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Counting preimages

Published online by Cambridge University Press:  24 January 2017

MICHAŁ MISIUREWICZ  and
ANA RODRIGUES
MICHAŁ MISIUREWICZ
Affiliation:
Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, IN 46202, USA email [email protected]
ANA RODRIGUES
Affiliation:
Department of Mathematics, University of Exeter, Harrison Building, Streatham Campus, North Park Road, Exeter, EX4 4QF, UK email [email protected]
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Abstract

For non-invertible maps, subshifts that are mainly of finite type and piecewise monotone interval maps, we investigate what happens if we follow backward trajectories, which are random in the sense that, at each step, every preimage can be chosen with equal probability. In particular, we ask what happens if we try to compute the entropy this way. It turns out that, instead of the topological entropy, we get the metric entropy of a special measure, which we call the fair measure. In general, this entropy (the fair entropy) is smaller than the topological entropy. In such a way, for the systems that we consider, we get a new natural measure and a new invariant of topological conjugacy.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 5 , August 2018 , pp. 1837 - 1856
Copyright
© Cambridge University Press, 2017 

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References

Alsedà, Ll., Llibre, J. and Misiurewicz, M.. Combinatorial Dynamics and Entropy in Dimension One (Advanced Series in Nonlinear Dynamics, 5) , 2nd edn. World Scientific, Singapore, 2000.Google Scholar
Alsedà, Ll. and Misiurewicz, M.. Semiconjugacy to a map of a constant slope. Discrete Contin. Dyn. Syst. Ser. B 20 (2015), 3403–3413.Google Scholar
Barnsley, M. F.. Fractals Everywhere, 2nd edn. Academic Press Professional, Boston, MA, 1993.Google Scholar
Block, L. and Coppel, W.. Dynamics in One Dimension (Lecture Notes in Mathematics, 1513) . Springer, Berlin, 1992.CrossRefGoogle Scholar
Block, L., Keesling, J., Li, S. H. and Peterson, K.. An improved algorithm for computing topological entropy. J. Statist. Phys. 55 (1989), 929–939.Google Scholar
Bruin, H.. An algorithm to compute the topological entropy of a unimodal map. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), 1881–1882.CrossRefGoogle Scholar
Dilao, R. and Amigo, J.. Computing the topological entropy of unimodal maps. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22 (2012), 1250152, 14pp.CrossRefGoogle Scholar
Hawkins, J. and Taylor, M.. Maximal entropy measure for rational maps and a random iteration algorithm. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 1442–1447.Google Scholar
Hofbauer, F. and Keller, G.. Equilibrium states for piecewise monotonic transformations. Ergod. Th. & Dynam. Sys. 2(1) (1982), 23–43.Google Scholar
Keane, M.. Strongly mixing g-measures. Invent. Math. 16 (1972), 309–324.Google Scholar
Ledrappier, F.. Principe variationnel et systmes dynamiques symboliques. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 30 (1974), 185–202.Google Scholar
Misiurewicz, M. and Roth, S.. No semiconjugacy to a map of constant slope. Ergod. Th. & Dynam. Sys. 36 (2016), 875–889.CrossRefGoogle Scholar
Parry, W.. Entropy and Generators in Ergodic Theory. W. A. Benjamin, Inc., New York, 1969.Google Scholar
Walters, P.. Ruelle’s operator theorem and g-measures. Trans. Amer. Math. Soc. 214 (1975), 375–387.Google Scholar