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- Title
Cone exchange transformations and boundedness of orbits.
- Authors
Ashwin, Peter; Goetz, Arek
- Abstract
We introduce a class of two-dimensional piecewise isometries on the plane that we refer to as cone exchange transformations (CETs). These are generalizations of interval exchange transformations (IETs) to 2D unbounded domains. We show for a typical CET that boundedness of orbits is determined by ergodic properties of an associated IET and a quantity we refer to as the 'flux at infinity'. In particular we show, under an assumption of unique ergodicity of the associated IET, that a positive flux at infinity implies unboundedness of almost all orbits outside some bounded region, while a negative flux at infinity implies boundedness of all orbits. We also discuss some examples of CETs for which the flux is zero and/or we do not have unique ergodicity of the associated IET; in these cases (which are of great interest from the point of view of applications such as dual billiards) it remains an outstanding problem to find computable necessary and sufficient conditions for boundedness of orbits.
- Subjects
CONES; ERGODIC theory; MATHEMATICAL transformations; ITERATIVE methods (Mathematics); INTERVAL analysis; MATHEMATICS
- Publication
Ergodic Theory & Dynamical Systems, 2010, Vol 30, Issue 5, p1311
- ISSN
0143-3857
- Publication type
Article
- DOI
10.1017/S0143385709000625