We present the first natural and visible examples of Hamiltonian systems with divided phase space allowing a rigorous mathematical analysis. The simplest such family (mushrooms) demonstrates a continuous transition from a completely chaotic system (stadium) to a completely integrable one (circle). In the course of this transition, an integrable island appears, grows and finally occupies the entire phase space. We also give the first examples of billiards with a “chaotic sea” (one ergodic component) and an arbitrary (finite or infinite) number of KAM islands and the examples with arbitrary (finite or infinite) number of chaotic (ergodic) components with positive measure coexisting with an arbitrary number of islands. Among other results is the first example of completely understood (rigorously studied) billiards in domains with a fractal boundary.

1.
L.
Markus
and
K. R.
Meyer
, “
Generic Hamiltonian dynamical systems are neither integrable nor ergodic
,”
Mem. Am. Math. Soc.
144
,
1
–
52
(
1978
).
Google Scholar
 
2.
L. A.
Bunimovich
and
J.
Rehacek
, “
How high-dimensional stadia look like
,”
Commun. Math. Phys.
197
,
277
–
301
(
1998
).
Google Scholar
Crossref
Search ADS
 
3.
L. A.
Bunimovich
and
J.
Rehacek
, “
On the ergodicity of high-dimensional focusing billiards
,”
Ann. Inst. Henri Poincaré
68
,
421
–
448
(
1999
).
Google Scholar
 
4.
L. A.
Bunimovich
, “
A theorem on ergodicity of two-dimensional hyperbolic billiards
,”
Commun. Math. Phys.
130
,
599
–
621
(
1990
).
Google Scholar
Crossref
Search ADS
 
5.
N. I.
Chernov
and
C.
Haskell
, “
Non-uniformly hyperbolic K-systems are Bernoulli
,”
Ergod. Theory Dyn. Syst.
16
,
19
–
44
(
1996
).
Google Scholar
Crossref
Search ADS
 
6.
R. Markarian, “Billiards with polynomial decay of correlations,” preprint, Universidad de la Republica, Montevideo, 2001.
7.
L. A. Bunimovich (in preparation).
This content is only available via PDF.
© 2001 American Institute of Physics.
2001
American Institute of Physics
You do not currently have access to this content.