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29 March 2024
 
  » arxiv » 0711.3821

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Unique ergodicity of circle and interval exchange transformations with flips
C. Gutierrez ; S. Lloyd ; V. Medvedev ; B. Pires ; E. Zhuzhoma ;
Date 24 Nov 2007
AbstractWe study circle exchange transformations on three subintervals, that reverse orientation on at least one of the subintervals (a ’flip’). We prove that if such a transformation has exactly one flip, then it has a periodic orbit and thus no dense orbit. For the cases of two and three flips, we construct uniquely ergodic minimal examples by finding a periodic point of an adapted Rauzy--Veech renormalisation operator. Finally, by means of an inductive reasoning, we show that for any $n geq 4$ and $1 leq f leq n$, there exist uniquely ergodic transitive circle exchange transformations on $n$ subintervals with $f$ flips.
Source arXiv, 0711.3821
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