Upcoming seminar:
A classical insight in dynamical systems is that strongly chaotic deterministic dynamics often behaves, in many respects, like a random process. Most results in this direction are asymptotic: they describe what happens as time tends to infinity. In this talk, I will discuss a complementary question: how much can one say about finite-time behavior in chaotic and random systems?
Focusing on first hitting probabilities for different parts of phase space, I will explain how this problem can be translated into symbolic dynamics, and in fair-dice-like systems into a combinatorial problem on words. Previous work [1] showed that, for words of the same length, there is an initial interval during which the hierarchy of first hitting probabilities is fixed, and that the length of this interval grows at least linearly with the word length. We prove that this growth is in fact exponential [2]. I will also briefly discuss the role of autocorrelation, the main ideas of the proof, and possible extensions.
Joint work with Leonid Bunimovich.
[1] M. Bolding and L. A. Bunimovich, Nonlinearity 32 (2019), 1731.
[2] L. Bunimovich and K. Kovalenko, "For how long time evolution of chaotic or random systems can be predicted," arXiv:2512.16186, 2025.