Ratner’s property and mild mixing for smooth flows on surfaces

Let${\mathcal{T}}=(T_{t}^{f})_{t\in \mathbb{R}}$be a special flow built over an IET$T:\mathbb{T}\rightarrow \mathbb{T}$of bounded type, under a roof function$f$with symmetric logarithmic singularities at a subset of discontinuities of$T$. We show that${\mathcal{T}}$satisfies the so-called switchable Ratner’s property which was introduced in Fayad and Kanigowski [On multiple mixing for a class of conservative surface flows.Invent. Math.to appear]. A consequence of this fact is that such flows are mildly mixing (before, they were only known to be weakly mixing [Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations.J. Mod. Dynam.3(2009), 35–49] and not mixing [Ulcigrai. Absence of mixing in area-preserving flows on surfaces.Ann. of Math.(2)173(2011), 1743–1778]). Thus, on each compact, connected, orientable surface of genus greater than one there exist flows that are mildly mixing and not mixing.

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Singularity of the Spectrum for Smooth Area-Preserving Flows in Genus Two and Translation Surfaces Well Approximated by Cylinders

Communications in Mathematical Physics ◽

10.1007/s00220-020-03895-x ◽

2020 ◽

Author(s):

Jon Chaika ◽

Krzysztof Frączek ◽

Adam Kanigowski ◽

Corinna Ulcigrai

Keyword(s):

Full Measure ◽

Interval Exchange ◽

Connected Component ◽

Translation Surfaces ◽

Hamiltonian Flows ◽

Interval Exchange Transformations ◽

Logarithmic Singularities ◽

Area Preserving ◽

Special Case ◽

Orientable Surfaces

AbstractWe consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that, when the genus of the surface is two, almost every such locally Hamiltonian flow with two non-degenerate isomorphic saddles has singular spectrum. More in general, singularity of the spectrum holds for special flows over a full measure set of interval exchange transformations with a hyperelliptic permutation (of any number of exchanged intervals), under a roof with symmetric logarithmic singularities. The result is proved using a criterion for singularity based on tightness of Birkhoff sums with exponential tails decay. A key ingredient in the proof, which is of independent interest, is a result on translation surfaces well approximated by single cylinders. We show that for almost every translation surface in any connected component of any stratum there exists a full measure set of directions which can be well approximated by a single cylinder of area arbitrarily close to one. The result, in the special case of the stratum $$\mathcal {H}(1,1)$$ H ( 1 , 1 ) , yields rigidity sets needed for the singularity result.

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Invariant measures with bounded variation densities for piecewise area preserving maps

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385711001167 ◽

2012 ◽

Vol 33 (2) ◽

pp. 624-642 ◽

Cited By ~ 1

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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Sobolev regularity of solutions of the cohomological equation

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2019.108 ◽

2020 ◽

pp. 1-105 ◽

Cited By ~ 2

Author(s):

GIOVANNI FORNI

Keyword(s):

Interval Exchange ◽

Regularity Of Solutions ◽

Sobolev Regularity ◽

University Of Toronto ◽

Interval Exchange Transformations ◽

Cohomological Equation ◽

Higher Genus ◽

The University ◽

Area Preserving ◽

Teichmuller Dynamics

In this survey we prove the sharpest results on the loss of Sobolev regularity for solutions of the cohomological equation for translation flows on translation surfaces, available to the methods developed by the author in Forni [Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Ann. of Math. (2)146(2) (1997), 295–344] and Forni [Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. of Math. (2)155(1) (2002), 1–103]. The paper was mostly written between 2005 and 2006 while the author was at the University of Toronto, Canada, and was posted on arXiv in July 2007 [Forni. Sobolev regularity of solutions of the cohomological equation. Preprint, 2007, arXiv:0707.0940v2]. In an updated introduction we describe our results, taking into account later work on the problem and relevant recent progress in the field of Teichmüller dynamics, interval exchange transformations and translation flows.

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