On the Relation between Topological Entropy and Entropy Dimension

P. S. Saltykov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: For the Lipschitz mapping of a metric compact set into itself, there is a classical upper bound on topological entropy, namely, the product of the entropy dimension of the compact set by the logarithm of the Lipschitz constant. The Ghys conjecture is that, by varying the metric, one can approximate the upper bound arbitrarily closely to the exact value of the topological entropy. In the present paper, we obtain a criterion for the validity of the Ghys conjecture for an individual mapping. Applying this criterion, we prove the Ghys conjecture for hyperbolic mappings.

Keywords: topological entropy, topological dimension, Lipschitz mapping, Ghys conjecture, hyperbolic mapping, hyperbolic homeomorphism.

UDC: 517

Received: 24.10.2006
Revised: 12.05.2008

DOI: 10.4213/mzm4007

 English version: Mathematical Notes, 2009, 86:2, 255–263

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