# Topological Theory of Dynamical Systems

**Recent Advances**

**By**

- N. Aoki, Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo, Japan
- K. Hiraide, Department of Mathematics, Faculty of Science, Ehime University, Matsuyama, Japan

This monograph aims to provide an advanced account of some aspects of dynamical systems in the framework of general topology, and is intended for use by interested graduate students and working mathematicians. Although some of the topics discussed are relatively new, others are not: this book is not a collection of research papers, but a textbook to present recent developments of the theory that could be the foundations for future developments.

This book contains a new theory developed by the authors to deal with problems occurring in diffentiable dynamics that are within the scope of general topology. To follow it, the book provides an adequate foundation for topological theory of dynamical systems, and contains tools which are sufficiently powerful throughout the book.

Graduate students (and some undergraduates) with sufficient knowledge of basic general topology, basic topological dynamics, and basic algebraic topology will find little difficulty in reading this book.

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### Book information

- Published: June 1994
- Imprint: NORTH-HOLLAND
- ISBN: 978-0-444-89917-0

### Table of Contents

Introduction. **1. Some Properties of Anosov Systems. **Toral endomorphisms. **2. Dynamics of Continuous Maps. **Self-covering maps. Expansivity. Psuedo orbit tracing property. Topological Anosov maps (TA-maps). **3. Nonwandering Sets. **Chain recurrent sets. Stable and unstable sets. Recurrent sets and Birkhoff centers. Nonwandering sets of TA-maps. Inverse limit systems. **4. Markov Partitions. **Markov partitions and subshifts. Construction of Markov partitions. Symbolic dynamics. **5. Local Product Structures. **Stable sets in strong sense. Local product structures for TA-covering maps. Expanding factors of TA-Maps. Subclasses of the class of TA-maps. **6. TA-Covering Maps. **Fundamental groups. Universal covering spaces. Covering transformation groups. S-injectivity of TA-covering maps. Structure groups for inverse limit systems. Lifting of local product structures. TA-covering maps of closed topological manifolds. Classification of TA-covering maps on tori. **7. Solenoidal Groups and Self-Covering Maps. **Geometrical structures of solenoidal groups. Inverse limit systems of self-covering maps on tori. **8. TA-Covering Maps of Tori. **Toral endomorphisms homotopic to TA-covering maps. Construction of semi-conjugacy maps. Nonwandering sets. Injectivity of semi-conjugacy maps. Proof of theorem 6.8.1. Proof of theorem 6.8.2. Remarks. **9. Perturbations of Hyperbolic Toral Endomorphisms. **TA-C∞ regular maps that are not Anosov. One-parameter families of homeomorphisms. **10. Fixed Point Indices. **Chain complexes. Singular homology. Euclidean neighbourhood retracts (ENRs). Fixed point indices. Lefschetz numbers. Orientability of manifolds. Orientability of generalized foliations. Fixed point indices of expanding maps. Fixed point indices of TA-covering maps. **11. Foundations of Ergodic Theory. **Measure theory. Measure-preserving transformations. Ergodic theorems. Probability measures of compact metric spaces. Applications to topological dynamics. **References. Index.**