On Some Aspects of the Theory of Anosov Systems: With a Survey by Richard Sharp: "Periodic Orbits of Hyperbolic Flows" (Springer Monographs in Mathematics) Review

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On Some Aspects of the Theory of Anosov Systems: With a Survey by Richard Sharp: "Periodic Orbits of Hyperbolic Flows" (Springer Monographs in Mathematics) ReviewThis is Gregory Margulis' long-unpublished thesis, which contains much of subsequent work in dynamics (can't blame the practitioners, since even the Russian speakers among them could not get their hands on the thesis) -- for example, after a very well-known paper (one of whose authors later won the Fields medal, and the other probably should have) was published in the early 1990s, one of the authors found at his doorstep a copy of the original Russian version of the thesis, and was slightly embarrassed to find himself scooped by Margulis by a quarter century.
Sharp's survey is quite nice, though the reader is encouraged to find the Parry/Pollicott Asterisque volume on dynamical zeta functions in his/her university library.On Some Aspects of the Theory of Anosov Systems: With a Survey by Richard Sharp: "Periodic Orbits of Hyperbolic Flows" (Springer Monographs in Mathematics) OverviewThe seminal 1970 Moscow thesis of Grigoriy A. Margulis, 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.

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