Dynamics of a piecewise rotation

We investigate the dynamics of systems generalizing interval exchanges to planar mappings. Unlike interval exchanges and translations, our mappings, despite the lack of hyperbolicity, exhibit many features of attractors. The main result states that for a certain class of noninvertible piecewise isometries, orbits visiting both atoms infinitely often must accumulates on the boundaries of the attractor consisting of two maximal invariant discs $D_0 \cup D_1$ fixed by $T$. The key new idea is a dynamical and geometric observation about the monotonic behavior of orbits of a certain first-return map. Our model emerges as the local map for other piecewise isometries and can be the basis for the construction of more complicated molecular attractors.

[1]  C. Boldrighini,et al.  Billiards in Polygons , 1978 .

[2]  Harvey B. Keynes,et al.  A “minimal”, non-uniquely ergodic interval exchange transformation , 1976 .

[3]  M. Keane,et al.  Non-ergodic interval exchange transformations , 1977 .

[4]  A. Katok.  Interval exchange transformations and some special flows are not mixing , 1980 .

[5]  S. Kerckhoff.  Simplicial systems for interval exchange maps and measured foliations , 1985, Ergodic Theory and Dynamical Systems.

[6]  H. Masur.  Interval Exchange Transformations and Measured Foliations , 1982 .

[7]  M. Boshernitzan,et al.  Interval translation mappings , 1995, Ergodic Theory and Dynamical Systems.

[8]  G. Cairns,et al.  Riemannian Foliations , 1988 .

[9]  A. Katok.  The growth rate for the number of singular and periodic orbits for a polygonal billiard , 1987 .

[10]  A. Goetz.  Dynamics of piecewise isometries , 2000 .

[11]  Hans Haller.  Rectangle exchange transformations , 1981 .

[12]  M. Boshernitzan.  A condition for minimal interval exchange maps to be uniquely ergodic , 1985 .

[13]  D. Ornstein,et al.  Cutting and stacking, interval exchanges and geometric models , 1985 .

[14]  M. Keane.  Interval exchange transformations , 1975 .

[15]  W. Veech.  Gauss measures for transformations on the space of interval exchange maps , 1982 .

[16]  N. Haydn,et al.  Topological entropy of generalized polygon exchanges , 1995, math/9501227.

[17]  M. Rees.  An alternative approach to the ergodic theory of measured foliations on surfaces , 1981, Ergodic Theory and Dynamical Systems.

[18]  Gilbert Levitt,et al.  La dynamique des pseudogroupes de rotations , 1993 .

[19]  N. Haydn,et al.  Topological entropy of polygon exchange transformations and polygonal billiards , 1997, Ergodic Theory and Dynamical Systems.