On the dual of Rauzy induction

2016 ◽  
Vol 37 (5) ◽  
pp. 1492-1536 ◽  
Author(s):  
KAE INOUE ◽  
HITOSHI NAKADA
Keyword(s):  
Continued Fractions ◽  
Natural Extension ◽  
Gauss Map ◽  
Interval Exchange ◽  
Dual Relationship ◽  
Rauzy Induction ◽  
Interval Exchange Maps

We investigate a certain dual relationship between piecewise rotations of a circle and interval exchange maps. In 2005, Cruz and da Rocha [A generalization of the Gauss map and some classical theorems on continued fractions. Nonlinearity18 (2005), 505–525]  introduced a notion of ‘castles’ arising from piecewise rotations of a circle. We extend their idea and introduce a continuum version of castles, which we show to be equivalent to Veech’s zippered rectangles [Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982), 201–242]. We show that a fairly natural map defined on castles represents the inverse of the natural extension of the Rauzy map.

scholarly journals Renewal-type limit theorem for continued fractions with even partial quotients

2009 ◽  
Vol 29 (5) ◽  
pp. 1451-1478 ◽  
Author(s):  
FRANCESCO CELLAROSI
Keyword(s):  
Limit Theorem ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Type Theorem ◽  
Natural Extension ◽  
Gauss Map ◽  
Continued Fraction Expansion ◽  
Special Flow ◽  
Partial Quotients ◽  
Continued Fraction Expansions

AbstractWe prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by Schweiger [Continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg4 (1982), 59–70; On the approximation by continues fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg1–2 (1984), 105–114] and studied also by Kraaikamp and Lopes [The theta group and the continued fraction expansion with even partial quotients. Geom. Dedicata59(3) (1996), 293–333]. Our main result is proven following the strategy used by Sinai and Ulcigrai [Renewal-type limit theorem for the Gauss map and continued fractions. Ergod. Th. & Dynam. Sys.28 (2008), 643–655] in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.

scholarly journals Renewal-type limit theorem for the Gauss map and continued fractions

2008 ◽  
Vol 28 (2) ◽  
pp. 643-655 ◽  
Author(s):  
YAKOV G. SINAI ◽  
CORINNA ULCIGRAI
Keyword(s):  
Limit Theorem ◽  
Continued Fractions ◽  
Natural Extension ◽  
Gauss Map ◽  
Limiting Distribution ◽  
Special Flow

AbstractIn this paper we prove a renewal-type limit theorem. Given $\alpha \in (0,1)\backslash \mathbb {Q}$ and R>0, let qnR be the first denominator of the convergents of α which exceeds R. The main result in the paper is that the ratio qnR/R has a limiting distribution as R tends to infinity. The existence of the limiting distribution uses mixing of a special flow over the natural extension of the Gauss map.

scholarly journals Dynamics of non-classical interval exchanges

2011 ◽  
Vol 32 (6) ◽  
pp. 1930-1971 ◽  
Author(s):  
VAIBHAV S. GADRE
Keyword(s):  
Dynamical Property ◽  
Natural Generalization ◽  
Interval Exchange ◽  
Flat Surfaces ◽  
Interval Exchanges ◽  
Refinement Process ◽  
Train Tracks ◽  
Rauzy Induction ◽  
Interval Exchange Maps ◽  
Uniquely Ergodic

AbstractA natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira [Measured foliations on non-orientable surfaces.Ann. Sci. Éc. Norm. Supér.(4)26(6) (1993), 645–664]. Recurrent train tracks with a single switch provide a subclass of linear involutions. We call such linear involutions non-classical interval exchanges. They are related to measured foliations on orientable flat surfaces. Non-classical interval exchanges can be studied as a dynamical system by considering Rauzy induction in this context. This gives a refinement process on the parameter space similar to Kerckhoff’s simplicial systems. We show that the refinement process gives an expansion that has a key dynamical property calleduniform distortion. We use uniform distortion to prove normality of the expansion. Consequently, we prove an analog of Keane’s conjecture: almost every non-classical interval exchange is uniquely ergodic. Uniform distortion has been independently shown in [A. Avila and M. Resende. Exponential mixing for the Teichmüller flow in the space of quadratic differentials, http://arxiv.org/abs/0908.1102].

scholarly journals A Sufficient Condition for Absolute Continuity of Conjugations between Interval Exchange Maps

2019 ◽  
pp. 276-284
Author(s):  
Abdumajid S. Begmatov
Keyword(s):  
Absolute Continuity ◽  
Interval Exchange ◽  
Sufficient Condition ◽  
Genus One ◽  
Interval Exchange Maps

A class of topological equivalent generalized interval exchange maps of genus one and of the same bounded combinatorics is considered in the paper. A sufficient condition for absolute continuity of the conjugation between two maps from this class is provided

The recurrence time for interval exchange maps

Nonlinearity ◽  
2008 ◽  
Vol 21 (9) ◽  
pp. 2201-2210 ◽  
Author(s):  
Dong Han Kim ◽  
Stefano Marmi
Keyword(s):  
Interval Exchange ◽  
Recurrence Time ◽  
Interval Exchange Maps

scholarly journals Dynamics and geometry of the Rauzy–Veech induction for quadratic differentials

2009 ◽  
Vol 29 (3) ◽  
pp. 767-816 ◽  
Author(s):  
CORENTIN BOISSY ◽  
ERWAN LANNEAU
Keyword(s):  
Geodesic Flow ◽  
Natural Generalization ◽  
Interval Exchange ◽  
Connected Components ◽  
Quadratic Differentials ◽  
Flat Surfaces ◽  
Interval Exchange Transformations ◽  
Transverse Segment ◽  
First Return Maps ◽  
Interval Exchange Maps

AbstractInterval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy–Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmüller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely linear involutions. These maps are related to general measured foliations on surfaces (whether orientable or not). In this paper we are interested by such maps related to geodesic flow on (orientable) flat surfaces with ℤ/2ℤ linear holonomy. We relate geometry and dynamics of such maps to the combinatorics of generalized permutations. We study an analogue of the Rauzy–Veech induction and give an efficient combinatorial characterization of its attractors. We establish a natural bijection between the extended Rauzy classes of generalized permutations and connected components of the strata of meromorphic quadratic differentials with at most simple poles, which allows us, in particular, to classify the connected components of all exceptional strata.

scholarly journals Linearization of generalized interval exchange maps

2012 ◽  
Vol 176 (3) ◽  
pp. 1583-1646 ◽  
Author(s):  
Stefano Marmi ◽  
Pierre Moussa ◽  
Jean-Christophe Yoccoz
Keyword(s):  
Interval Exchange ◽  
Interval Exchange Maps

scholarly journals Simplicial systems for interval exchange maps and measured foliations

1985 ◽  
Vol 5 (2) ◽  
pp. 257-271 ◽  
Author(s):  
S. P. Kerckhoff
Keyword(s):  
General Theorem ◽  
Interval Exchange ◽  
Alternative Proof ◽  
Simple Assumption ◽  
Refinement Process ◽  
Almost All ◽  
Interval Exchange Maps ◽  
Uniquely Ergodic

AbstractThe spaces of interval exchange maps and measured foliations are considered and an alternative proof that almost all interval exchange maps and measured foliations are uniquely ergodic is given. These spaces are endowed with a refinement process, called a simplicial system, which is studied abstractly and is shown to be normal under a simple assumption. The results follow and thus are a corollary of a more general theorem in a broader setting.

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