scholarly journals Cohomological equation and local conjugacy class of Diophantine interval exchange maps

2019 ◽  
pp. 1
Author(s):  
Giovanni Forni ◽  
Stefano Marmi ◽  
Carlos Matheus
Keyword(s):  
Conjugacy Class ◽  
Interval Exchange ◽  
Cohomological Equation ◽  
Local Conjugacy ◽  
Interval Exchange Maps

scholarly journals On the cohomological equation for interval exchange maps

2003 ◽  
Vol 336 (11) ◽  
pp. 941-948 ◽  
Author(s):  
Stefano Marmi ◽  
Pierre Moussa ◽  
Jean-Christophe Yoccoz
Keyword(s):  
Interval Exchange ◽  
Cohomological Equation ◽  
Interval Exchange Maps

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

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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