动力系统系列讨论班—Finite pseudotrajectories and probabilistic aspect
主 题: 动力系统系列讨论班—Finite pseudotrajectories and probabilistic aspect
报告人: Sergey Tikhomirov (St. Petersburg Univ., Russia.)
时 间: 2017-04-18 15:10-17:00
地 点: 三教108
Abstract: The celebrated shadowing lemma states that in a neighborhood of a hyperbolic set diffeomorphism satisfies the shadowing property. In fact this propetry is Lipschitz. In 2010 jointly with Pilyugin we proved that Lipschitz shadowing is equivalent to hyperbolicity (structural stability). This statement raises a question which type of shadowing can have a non-hyperbolic system.
Based on numerical simulation Hammel, Grebogi and Yorke conjectured that for a wide class of nonuniformly hyperbolic systems a $d$-pseudotrajectory of length $1/\sqrt{d}$ can be $\sqrt{d}$ shadowed. In this talk we show that this conjecture cannot be improved. We also present a probabilistic approach to this problem which gives a hope to prove a statement similar to Hammel-Grebogi-Yorke conjecture and prove its weak form for a special case of linear skew product.
