Yuval Peres: Highlights from the History of Ergodic Theory | Tsinghua University, Beijing скачать в хорошем качестве

Yuval Peres: Highlights from the History of Ergodic Theory | Tsinghua University, Beijing

Speaker: Yuval Peres, BIMSA Date: April, 2025 Abstract: Ergodic Theory studies the long term behavior of dynamical systems. It originated from two main themes: the phenomenon of recurrence, discovered by Poincare (1890), stating that almost all orbits return arbitrarily near their starting points, and the ergodic hypothesis of Boltzman (1887), that time averages converge to space averages; the latter was famously established by Von Neumann and Birkhoff in 1932. Levy (1935) applied the ergodic theorem to the Gauss map and obtained key results on the continued fraction expansions of typical real numbers. The problem of identifying which dynamical systems are isomorphic, was solved in many cases by Kolmogorov, Sinai and Ornstein between 1955 and 1970 using entropy (building on the ideas of Shannon from information theory). The Markov partitions pioneered by Adler and Weiss (1967) were extended by Sinai (1968) and Bowen (1973) to a powerful tool in smooth dynamics. Kingman's (1973) subadditive ergodic theorem yielded important connections to probability theory, and Furstenberg's (1977) refinements of Poincare recurrence uncovered rich connections to combinatorial number theory. In the lecture I will discuss these breakthroughs and some of their modern developments