Weak mixing for interval exchange transformations and translation flows

By Artur Avila, G. Forni. 2004

213 citations recorded by the source corpus.

We prove that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation. We also conclude that a typical translation flow on a typical translation surface of genus g ≥ 2 (with prescribed singularity types) is weakly mixing.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1aa686f6b956b9272a939c48a7dab0098f443156

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Citations in the corpus

Dynamics and geometry of the Rauzy–Veech induction for quadratic differentials (2007, background): “Avila and Forni proved the weak mixing for almost all interval exchange transformations and translation flows [ AF07 ].”
Exponential mixing for the Teichmüller flow (2005, background): “For instance, the proof of typical weak mixing in [ AF ] can be made more transparent using such estimates.”
Beyond substitutive dynamical systems: S-adic expansions (2013, background): “Let us just mention the most striking ones: the existence of nice invariant measures on the space of interval exchanges \lfloor 108, 131], which leads to the resolution of Keanes conjecture on the genericity of unique ergodicity, the prevalence of weak‐mixing [21], continued fractions and associate”
A geometric criterion for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle (2010, background): “Avila and the author to weak mixing for translation flows and interval-exchange transformations [3].”
Stability of Operators and Operator Semigroups (2010, background): “, Cornfeld, Fomin, Sinai [53, Chapter 14], Katok [141], Katok, Stepin [144], Avila, Forni [11], Ulcigrai [244], Scheglov [228], Kulaga [159] and also Example 3.”
Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials (2007, methodology): “This technique was developed and applied in [3, 4, 5, 7, 19, 32], and in other papers; see also surveys [29, 30, 31, 33].”
Chapter 8 - On the Lyapunov Exponents of the Kontsevich–Zorich Cocycle (2006, background): “Avila and the author [7] have recently shown that the positivity of the second exponent (for surfaces of higher genus) implies that almost every i.”