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 ρ = (2/π) arccos r. Almost all ρ 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 ρ is MUPOfree 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 ρ has an eventually periodic continued fraction expansion, that is, a quadratic irrational, t^2 P(t) converges to a logperiodic 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.
Original language  English 

Title of host publication  Dynamical Systems, Ergodic Theory, and Probability: in Memory of Kolya Chernov 
Subtitle of host publication  Conference Dedicated to the Memory of Nikolai Chernov Dynamical Systems, Ergodic Theory, and Probability May 18–20, 2015 University of Alabama at Birmingham, Birmingham, Alabama 
Editors  Alexander M Bloch, Leonid A Bunimovich, Paul H Jung, Lex G Oversteegen, Yakov G Sinai 
Publisher  American Mathematical Society 
Pages  111128 
Number of pages  18 
ISBN (Electronic)  9781470442248 
ISBN (Print)  9781470427733 
DOIs  
Publication status  Published  30 Sep 2017 
Publication series
Name  Contemporary Mathematics 

Publisher  American Mathematical Society 
Volume  698 
ISSN (Print)  02714132 
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Professor Carl P Dettmann
 Probability, Analysis and Dynamics
 School of Mathematics  Professor of Applied Mathematics
 Mathematical Physics
Person: Academic , Member, Group lead