Exchange transformations reversing orientation

C. Gutierrez; S. Lloyd; B. Pires; E. V. Zhuzhoma; V. S. Medvedev

doi:10.1134/S106456240804008X

Exchange transformations reversing orientation

C. Gutierrez^*
, S. Lloyd
, B. Pires
, E. V. Zhuzhoma
, V. S. Medvedev

^*Corresponding author for this work

Universidade de São Paulo
University of New South Wales
Minin Nizhny Novgorod State Pedagogical University
Lobachevsky State University of Nizhni Novgorod

Research output: Contribution to journal › Article › peer-review

Abstract

Exchange transformations have emerged as an essential object of study in the theory of dynamical systems and ergodic theory. These theories arise in studying flat metrics, measurable foliations, and polygonal billiards. It is observed that any exchange transformation has an invariant Lebesgue measure, coinciding with the standard measure on the interval or the circle. Several aspects associated with the existence of ergodic interval exchange transformations, with flips on the interval are also explained. The definition of an exchange transformation needs to be modified in the presence of flips, to ensure its single-value at the discontinuity points.

Original language	English
Pages (from-to)	500-502
Number of pages	3
Journal	Doklady Mathematics
Volume	78
Issue number	1
DOIs	https://doi.org/10.1134/S106456240804008X
Publication status	Published - Aug 2008
Externally published	Yes

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title = "Exchange transformations reversing orientation",

abstract = "Exchange transformations have emerged as an essential object of study in the theory of dynamical systems and ergodic theory. These theories arise in studying flat metrics, measurable foliations, and polygonal billiards. It is observed that any exchange transformation has an invariant Lebesgue measure, coinciding with the standard measure on the interval or the circle. Several aspects associated with the existence of ergodic interval exchange transformations, with flips on the interval are also explained. The definition of an exchange transformation needs to be modified in the presence of flips, to ensure its single-value at the discontinuity points.",

author = "C. Gutierrez and S. Lloyd and B. Pires and Zhuzhoma, \{E. V.\} and Medvedev, \{V. S.\}",

note = "Funding Information: C. Gutierrez acknowledges the support of FAPESP (grant no. 03/03107-9), CNPq (grant no. 470957/2006-9), and Brazil (grant no. 306328/2006-2). S. Lloyd acknowl- edges the support of FAPESP, grant no. 06/52650-5. B. Pires acknowledges the support of FAPESP, grant no. 06/60600-8 (Brazil). E. V. Zhuzhoma and V. S. Med-vedev acknowledge the support of the Russian Foundation for Basic Research, project nos. 08-01-00547a and 05-01-00501. Funding Information: This work began during E. V. Zhuzhoma{\textquoteright}s visit to Institute of Mathematical and Computer Sciences, University of S{\~a}o Paulo (Brazil) in March of 2007. E. V. Zhuzhoma thanks C. Gutierrez for hospitality and the University of S{\~a}o Paulo for financial support of the visit.",

year = "2008",

month = aug,

doi = "10.1134/S106456240804008X",

language = "English",

volume = "78",

pages = "500--502",

journal = "Doklady Mathematics",

issn = "1064-5624",

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AU - Gutierrez, C.

AU - Lloyd, S.

AU - Pires, B.

AU - Zhuzhoma, E. V.

AU - Medvedev, V. S.

N1 - Funding Information: C. Gutierrez acknowledges the support of FAPESP (grant no. 03/03107-9), CNPq (grant no. 470957/2006-9), and Brazil (grant no. 306328/2006-2). S. Lloyd acknowl- edges the support of FAPESP, grant no. 06/52650-5. B. Pires acknowledges the support of FAPESP, grant no. 06/60600-8 (Brazil). E. V. Zhuzhoma and V. S. Med-vedev acknowledge the support of the Russian Foundation for Basic Research, project nos. 08-01-00547a and 05-01-00501. Funding Information: This work began during E. V. Zhuzhoma’s visit to Institute of Mathematical and Computer Sciences, University of São Paulo (Brazil) in March of 2007. E. V. Zhuzhoma thanks C. Gutierrez for hospitality and the University of São Paulo for financial support of the visit.

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N2 - Exchange transformations have emerged as an essential object of study in the theory of dynamical systems and ergodic theory. These theories arise in studying flat metrics, measurable foliations, and polygonal billiards. It is observed that any exchange transformation has an invariant Lebesgue measure, coinciding with the standard measure on the interval or the circle. Several aspects associated with the existence of ergodic interval exchange transformations, with flips on the interval are also explained. The definition of an exchange transformation needs to be modified in the presence of flips, to ensure its single-value at the discontinuity points.

AB - Exchange transformations have emerged as an essential object of study in the theory of dynamical systems and ergodic theory. These theories arise in studying flat metrics, measurable foliations, and polygonal billiards. It is observed that any exchange transformation has an invariant Lebesgue measure, coinciding with the standard measure on the interval or the circle. Several aspects associated with the existence of ergodic interval exchange transformations, with flips on the interval are also explained. The definition of an exchange transformation needs to be modified in the presence of flips, to ensure its single-value at the discontinuity points.

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Exchange transformations reversing orientation

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