Mendoza D'Paola, Leonardo (1983) Entropy of diffeomorphisms of surfaces. PhD thesis, University of Warwick.

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Abstract

We study the measure-theoretic and topological entropies of diffeo- morphisms of surfaces. In the measure theoretic case we look for relations between Lyapunov exponents, Hausdorff dimension and the entropy of ergodic invariant measures.

First we describe the concept of measure-theoretic entropy in topological terms and discuss a general method of relating it with the Hausdorff dimension of ergodic invariant measures. This is done in a general setting, namely Lipschitz maps of compact metric spaces. The rest of the thesis is mainly directed to the study of diffeomorphisms of surfaces.

To apply a refinement of this general method to C2 diffeomorphisms of surfaces we need Pesin's theory of non-uniform hyperbolicity, which we review in Chapter 2. Also in this chapter, we prove that the topological pressure of certain functions can be approximated by its restriction to the hyperbolic sets of the diffeomorphisms. This result is used in Chapter 3 to study the size of sets of generic points of ergodic measures supported on hyperbolic sets.

The main result of Chapter 3 is that if µ is an ergodic Borel f- invariant measure for a diffeomorphism f:M -► M of a surface M . Then, provided the entropy hµ(f) > 0 , the Hausdorff dimension of the set of generic points of µ is at least 1 + hµ (f)xtµ , where xtµ is the positive Lyapunov exponent of µ.

In Chapter 4 we prove that if the family of local stable manifolds is Lipschitz, then for an ergodic measure µ, hµ (f)=HD(µϩx)X+ µ for almost every X ε M . Here f is as 1n Chapter 3 and µϩx is a quotient measure defined by the family of local stable manifolds.

Finally, Chapter 5 is devoted to study the topological entropy of homoclinic closures by 'counting1 homoclinic orbits.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Diffeomorphisms, Surfaces
Official Date:	June 1983
Dates:	Date Event June 1983 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Manning, Anthony, 1946-
Sponsors:	Fundación Gran Mariscal de Ayacucho‏ ; Universidad Centro Occidental Lisandro Alvarado
Format of File:	pdf
Extent:	[5], vi, 131 leaves : illustrations
Language:	eng

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