scholarly journals Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents

1996 ◽  
Vol 46 (2) ◽  
pp. 325-370 ◽  
Author(s):  
Anton Zorich
Keyword(s):  
Lyapunov Exponents ◽  
Interval Exchange ◽  
Gauss Measure ◽  
Interval Exchange Transformations

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 Invariant measures with bounded variation densities for piecewise area preserving maps

2012 ◽  
Vol 33 (2) ◽  
pp. 624-642 ◽  
Author(s):  
YIWEI ZHANG ◽  
CONGPING LIN
Keyword(s):  
Bounded Variation ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Topological Transitivity ◽  
Borel Measures ◽  
Interval Exchange Transformations ◽  
Area Preserving Maps ◽  
Piecewise Isometries ◽  
Area Preserving ◽  
Absolutely Continuous Invariant

AbstractWe investigate the properties of absolutely continuous invariant probability measures (ACIPs), especially those measures with bounded variation densities, for piecewise area preserving maps (PAPs) on ℝd. This class of maps unifies piecewise isometries (PWIs) and piecewise hyperbolic maps where Lebesgue measure is locally preserved. Using a functional analytic approach, we first explore the relationship between topological transitivity and uniqueness of ACIPs, and then give an approach to construct invariant measures with bounded variation densities for PWIs. Our results ‘partially’ answer one of the fundamental questions posed in [13]—to determine all invariant non-atomic probability Borel measures in piecewise rotations. When restricting PAPs to interval exchange transformations (IETs), our results imply that for non-uniquely ergodic IETs with two or more ACIPs, these ACIPs have very irregular densities, i.e. they have unbounded variation.

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