Minimality and the Rauzy – Veech algorithm for interval exchange transformations with flips

Hernández, Carlos Angosto; López, Gabriel Soler

doi:10.1080/14689367.2013.824950

Cited by 6 publications

(8 citation statements)

References 10 publications

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Section: Rauzy Inductionmentioning

confidence: 99%

“…Proof This was proven [, Theorem A]. Our notation differs slightly from that of , for us infiniteness of the path means that the Rauzy induction is applicable infinite amount of times without finishing in the hole. Our assumption is exactly equivalent to the two assumptions of Theorem A that all the permutations obtained during the Rauzy induction application are irreducible and the Rauzy induction in their sense is applicable an infinite number of times.

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…”

Section: Rauzy Inductionmentioning

confidence: 99%

“…Proof An interval gets smaller at a step of the Rauzy induction if and only if the letter labeling this interval is a winner. [, Proposition 10] implies that all the intervals are getting smaller if the path is long enough, thus each letter in the alphabet must be a winner at least once. On the other hand, every path which is a concatenation of at least

2 n - 3

complete paths is positive.…”

Section: Rauzy Inductionmentioning

confidence: 99%

“…The first example of a minimal fIET was constructed in [18]. Some examples of minimal and uniquely ergodic fIETs can be also found in [1,12], and a analyze of the number of periodic and minimal components of fIETs can be found in [19].…”

Section: Interval Exchange Transformations With Flipsmentioning

confidence: 99%

“…A detailed description of Rauzy induction for fIETs can be found in [1,12,18]. Here we present a brief scheme of the induction algorithm.…”

Section: Rauzy Inductionmentioning

confidence: 99%

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On the Hausdorff dimension of minimal interval exchange transformations with flips

Skripchenko

Troubetzkoy

2018

J. London Math. Soc.

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Section: Rauzy Inductionmentioning

confidence: 99%

□

…”

Section: Rauzy Inductionmentioning

confidence: 99%

2 n - 3

complete paths is positive.…”

Section: Rauzy Inductionmentioning

confidence: 99%

Section: Interval Exchange Transformations With Flipsmentioning

confidence: 99%

“…A detailed description of Rauzy induction for fIETs can be found in [1,12,18]. Here we present a brief scheme of the induction algorithm.…”

Section: Rauzy Inductionmentioning

confidence: 99%

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On the Hausdorff dimension of minimal interval exchange transformations with flips

Skripchenko

Troubetzkoy

2018

J. London Math. Soc.

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Minimal non uniquely ergodic IETs with flips

Bas

López

2023

Journal of Differential Equations

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Minimal interval exchange transformations with flips

Bas¹,

López²

2017

Ergod. Th. Dynam. Sys.

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We consider interval exchange transformations of $n$ intervals with $k$ flips, or $(n,k)$-IETs for short, for positive integers $k,n$ with $k\leq n$. Our main result establishes the existence of minimal uniquely ergodic $(n,k)$-IETs when $n\geq 4$; moreover, these IETs are self-induced for $2\leq k\leq n-1$. This result extends the work on transitivity in Gutierrez et al [Transitive circle exchange transformations with flips. Discrete Contin. Dyn. Syst. 26(1) (2010), 251–263]. In order to achieve our objective we make a direct construction; in particular, we use the Rauzy induction to build a periodic Rauzy graph whose associated matrix has a positive power. Then we use a result in the Perron–Frobenius theory [Pullman, A geometric approach to the theory of non-negative matrices. Linear Algebra Appl. 4 (1971) 297–312] which allows us to ensure the existence of these minimal self-induced and uniquely ergodic $(n,k)$-IETs, $2\leq k\leq n-1$. We then find other permutations in the same Rauzy class generating minimal uniquely ergodic $(n,1)$- and $(n,n)$-IETs.

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Minimality and the Rauzy – Veech algorithm for interval exchange transformations with flips

Cited by 6 publications

References 10 publications

On the Hausdorff dimension of minimal interval exchange transformations with flips

On the Hausdorff dimension of minimal interval exchange transformations with flips

Minimal non uniquely ergodic IETs with flips

Minimal interval exchange transformations with flips

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