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In this book the seminal 1970 Moscow thesis of Grigoriy A. Margulis is published for the first time. Entitled "On Some Aspects of the Theory of Anosov Systems", it uses ergodic theoretic techniques to study the distribution of periodic orbits of Anosov flows. The thesis introduces the "Margulis measure" and uses it to obtain a precise asymptotic formula for counting periodic orbits. This has an immediate application to counting closed geodesics on negatively curved manifolds. The thesis also contains asymptotic formulas for the number of lattice points on universal coverings of compact manifolds of negative curvature.
The thesis is complemented by a survey by Richard Sharp, discussing more recent developments in the theory of periodic orbits for hyperbolic flows, including the results obtained in the light of Dolgopyat's breakthroughs on bounding transfer operators and rates of mixing.
List of contents
G. Margulis: On Some Aspects of the Theory of Anosov Systems: 1. Some Preliminaries on Anosov Flows.- 2. Behaviour of Lebesgue Measures on Leaves of ~$mathfrak{S}^{l+1}$ under the Action of Anosov Flows.- 3. Construction of Special Measures on Leaves of ~$mathfrak{S}^{l+1}$, $mathfrak{S}^{k+1}$, $mathfrak{S}^l$ and ~$mathfrak{S}^k$.- 4. Construction of a Special Measure on Wn and the properties of the flow {Tt} with this Measure.- 5. Ergodic Properties of.~$mathfrak{S}^k$.- 6. Asymptotics of the Number of Periodic Trajectories.- 7. Some Asymptotical Properties of the Anosov Systems.- Appendix. References. R. Sharp: Periodic Orbits of Hyperbolic Flows: 0. Introduction.- 1. Definition and Results.- 2. Zeta Functions.- 3. Subshifts of Finite Type and Suspended Flows.- 4. Ruelle Transfer Operators.- 5. Extending Zeta Funktions.- 6. Meromorphic Extensions.- 7. Bounds on the Zeta Function and Exponential Error Terms.- 8. Polynomial Error Terms.- 9. Equidistribution Results.- 10. Finite Group Extensions.- 11. Counting with Homological Constraints.- 12. Lalley's Theorem.- 13. Lattice Point Counting.- 14. Manifolds of Non-Positive Curvature.- Appendix A: Symbolic Dynamics.- Appendix B: Livsic Theorems: Cohomology and Periodic Orbits.
