scholarly journals Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards

2015 ◽  
Vol 8 (3/4) ◽  
pp. 271-436 ◽  
Author(s):  
Giovanni Forni ◽  
Carlos Matheus
Keyword(s):  
Interval Exchange ◽  
Teichmuller Theory ◽  
Interval Exchange Transformations ◽  
Teichmüller Theory ◽  
Flows On Surfaces

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

2015 ◽  
Vol 36 (8) ◽  
pp. 2512-2537 ◽  
Author(s):  
ADAM KANIGOWSKI ◽  
JOANNA KUŁAGA-PRZYMUS
Keyword(s):  
Orientable Surface ◽  
Interval Exchange ◽  
Special Flow ◽  
Interval Exchange Transformations ◽  
Multiple Mixing ◽  
Weakly Mixing ◽  
Logarithmic Singularities ◽  
Area Preserving ◽  
Flows On Surfaces ◽  
Roof Function

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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