Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-13T22:53:00.234Z Has data issue: false hasContentIssue false
  • English
  • Français

Lifting measures to inducing schemes

Published online by Cambridge University Press:  01 April 2008

YA. B. PESIN,
S. SENTI and
K. ZHANG
YA. B. PESIN
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA (email: pesin@math.psu.edu)
S. SENTI
Affiliation:
Instituto de Matematica, Universidade Federal do Rio de Janeiro, C.P. 68 530, CEP 21945-970, R.J., Brazil (email: senti@impa.br)
K. ZHANG
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA (email: kezhang@math.umd.edu)
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.

In this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [Y. Pesin and S. Senti. Thermodynamical formalism associated with inducing schemes for one-dimensional maps. Mosc. Math. J.5(3) (2005), 669–678; Y. Pesin and S. Senti. Equilibrium measures for maps with inducing schemes. Preprint, 2007]. We show that under some natural assumptions on the inducing schemes—which hold for many known examples—any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the inducing scheme. The argument uses the construction of connected Markov extensions due to Buzzi [J. Buzzi. Markov extensions for multi-dimensional dynamical systems. Israel J. Math.112 (1999), 357–380], his results on the liftability of measures of large entropy, and a generalization of some results by Bruin [H. Bruin. Induced maps, Markov extensions and invariant measures in one-dimensional dynamics. Comm. Math. Phys.168(3) (1995), 571–580] on relations between inducing schemes and Markov extensions. We apply our results to study the liftability problem for one-dimensional cusp maps (in particular, unimodal and multi-modal maps) and for some multi-dimensional maps.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008, pp. 553 - 574
Copyright
Copyright © Cambridge University Press 2008

References

[1]Blokh, A. and Levin, G.. An inequality for laminations, Julia sets and ‘growing trees’. Ergod. Th. & Dynam. Sys. 22(1) (2002), 63–97.CrossRefGoogle Scholar
[2]Bruin, H.. Induced maps, Markov extensions and invariant measures in one-dimensional dynamics. Comm. Math. Phys. 168(3) (1995), 571–580.CrossRefGoogle Scholar
[3]Bruin, H. and Todd, M.. Markov extensions and lifting measures for complex polynomials. Ergod. Th. & Dynam. Sys. 27(3) (2007), 743–768.CrossRefGoogle Scholar
[4]Buzzi, J.. Markov extensions for multi-dimensional dynamical systems. Israel J. Math. 112 (1999), 357–380.CrossRefGoogle Scholar
[5]Carleson, L. and Gamelin, T. W.. Complex Dynamics (Universitext: Tracts in Mathematics). Springer, New York, 1993.CrossRefGoogle Scholar
[6]de Melo, W. and van Strien, S.. One-Dimensional Dynamics. Springer, Berlin, 1993.Google Scholar
[7]Dobbs, N.. Critical points, cusps and induced expansion in dimension one. PhD Thesis, Université de Paris-Sud, Orsay, 2006.Google Scholar
[8]Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel J. Math. 34(3) (1979), 213–237.CrossRefGoogle Scholar
[9]Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. II. Israel J. Math. 38(1–2) (1981), 107–115.CrossRefGoogle Scholar
[10]Keller, G.. Lifting measures to Markov extensions. Monatsh. Math. 108(2–3) (1989), 183–200.CrossRefGoogle Scholar
[11]Martens, M.. Distortion results and invariant Cantor sets of unimodal maps. Ergod. Th. & Dynam. Sys. 14(2) (1994), 331–349.CrossRefGoogle Scholar
[12]Pesin, Y. and Senti, S.. Thermodynamical formalism associated with inducing schemes for one-dimensional maps. Mosc. Math. J. 5(3) (2005), 669–678.CrossRefGoogle Scholar
[13]Pesin, Y. and Senti, S.. Equilibrium measures for maps with inducing schemes. Preprint, 2007.Google Scholar
[14]Pesin, Y. and Zhang, K.. Thermodynamics associated with inducing schemes and liftability of measures. Proc. Fields Inst. (2007).Google Scholar
[15]Senti, S.. Dimension of weakly expanding points for quadratic maps. Bull. Soc. Math. France 131(3) (2003), 399–420.CrossRefGoogle Scholar
[16]Yoccoz, J.-C.. Jakobson’s theorem. Manuscript of Course at Collège de France, 1997.Google Scholar
[17]Zweimüller, R.. Invariant measures for general(ized) induced transformations. Proc. Amer. Math. Soc. 133(8) (2005), 2283–2295 (electronic).CrossRefGoogle Scholar