Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-02-11T14:04:28.728Z Has data issue: false hasContentIssue false
  • English
  • Français

Absolutely continuous invariant measures for some piecewise hyperbolic affine maps

Published online by Cambridge University Press:  01 February 2008

TOMAS PERSSON
TOMAS PERSSON*
Affiliation:
Centre for Mathematical Sciences, Lund University, Box 118, SE-22100 Lund, Sweden (email: tomasp@maths.lth.se)
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.

A class of piecewise affine hyperbolic maps on a bounded subset of the plane is considered. It is shown that if a map from this class is sufficiently area-expanding then almost surely this map has an absolutely continuous invariant measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 1 , February 2008 , pp. 211 - 228
Copyright
Copyright © Cambridge University Press 2007

References

[1]Alexander, J. C. and Yorke, J. A.. Fat baker’s transformations. Ergod. Th. & Dynam. Sys. 4 (1984), 1–23.CrossRefGoogle Scholar
[2]Belykh, V. P.. Models of discrete systems of phase synchronization. Systems of Phase Synchronization. Eds. V. V. Shakhildyan and L. N. Belynshina. Radio i Svyaz, Moscow, 1982, pp. 61–176.Google Scholar
[3]Buzzi, J.. Intrinsic ergodicity of affine maps in [0,1]d. Monatsh. Math. 124 (1997), 97–118.CrossRefGoogle Scholar
[4]Buzzi, J.. Absolutely continuous invariant measures for generic multi-dimensional piecewise affine expanding maps. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), 1743–1750.CrossRefGoogle Scholar
[5]Buzzi, J.. Absolutely continuous invariant probability measures for arbitrary expanding piecewise -analytic mappings of the plane. Ergod. Th. & Dynam. Sys. 20 (2000), 697–708.CrossRefGoogle Scholar
[6]Chernov, N.. Statistical properties of piecewise smooth hyperbolic systems in high dimensions. Discrete Contin. Dyn. Syst. 5 (1999), 425–448.CrossRefGoogle Scholar
[7]Cornfeld, I. P., Fomin, S. V. and Sinaĭ, Ya. G.. Ergodic Theory (Grundlehren der Mathematischen Wissenschaften, 245). Springer, New York, 1982.CrossRefGoogle Scholar
[8]Kruglikov, B. and Rypdal, M.. Entropy via multiplicity. Discrete Contin. Dyn. Syst. 16(2) (2006), 395–410.CrossRefGoogle Scholar
[9]Neunhäuserer, J.. Dimension theoretical properties of generalized Baker’s transformations. Nonlinearity 15 (2002), 1299–1307.CrossRefGoogle Scholar
[10]Peres, Y. and Solomyak, B.. Absolute continuity of Bernoulli convolutions, a simple proof. Math. Res. Lett. 3 (1996), 231–236.CrossRefGoogle Scholar
[11]Persson, T.. A piecewise hyperbolic map with absolutely continuous invariant measure. Dyn. Syst. 21(3) (2006), 363–378.CrossRefGoogle Scholar
[12]Pesin, Ya. B.. Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Ergod. Th. & Dynam. Sys. 12 (1992), 123–151.CrossRefGoogle Scholar
[13]Sataev, E. A.. Invariant measures for hyperbolic maps with singularities. Russian Math. Surv. 47 (1992), 191–251.CrossRefGoogle Scholar
[14]Schmeling, J.. A dimension formula for endomorphisms—the Belykh family. Ergod. Th. & Dynam. Sys. 18 (1998), 1283–1309.CrossRefGoogle Scholar
[15]Schmeling, J. and Troubetzkoy, S.. Dimension and invertibility of hyperbolic endomorphisms with singularities. Ergod. Th. & Dynam. Sys. 18 (1998), 1257–1282.CrossRefGoogle Scholar
[16]Solomyak, B.. On the random series (an Erdös problem). Ann. of Math. (2) 142 (1995), 611–625.CrossRefGoogle Scholar
[17]Tsujii, M.. Absolutely continuous invariant measures for piecewise real-analytic expanding maps on the plane. Comm. Math. Phys. 208 (2000), 605–622.CrossRefGoogle Scholar
[18]Tsujii, M.. Absolutely continuous invariant measures for expanding piecewise linear maps. Invent. Math. 143 (2001), 349–373.CrossRefGoogle Scholar
[19]Tsujii, M.. Fat solenoidal attractors. Nonlinearity 14 (2001), 1011–1027.CrossRefGoogle Scholar
[20]Tsujii, M.. Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194(1) (2005), 37–132.CrossRefGoogle Scholar
[21]Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 (1998), 585–650.CrossRefGoogle Scholar