Rotational Entropy for Random Torus Maps

Abstract

In this paper, the rotational entropy \(h_r(\varphi )\) is investigated for a random dynamical system \(\varphi \) on the torus. The formula of \(h_r(\varphi )\) is obtained for \(\varphi \) which satisfies certain assumptions, and the lower and upper bounds of \(h_r(\varphi )\) are given for more general \(\varphi \). Several examples are presented to show that these results may not hold without the assumptions. This work can be seen as a random version of the previous work (Jiang et al. in J Differ Equ 379:862–883, 2024), in which the rotational entropy was introduced and investigated as a homotopy invariant for any torus map.