Piecewise linear maps, Liapunov exponents and entropy

Jonq Juang, Shih Feng Shieh

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Let LA = {fA, x : x is a partition of [0, 1]} be a class of piecewise linear maps associated with a transition matrix A. In this paper, we prove that if fA, x ∈ LA, then the Liapunov exponent λ (x) of fA, x is equal to a measure theoretic entropy hmA, x of fA, x, where mA, x is a Markov measure associated with A and x. The Liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix A is symmetric. Moreover, we also show that maxx λ (x) = maxx hmA, x = log (λ1), where λ1 is the maximal eigenvalue of A.

Original languageEnglish
Pages (from-to)358-364
Number of pages7
JournalJournal of Mathematical Analysis and Applications
Volume338
Issue number1
DOIs
Publication statusPublished - 2008 Feb 1

Fingerprint

Piecewise Linear Map
Transition Matrix
Lyapunov Exponent
Entropy
Eigenvalue Problem
Partition
Eigenvalue
Class

Keywords

  • Entropy
  • Ergodic theory
  • Liapunov exponents
  • Piecewise linear map

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Piecewise linear maps, Liapunov exponents and entropy. / Juang, Jonq; Shieh, Shih Feng.

In: Journal of Mathematical Analysis and Applications, Vol. 338, No. 1, 01.02.2008, p. 358-364.

Research output: Contribution to journalArticle

@article{a10da8aed7f948ab958b311585f7d839,
title = "Piecewise linear maps, Liapunov exponents and entropy",
abstract = "Let LA = {fA, x : x is a partition of [0, 1]} be a class of piecewise linear maps associated with a transition matrix A. In this paper, we prove that if fA, x ∈ LA, then the Liapunov exponent λ (x) of fA, x is equal to a measure theoretic entropy hmA, x of fA, x, where mA, x is a Markov measure associated with A and x. The Liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix A is symmetric. Moreover, we also show that maxx λ (x) = maxx hmA, x = log (λ1), where λ1 is the maximal eigenvalue of A.",
keywords = "Entropy, Ergodic theory, Liapunov exponents, Piecewise linear map",
author = "Jonq Juang and Shieh, {Shih Feng}",
year = "2008",
month = "2",
day = "1",
doi = "10.1016/j.jmaa.2007.05.035",
language = "English",
volume = "338",
pages = "358--364",
journal = "Journal of Mathematical Analysis and Applications",
issn = "0022-247X",
publisher = "Academic Press Inc.",
number = "1",

}

TY - JOUR

T1 - Piecewise linear maps, Liapunov exponents and entropy

AU - Juang, Jonq

AU - Shieh, Shih Feng

PY - 2008/2/1

Y1 - 2008/2/1

N2 - Let LA = {fA, x : x is a partition of [0, 1]} be a class of piecewise linear maps associated with a transition matrix A. In this paper, we prove that if fA, x ∈ LA, then the Liapunov exponent λ (x) of fA, x is equal to a measure theoretic entropy hmA, x of fA, x, where mA, x is a Markov measure associated with A and x. The Liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix A is symmetric. Moreover, we also show that maxx λ (x) = maxx hmA, x = log (λ1), where λ1 is the maximal eigenvalue of A.

AB - Let LA = {fA, x : x is a partition of [0, 1]} be a class of piecewise linear maps associated with a transition matrix A. In this paper, we prove that if fA, x ∈ LA, then the Liapunov exponent λ (x) of fA, x is equal to a measure theoretic entropy hmA, x of fA, x, where mA, x is a Markov measure associated with A and x. The Liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix A is symmetric. Moreover, we also show that maxx λ (x) = maxx hmA, x = log (λ1), where λ1 is the maximal eigenvalue of A.

KW - Entropy

KW - Ergodic theory

KW - Liapunov exponents

KW - Piecewise linear map

UR - http://www.scopus.com/inward/record.url?scp=34548835002&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548835002&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2007.05.035

DO - 10.1016/j.jmaa.2007.05.035

M3 - Article

AN - SCOPUS:34548835002

VL - 338

SP - 358

EP - 364

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -