Hostname: page-component-cb9f654ff-mwwwr Total loading time: 0 Render date: 2025-08-19T02:42:39.921Z Has data issue: false hasContentIssue false
  • English
  • Français

Discontinuity of topological entropy for Lozi maps

Published online by Cambridge University Press:  16 September 2011

IZZET BURAK YILDIZ
IZZET BURAK YILDIZ*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA (email: yildiz@cbs.mpg.de)
Get access Rights & Permissions [Opens in a new window]

Abstract

Recently, Buzzi [Maximal entropy measures for piecewise affine surface homeomorphisms. Ergod. Th. & Dynam. Sys.29 (2009), 1723–1763] showed in the compact case that the entropy map f→htop(f) is lower semi-continuous for all piecewise affine surface homeomorphisms. We prove that topological entropy for Lozi maps can jump from zero to a value above 0.1203 as one crosses a particular parameter and hence it is not upper semi-continuous in general. Moreover, our results can be extended to a small neighborhood of this parameter showing the jump in the entropy occurs along a line segment in the parameter space.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 5: Daniel J. Rudolph – in Memoriam , October 2012 , pp. 1783 - 1800
Copyright
Copyright © Cambridge University Press 2011

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.)

Article purchase

Temporarily unavailable

References

[1]Buzzi, J.. Maximal entropy measures for piecewise affine surface homeomorphisms. Ergod. Th. & Dynam. Sys. 29 (2009), 1723–1763.Google Scholar
[2]Galias, Z.. Obtaining rigorous bounds for topological entropy for discrete time dynamical systems. Proceedings of the International Symposium on Nonlinear Theory and Applications (NOLTA’02), Xi’an, PR China, 2002, pp. 619–622.Google Scholar
[3]Galias, Z. and Zygliczyński, P.. Abundance of homoclinic and heteroclinic orbits and rigorous bounds for the topological entropy for the Hénon map. Nonlinearity 14 (2001), 909–932.Google Scholar
[4]Hénon, M.. A two-dimensional mapping with a strange attractor. Comm. Math. Phys. 50 (1976), 69–77.Google Scholar
[5]Ikeda, K., Daido, H. and Akimoto, O.. Optical turbulence: chaotic behavior of transmitted light from a ring cavity. Phys. Rev. Lett. 45 (1980), 709–712.CrossRefGoogle Scholar
[6]Ishii, Y. and Sands, D.. Monotonicity of the Lozi family near the tent-maps. Comm. Math. Phys. 198 (1998), 397–406.CrossRefGoogle Scholar
[7]Ishii, Y. and Sands, D.. Lap number entropy formula for piecewise affine and projective maps in several dimensions. Nonlinearity 20(18) (2007), 2755–2772.CrossRefGoogle Scholar
[8]Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51(1) (1980), 137–173.Google Scholar
[9]Katok, A.. Nonuniform hyperbolicity and structure of smooth dynamical systems. Proc. Intl. Congress of Math. 2 (1983), 1245–1254.Google Scholar
[10]Misiurewicz, M.. On non-continuity of topological entropy. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astro. Phys. 19(4) (1971), 319–320.Google Scholar
[11]Misiurewicz, M.. Diffeomorphisms without any measure with maximal entropy. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astro. Phys. 21(10) (1973), 903–910.Google Scholar
[12]Misiurewicz, M.. Jumps of entropy in one dimension. Fund. Math. 132(3) (1989), 215–226.CrossRefGoogle Scholar
[13]Misiurewicz, M. and Szlenk, W.. Entropy of piecewise monotone mappings. Studia Math. 67(1) (1980), 45–63.CrossRefGoogle Scholar
[14]Newhouse, S.. Continuity properties of entropy. Ann. of Math. (2) 129 (1989), 215–235.CrossRefGoogle Scholar
[15]Newhouse, S., Berz, M., Grote, J. and Makino, K.. On the estimation of topological entropy on surfaces. Contemp. Math. 469 (2008), 243–270.Google Scholar
[16]Rees, M.. A minimal positive entropy homeomorphism of the 2-torus. J. Lond. Math. Soc. (2) 23 (1981), 537–550.Google Scholar
[17]Yildiz, I. B.. Monotonicity of the Lozi family and the zero entropy locus. Nonlinearity 24 (2011), 1613–1628.Google Scholar
[18]Yomdin, Y.. Volume growth and entropy. Israel J. Math. 57 (1987), 285–300.Google Scholar
[19]Zygliczyński, P.. Computer assisted proof of chaos in the Rossler equations and the Hénon map. Nonlinearity 10(1) (1997), 243–252.Google Scholar