Measures of intermediate entropies for skew product diffeomorphisms
Peng Sun
Discrete & Continuous Dynamical Systems - A 2010, 27(3): 1219-1231 doi: 10.3934/dcds.2010.27.1219
In this paper we study a skew product map $F$ preserving an ergodic measure $\mu$ of positive entropy. We show that if on the fibers the map are $C^{1+\alpha}$ diffeomorphisms with nonzero Lyapunov exponents, then $F$ has ergodic measures of arbitrary intermediate entropies. To construct these measures we find a set on which the return map is a skew product with horseshoes along fibers. We can control the average return time and show the maximal entropy of these measures can be arbitrarily close to $h_\mu(F)$.
keywords: skew product ergodic measure Pesin theory. entropy return map Lyapunov exponents horseshoe
Exponential decay of Lebesgue numbers
Peng Sun
Discrete & Continuous Dynamical Systems - A 2012, 32(10): 3773-3785 doi: 10.3934/dcds.2012.32.3773
We study the exponential rate of decay of Lebesgue numbers of open covers in topological dynamical systems. We show that topological entropy is bounded by this rate multiplied by dimension. Some corollaries and examples are discussed.
keywords: Lipschitz constant topological entropy exponential decay. open cover box dimension Lebesgue number Hausdorff dimension

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