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A Vision for Dynamics in the 21st Century
  • The Legacy of Anatole Katok
  • Edited by Danijela Damjanovic, Boris Hasselblatt, Andrey Gogolev, Yakov Pesin
  • Published online:
    01 February 2024
    Print publication:
    08 February 2024
  • A large international conference celebrated the 50-year career of Anatole Katok and the body of research across smooth dynamics and ergodic theory that he touched. In this book many leading experts provide an account of the latest developments at the research frontier and together set an agenda for future work, including an explicit problem list. This includes elliptic, parabolic, and hyperbolic smooth dynamics, ergodic theory, smooth ergodic theory, and actions of higher-rank groups. The chapters are written in a readable style and give a broad view of each topic; they blend the most current results with the developments leading up to them, and give a perspective on future work. This book is ideal for graduate students, instructors and researchers across all research areas in dynamical systems and related subjects.

  • Hadamard–Perron theorems and effective hyperbolicity

  • VAUGHN CLIMENHAGA, YAKOV PESIN
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 36 / Issue 1 / February 2016
    Published online by Cambridge University Press:
    16 September 2014, pp. 23-63
    Print publication:
    February 2016

  • We prove several new versions of the Hadamard–Perron theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard–Perron theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of ‘effective hyperbolicity’ and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite-orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.

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