Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T09:25:22.617Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant measures for piecewise linear fractional maps

Part of: Measure-theoretic ergodic theory

Published online by Cambridge University Press:  09 April 2009

F. Schweiger
F. Schweiger
Affiliation:
Department of Mathematics Monash UniversityClayton, Victoria, 3168, Australia Institut für Mathematik Universität SalzburgSalzburg Oesterreich
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let T: [0,1] → [0,1] be a map which is given piecewise as a linear fractional map such that T0 = T] = 0 and T'0 < 1. Then T is ergodic and admits an invariant measure which can be calculated explicitly.

MSC classification

Secondary: 28D05: Measure-preserving transformations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 34 , Issue 1 , February 1983 , pp. 55 - 59
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Fischer, R., ‘Ergodische Theorie von Zifferentwicklungen in Wahrscheinlichkeitsräumen.’ Math. Z. 128 (1972), 217–230.CrossRefGoogle Scholar
[2]Nakada, H., ‘On the invariant measures and the entropies for continued fraction transformations,’ Keio Math. Sem. Rep. 5 (1980), 37–44.Google Scholar
[3]Schweiger, F., ‘Dual algorithms and invariant measures,’ Preprint, Salzburg 1980.Google Scholar
[4]Schweiger, F., ‘Metrische Theorie einer Klasse zahlentheoretischer Transformationen,’ Acta Arith. 15 (1968), 1–18.CrossRefGoogle Scholar
Corrigendum Acta Arith. 16 (1969), 217–219.Google Scholar
[5]Schweiger, F., ‘Some remarks on ergodicity and invariant measures,’ Michigan Math. J. 22 (1975), 181–187.CrossRefGoogle Scholar
[6]Tanaka, S. and Ito, Sh., ‘On a family of continued fraction expansions and their ergodic properties,’ Preprint.Google Scholar