Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-17T15:02:17.570Z 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: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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