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Conditional stability and real analytic pseudo-Anosov maps. (English) Zbl 0572.58018

Mem. Am. Math. Soc. 321, 116 p. (1985).
In this book the author studies the class of smooth pseudo-Anosov maps on surfaces. She proves that they are structurally stable if the perturbations are restricted by the condition that certain k-jets at singular points are fixed. This result allows the construction of real analytic models for pseudo-Anosov maps. Further on any surface there are real analytic diffeomorphisms having Bernoullian property with respect to a smooth measure. Similar results are obtained in the case of the two dimensional disk: there exists a real analytic Bernoullian diffeomorphism of the disk. Hence neither the topology of the underlying manifold nor real analyticity restricts a diffeomorphism from having the Bernoullian property.
Previous results in this direction were obtained by Katok and the author. The exposition in the book is very clear and largely independent. The methods used belong to the theory of nonuniform hyperbolicity.
Reviewer: M.Wojtkowski

MSC:

37D99 Dynamical systems with hyperbolic behavior
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37A99 Ergodic theory
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