scholarly journals Rank two interval exchange transformations

1988 ◽  
Vol 8 (3) ◽  
pp. 379-394 ◽  
Author(s):  
Michael D. Boshernitzan
Keyword(s):  
Unique Ergodicity ◽  
Interval Exchange ◽  
Linear Rank ◽  
Interval Exchange Transformations ◽  
Rank 2

AbstractWe consider interval exchange transformations T for which the lengths of the exchanged intervals have linear rank 2 over the field of rationals. We prove that, for such T, minimality implies unique ergodicity. We also provide an algorithm which tests T for aperiodicity and minimality.

A condition for unique ergodicity of minimal symbolic flows

1992 ◽  
Vol 12 (3) ◽  
pp. 425-428 ◽  
Author(s):  
Michael D. Boshernitzan
Keyword(s):  
Unique Ergodicity ◽  
Interval Exchange ◽  
Sufficient Condition ◽  
Interval Exchange Transformations ◽  
Special Case

AbstractA sufficient condition for unique ergodicity of symbolic flows is provided. In an important but special case of interval exchange transformations, the condition has already been validated by W. Veech.

scholarly journals Cone exchange transformations and boundedness of orbits

2009 ◽  
Vol 30 (5) ◽  
pp. 1311-1330 ◽  
Author(s):  
PETER ASHWIN ◽  
AREK GOETZ
Keyword(s):  
Sufficient Conditions ◽  
Point Of View ◽  
Unique Ergodicity ◽  
Interval Exchange ◽  
Ergodic Properties ◽  
Interval Exchange Transformations ◽  
Piecewise Isometries ◽  
Almost All ◽  
Necessary And Sufficient ◽  
Negative Flux

AbstractWe 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.

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