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Transactions of the American Mathematical Society

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Regularity of the metric entropy for expanding maps

Author: Marek Rychlik
Journal: Trans. Amer. Math. Soc. 315 (1989), 833-847
MSC: Primary 28D05; Secondary 28D20, 58F11
MathSciNet review: 958899
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Abstract: The main result of the current paper is an estimate of the radius of the nonperipheral part of the spectrum of the Perron-Frobenius operator for expanding mappings. As a consequence, we are able to show that the metric entropy of an expanding map has modulus of continuity $ x\log (1/x)$ on the space of $ {C^2}$-expandings. We also give an explicit estimate of the rate of mixing for $ {C^1}$-functions in terms of natural constants.

It seems that the method we present can be generalized to other classes of dynamical systems, which have a distinguished invariant measure, like $ \operatorname{Axiom}$   A diffeomorphisms. It also can be adopted to show that the entropy of the quadratic family $ {f_\mu }(x) = 1 - \mu {x^2}$ computed with respect to the absolutely continuous invariant measure found in Jakobson's Theorem varies continuously (the last result is going to appear somewhere else).

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Article copyright: © Copyright 1989 American Mathematical Society

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