Systems
Date:2013
Digital Object Identifier (DOI):10.1017/S0143385711000836
ID Code:976825
Deposited By: DANIELLE DENNIE
Deposited On:29 Jan 2013 14:01
Last Modified:18 Jan 2018 17:43

References:

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