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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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