An entropy estimate for infinite interval exchange transformations

2011 ◽  
Vol 272 (1-2) ◽  
pp. 17-29 ◽  
Author(s):  
Frank Blume
Keyword(s):  
Interval Exchange ◽  
Infinite Interval ◽  
Interval Exchange Transformations ◽  
Entropy Estimate

scholarly journals Infinite interval exchange transformations from shifts

2016 ◽  
Vol 37 (6) ◽  
pp. 1935-1965 ◽  
Author(s):  
LUIS-MIGUEL LOPEZ ◽  
PHILIPPE NARBEL
Keyword(s):  
Topological Entropy ◽  
Interval Exchange ◽  
Infinite Interval ◽  
Linear Factor ◽  
Interval Exchange Transformations ◽  
Interval Exchanges ◽  
Finiteness Properties ◽  
Block Growth ◽  
Linear Block

We show that minimal shifts with zero topological entropy are topologically conjugate to interval exchange transformations, which are generally infinite. When these shifts have linear factor complexity (linear block growth), the conjugate interval exchanges are proved to satisfy strong finiteness properties.

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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