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 on the space of -expandings. We also give an explicit estimate of the rate of mixing for -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 A diffeomorphisms. It also can be adopted to show that the entropy of the quadratic family 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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DOI:
https://doi.org/10.1090/S0002-9947-1989-0958899-6

Article copyright:
© Copyright 1989
American Mathematical Society