Table of contents

CONTENTS Introduction § 1. Preliminaries from measure theory § 2. Isometric operators § 3. Measure-preserving transformations § 4. Entropy of a measurable partition § 5. Mean conditional entropy § 6. Spaces of partitions § 7. Fundamental lemmas § 8. Properties of the function h(T, ξ) § 9. Entropy of an endomorphism § 10. Existence of generators § 11. Automorphisms with zero entropy § 12. The theory of invariant partitions § 13. Endomorphisms with completely positive entropy § 14. Entropy and the spectrum § 15. Entropy and mixing § 16. Entropy and the isomorphism problem Appendix. S. A. Yuzvinskii, Metric properties of endomorphisms of locally-compact groups References

CONTENTS § 1. Basic definitions § 2. Some examples § 3. Cohomology of dynamical systems § 4. Dynamical systems and factors References

CONTENTS Introduction Part I. The method of approximations § 1. Definitions and examples § 2. Approximations, ergodicity and mixing § 3. Approximations and the spectrum § 4. Approximations and entropy § 5. Fibre bundles § 6. Flows § 7. Some unsolved problems Part II. Applications § 8. Shifting of intervals § 9. The group property of the spectrum § 10. Square roots of automorphisms § 11. Flows on a two-dimensional torus § 12. Entropy of classical dynamical systems References

CONTENTS Introduction Lecture 1. The Maupertuis-Lagrange-Jacobi principle and reduction of a dynamical system to a geodesic flow. Some general properties of smooth dynamical systems Lecture 2. Y-systems Lecture 3. Verification of the Y-conditions for a geodesic flow on manifolds of negative curvature Lecture 4. Transversal foliations Lecture 5. Measurability and absolute continuity of transversal foliations for Y-systems Conclusion Appendix. G. A. Margulis, Y-flows on three-dimensional manifolds References