How sticky is the chaos/order boundary?

TL;DR

This paper investigates the boundary stickiness in mushroom billiards, revealing how the probability of long trapping times depends on the Diophantine properties of the boundary parameter, with complex behaviors linked to continued fraction expansions.

Contribution

It provides a detailed analysis of how boundary stickiness varies with the Diophantine properties of the boundary parameter in mushroom billiards, especially for MUPO-free cases.

Findings

P(t) ~ C/t for MUPO-dominated cases

t^2 P(t) is bounded for MUPO-free cases with bounded partial quotients

Log-periodic convergence of t^2 P(t) for quadratic irrationals

Abstract

In dynamical systems with divided phase space, the vicinity of the boundary between regular and chaotic regions is often "sticky," that is, trapping orbits from the chaotic region for long times. Here, we investigate the stickiness in the simplest mushroom billiard, which has a smooth such boundary, but surprisingly subtle behaviour. As a measure of stickiness, we investigate P(t), the probability of remaining in the mushroom cap for at least time t given uniform initial conditions in the chaotic part of the cap. The stickiness is sensitively dependent on the radius of the stem r via the Diophantine properties of rho = (2/\pi) arccos r. Almost all rho give rise to families of marginally unstable periodic orbits (MUPOs) where P(t) ~ C/t, dominating the stickiness of the boundary. Here we consider the case where rho is MUPO-free and has continued fraction expansion with bounded partial quotients. We show that t^2 P(t) is bounded, varying infinitely often between values whose ratio is at least 32/27. When rho has an eventually periodic continued fraction expansion, that is, a quadratic irrational, t^2 P(t) converges to a log-periodic function. In general, we expect less regular behaviour, with upper and lower exponents lying between 1 and 2. The results may shed light on the parameter dependence of boundary stickiness in annular billiards and generic area preserving maps.