Mathematical Research Letters

Volume 17 (2010)

Number 5

The maximal entropy measure detects non-uniform hyperbolicity

Pages: 851 – 866



Juan Rivera-Letelier (Facultad de Matemáticas, P. Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile)


We characterize two of the most studied non-uniform hyperbolicity conditions for rational maps, semi-hyperbolicity and the topological Collet-Eckmann condition, in terms of the maximal entropy measure. With the same tools we give an extension of the result of Carleson, Jones and Yoccoz that semi-hyperbolicity characterizes those polynomial maps whose basin of attraction of infinity is a John domain, to rational maps having a completely invariant attracting basin.

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