Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds (London Mathematical Society Lecture Note Series)

Pesin Theory Consists Of The Study Of The Theory Of Non-uniformly Hyperbolic Diffeomorphisms. The Aim Of This Book Is To Provide The Reader With A Straightforward Account Of This Theory, Following The Approaches Of Katok And Newhouse. The Notes Are Divided Into Two Parts. The First Develops The Basic Theory, Starting With General Ergodic Theory And Introducing Liapunov Exponents. Part Two Deals With The Applications Of Pesin Theory And Contains An Account Of The Existence (and Distribution) Of Periodic Points. It Closes With A Look At Stable Manifolds, And Gives Some Results On Absolute Continuity. These Lecture Notes Provide A Unique Introduction To Pesin Theory And Its Applications. The Author Assumes That The Reader Has Only A Good Background Of Undergraduate Analysis And Nothing Further, So Making The Book Accessible To Complete Newcomers To The Field. Pt. I. The Basic Theory -- Ch. 1. Invariant Measures And Some Ergodic Theory -- 1.1. Invariant Measures -- 1.2. Poincare Recurrence -- 1.3. Ergodic Measures -- 1.4. Ergodic Decomposition -- 1.5. The Ergodic Theorem -- 1.6. Proof Of The Ergodic Theorem -- 1.7. Proof Of The Ergodic Decomposition Lemma -- Ch. 2. Ergodic Theory For Manifolds And Liapunov Exponents -- 2.1. The Subadditive Ergodic Theorem -- 2.2. The Subadditive Ergodic Theorem And Diffeomorphisms -- 2.3. Oseledec-type Theorems -- 2.4. Some Examples -- 2.5. Proof Of The Oseledec Theorem -- 2.6. Further Refinements Of The Oseledec Theorem -- 2.7. Proof Of The Subadditive Ergodic Theorem -- Ch. 3. Entropy -- 3.1. Measure Theoretic Entropy -- 3.2. Measure Theoretic Entropy And Liapunov Exponents -- 3.3. Topological Entropy -- 3.4. Topological Entropy And Liapunov Exponents -- 3.5. Equivalent Definitions Of Measure Theoretic Entropy -- 3.6. Proof Of The Pesin-ruelle Inequality --^ 3.7. Osceledec's Theorem, Topological Entropy And Lie Theory -- Ch. 4. The Pesin Set And Its Structure -- 4.1. The Pesin Set -- 4.2. The Pesin Set And Liapunov Exponents -- 4.3. Liapunov Metrics On The Pesin Set -- 4.4. Local Distortion -- 4.5. Proofs Of Propositions 4.1 And 4.2 -- 4.6. Liapunov Exponents With The Same Sign -- (a). Some Topical Examples -- (b). Uniformly Hyperbolic Diffeomorphisms -- (i). Shadowing -- (ii). Closing Lemma -- (iii). Stable Manifolds -- Pt. Ii. The Applications -- Ch. 5. Closing Lemmas And Periodic Points -- 5.1. Liapunov Neighborhoods -- 5.2. Shadowing Lemma -- 5.3. Uniqueness Of The Shadowing Point -- 5.4. Closing Lemmas -- 5.5. An Application Of The Closing Lemma -- Ch. 6. Structure Of Chaotic Diffeomorphisms -- 6.1. The Distribution Of Periodic Points -- 6.2. The Number Of Periodic Points -- 6.3. Homoclinic Points -- 6.4. Generalized Smale Horse-shoes -- 6.5. Entropy Stability -- 6.6. Entropy, Volume Growth And Yomdin's Inequality --^ 6.7. Examples Of Discontinuity Of Entropy -- 6.8. Proofs Of Propositions 6.1 And 6.2 -- Ch. 7. Stable Manifolds And More Measure Theory -- 7.1. Stable And Unstable Manifolds -- 7.2. Equality In The Pesin-ruelle Inequality -- 7.3. Foliations And Absolute Continuity -- 7.4. Ergodic Components -- 7.5. Proof Of Stable Manifold Theorem -- 7.6. Ergodic Components And Absolute Continuity -- Appendix A. Some Preliminary Measure Theory -- Appendix B. Some Preliminary Differential Geometry -- Appendix C. Geodesic Flows. Mark Pollicott. Includes Bibliographical References. Pesin theory consists of the study of non-uniformly hyperbolic diffeomorphisms. The aim of this book is to provide the reader with a straightforward account of this theory, following the approaches of Katok and Newhouse. The notes are divided into two parts: the first develops the basic theory, starting with general ergodic theory and introducing Liapunov exponents. Part II deals with the applications of Pesin theory and contains an account of the existence (and distribution) of periodic points; it closes with a look at stable manifolds, and gives some results on absolute continuity. These lecture notes provide a unique introduction to Pesin theory and its applications. The author only assumes that the reader has a good background of undergraduate analysis, making the book accessible to complete newcomers to the field. Pesin theory consists of the study of the theory of non-uniformly hyperbolic diffeomorphisms. The aim of this book is to provide the reader with a straightforward account of this theory, following the approaches of Katok and Newhouse. Emphasis is placed on generality and on the crucial role of measure theory, although no specialist knowledge of this subject is required.