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Invariant densities for piecewise linear maps of the unit interval

Published online by Cambridge University Press:  12 January 2009

PAWEŁ GÓRA
PAWEŁ GÓRA*
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, Montreal, Quebec, Canada H3G 1M8 (email: [email protected])
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Abstract

We find an explicit formula for the invariant density h of an arbitrary eventually expanding piecewise linear map τ of an interval [0,1]. We do not assume that the slopes of the branches are the same and we allow arbitrary number of shorter branches touching zero or touching one or hanging in between. The construction involves the matrix S which is defined in a way somewhat similar to the definition of the kneading matrix of a continuous piecewise monotonic map. Under some additional assumptions, we prove that if 1 is not an eigenvalue of S, then the dynamical system (τ,h⋅m) is ergodic with full support.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 5 , October 2009 , pp. 1549 - 1583
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Alves, J. F., Fachada, J. L. and Sousa Ramos, J.. Detecting topological transitivity of piecewise monotone interval maps. Topology Appl. 153(5–6) (2005), 680–697.CrossRefGoogle Scholar
[2]Boyarsky, A. and Góra, P.. Laws of chaos. Invariant measures and dynamical systems in one dimension. Probability and Its Applications. Birkhäuser, Boston, MA, 1997.Google Scholar
[3]Dajani, K., Hartono, Y. and Kraaikamp, C.. Mixing properties of (α,β)-expansions. Preprint.Google Scholar
[4]Dajani, K. and Kraaikamp, C.. Ergodic Theory of Numbers (Carus Mathematical Monographs, 29). Mathematical Association of America, Washington, DC, 2002.CrossRefGoogle Scholar
[5]Dajani, K. and Kalle, C.. Random β-expansions with deleted digits. Discrete Contin. Dyn. Syst. 18(1) (2007), 199–217.CrossRefGoogle Scholar
[6]Dajani, K. and Kalle, C.. A note on the greedy β-transformations with deleted digits. SMF Séminaires et Congres (19) (2008) to appear.Google Scholar
[7]Dajani, K. and Kalle, C.. A natural extension for the greedy β-transformation with three deleted digits. Preprint arXiv:0802.3571.Google Scholar
[8]Eslami, P.. Eventually expanding maps. Preprint,http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.4715v1.pdf.Google Scholar
[9]Gelfond, A. O.. A common property of number systems (Russian). Izv. Akad. Nauk SSSR. Ser. Mat. 23 (1959), 809–814.Google Scholar
[10]Góra, P.. Invariant densities for generalized β-transformations. Ergod. Th. & Dynam. Sys. 27(5) (2007), 1583–1598.CrossRefGoogle Scholar
[11]Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. II. Israel J. Math. 38(1–2) (1981), 107–115.CrossRefGoogle Scholar
[12]Islam, S.. Absolutely continuous invariant measures of linear interval maps. Int. J. Pure Appl. Math. 27(4) (2006), 449–464.Google Scholar
[13]Kopf, C.. Invariant measures for piecewise linear transformations of the interval. Appl. Math. Comput. 39(2) (1990), 123–144.Google Scholar
[14]Lasota, A. and Mackey, M. C.. Chaos, fractals, and noise. Stochastic Aspects of Dynamics, 2nd edn(Applied Mathematical Sciences, 97). Springer, New York, 1994.Google Scholar
[15]Lasota, A. and Yorke, J. A.. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973), 481–488.CrossRefGoogle Scholar
[16]Li, T. Y. and Yorke, J. A.. Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc. 235 (1978), 183–192.CrossRefGoogle Scholar
[17]Milnor, J. and Thurston, W.. On iterated maps of the interval. Dynamical Systems (College Park, MD, 1986–87) (Lecture Notes in Mathematics, 1342). Springer, Berlin, 1988, pp. 465–563.Google Scholar
[18]Henryk, M.. Nonnegative Matrices. Wiley, New York, 1988.Google Scholar
[19]Parry, W.. On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.CrossRefGoogle Scholar
[20]Parry, W.. Representations for real numbers. Acta Math. Acad. Sci. Hungar. 15 (1964), 95–105.CrossRefGoogle Scholar
[21]Pedicini, M.. Greedy expansions and sets with deleted digits. Theoret. Comput. Sci. 332(1–3) (2005), 313–336.CrossRefGoogle Scholar
[22]Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493.CrossRefGoogle Scholar