A unique ergodicity of minimal symbolic flows with linear block growth

Boshernitzan, Michael

doi:10.1007/bf02790191

Cited by 74 publications

(122 citation statements)

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“…A mainly combinatorial proof of this result, quite in the spirit of the present paper, can be deduced from [8] and [9]. When the solutions are non-uniquely ergodic, the analysis has been conducted in [22]: our induction path corresponds to any symmetric k-interval exchange whose vector of lengths lies in a convex set S, with extremal points [31].…”

Section: Proofmentioning

confidence: 98%

Structure of K-interval exchange transformations: Induction, trajectories, and distance theorems

Ferenczi

Zamboni

2010

JAMA

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Abstract. We define a new induction algorithm for k-interval exchange transformations associated to the "symmetric" permutation i → k − i + 1. Acting as a multi-dimensional continued fraction algorithm, it defines a sequence of generalized partial quotients given by an infinite path in a graph whose vertices, or states, are certain trees we call trees of relations. This induction is self-dual for the duality between the usual Rauzy induction and the da Rocha induction. We use it to describe those words obtained by coding orbits of points under a symmetric interval exchange, in terms of the generalized partial quotients associated with the vector of lengths of the k intervals. As a consequence, we improve a bound of Boshernitzan in a generalization of the three-distances theorem for rotations. However, a variant of our algorithm, applied to a class of interval exchange transformations with a different permutation, shows that the former bound is optimal outside the hyperelliptic class of permutations. PreliminariesInterval exchange transformations were originally introduced by Oseledec [27], following an idea of Arnold [2]; see also Katok and Stepin [20]. An exchange of k intervals, denoted throughout this paper by I, is given by a probability vector of k lengths (α 1 , . . . , α k ) together with a permutation π on k letters. The unit interval is partitioned into k subintervals of lengths α 1 , . . . , α k which are rearranged by I according to π. It was Rauzy [29] who first saw interval exchange transformations as a possible framework for generalizing the well-known interaction between circle rotations on one hand and Sturmian sequences on the other via the continued fraction algorithm (see, for example, the survey [7] [4]. But, in contrast with the Sturmian case, though this induction gives access to the symbolic dynamics of the trajectories, it is difficult to describe them explicitly and indeed the Rauzy induction was never actually used for that purpose. A partial description of the trajectories was given in [24], and it aroused interest by defining a different kind of induction; this da Rocha induction was then considered to be dual to the Rauzy induction: the Rauzy induction builds the points where the orbits of the discontinuities of I approximate 0, while the da Rocha induction builds the points where the orbit of 0 approximates the discontinuities of I −1 . Note that (non-constructive) combinatorial characterizations of the language of trajectories were given recently in [16] and [5].These inductions are also a fundamental tool in the study of the space of moduli of Riemann surfaces, and the various strata of its unit tangent bundle, through what has been a basic object of interest for the last 25 years, the Teichmüller flow on a stratum; in the case of interval exchanges, we consider the Teichmüller flow for the Riemann surface obtained by glueing parallel opposite sides of a polygon. The Rauzy induction chooses an initial segment of a horizontal separatrix and follows its vertical separatrix till it i...

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Section: Proofmentioning

confidence: 98%

Structure of K-interval exchange transformations: Induction, trajectories, and distance theorems

Ferenczi

Zamboni

2010

JAMA

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“…Theorem 2 is a direct consequence of Theorem 1 together with Theorem 1.5 of [Bos85] for the unique ergodicity statement (see also [FM10]). The weak convergence comes from the unique ergodicity.…”

Section: Theorem 2 (Frequencies and Convergence) Let W Be An S-adic mentioning

confidence: 90%

“…The proof relies on the fact that weak and strong bispecial factors are alternating in the sequence (ordered by increasing length) of non neutral bispecial factors. Then, by using a result of Boshernitzan [Bos85], we deduce the existence of (uniform) frequency of any factor, and thus of the letters. This provides a combinatorial proof of convergence for this multidimensional continued fraction algorithm.…”

Section: Introductionmentioning

confidence: 95%

Convergence and Factor Complexity for the Arnoux-Rauzy-Poincaré Algorithm

Berthé

Labbé

2013

Lecture Notes in Computer Science

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Abstract. We introduce a multidimensional continued fraction algorithm based on Arnoux-Rauzy and Poincaré algorithms, and we study its associated S-adic system. An S-adic system is made of infinite words generated by the composition of infinite sequences of substitutions with values in a given finite set of substitutions, together with some restrictions concerning the allowed sequences of substitutions, expressed in terms of a regular language. We prove that these words have a factor complexity p(n) with lim sup p(n)/n < 3, which provides a proof for the convergence of the associated algorithm by unique ergodicity.

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“…In [2,3] Boshernitzan introduced the following condition (which was later called condition (B) in [6]): the subshift Ω satisfies condition (B) if there exists an ergodic probability measure µ on Ω with lim sup…”

Section: Some Related Resultsmentioning

confidence: 99%