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 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 Diophantine type of interval exchange maps

2013 ◽  
Vol 34 (6) ◽  
pp. 1990-2017
Author(s):  
DONG HAN KIM
Keyword(s):  
Continued Fraction ◽  
Interval Exchange ◽  
Gauss Measure ◽  
Return Times ◽  
Interval Exchange Transformations ◽  
Cohomological Equation ◽  
Rotation Numbers ◽  
Interval Exchange Maps ◽  
Fraction Algorithm ◽  
Asymptotic Scaling

AbstractRoth type irrational rotation numbers have several equivalent arithmetical characterizations as well as several equivalent characterizations in terms of the dynamics of the corresponding circle rotations. In this paper we investigate how to generalize Roth-like Diophantine conditions to interval exchange maps. If one considers the dynamics in parameter space one can introduce two non-equivalent Roth type conditions, the first (condition (Z)) by means of the Zorich cocycle [Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier 46(2) (1996), 325–370], the second (condition (A)) by means of a further acceleration of the continued fraction algorithm by Marmi–Moussa–Yoccoz introduced in [The cohomological equation for Roth type interval exchange maps, J. Amer. Math. Soc. 18 (2005), 823–872]. A third very natural condition (condition (D)) arises by considering the distance between the discontinuity points of the iterates of the map. If one considers the dynamics of an interval exchange map in phase space then one can introduce the notion of Diophantine type by considering the asymptotic scaling of return times pointwise or with respect to uniform convergence (respectively conditions (R) and (U)). In the case of circle rotations all the above conditions are equivalent. For interval exchange maps of three intervals we show that (D) and (A) are equivalent and imply (Z), (U) and (R), which are equivalent among them. For maps of four intervals or more we prove several results; the only relation that we cannot decide is whether (Z) implies (R) or not.

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
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