Vered ROM-KEDAR and D. TURAEV

Department of Applied Mathematics 
and Computer Science
Weizmann institute of Science
P.O.B. 26, Rehovot, 76100
ISRAEL
vered@wisdom.weizmann.ac.il
 

Big islands appearing in near-ergodic flows

Abstract : We prove the existence and derive a rigorous estimate of the size of islands (in both phase space and parameter space) appearing in smooth Hamiltonian approximations of scattering billiards. The derivation includes the construction of a local return map near singular periodic orbits for an arbitrary scattering billiard and for the general smooth billiard potentials. Thus, {\it universality} classes for the local behavior are found. Moreover, for all scattering geometries and for many types of natural potentials which limit to the billiard flow as a parameter $\eps \goto 0$, islands of {\it polynomial} size in $\eps$ appear. This suggests that the loss of ergodicity via the introduction of the physically relevant effect of smoothening of the potential in modeling, for example, scattering molecules, may be of physically noticeable effect. Generalizations to higher dimensions will be discussed.