On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces

Conze, Jean-Pierre; Gutkin, Eugène

doi:10.1017/s0143385711001003

Cited by 21 publications

(33 citation statements)

References 35 publications

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“…The cocycles that arise in this setting are piecewise constant functions with values in Z d . First results in these geometric settings were only recently proved in [6,14,15,17,18].…”

Section: T ϕ (X Y) = (T X Y + ϕ(X))mentioning

confidence: 99%

“…We present here below a particular variation on the construction of Katok, using Rauzy-Veech induction (Definition 5.1), which allows us to obtain further properties (in particular Lemma 5.3) needed in the following sections. 6 Notation Let α ∈ A be such that π 0 (α) = 1, i.e. I α is the first of the intervals exchanged by T .…”

Section: Rigidity Sets With Large Oscillations Of Birkhoff Sumsmentioning

confidence: 99%

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Ergodic properties of infinite extensions of area-preserving flows

Frączek

Ulcigrai

2011

Math. Ann.

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We consider volume-preserving flows (Φ f t ) t∈R on S × R, where S is a compact connected surface of genus g ≥ 2 and (Φwhere (φ t ) t∈R is a locally Hamiltonian flow of hyperbolic periodic type on S and f is a smooth real valued function on S. We investigate ergodic properties of these infinite measure-preserving flows and prove that if f belongs to a space of finite codimension in C 2+ (S), then the following dynamical dichotomy holds: if there is a fixed point of (φ t ) t∈R on which f does not vanish, then (Φ f t ) t∈R is ergodic, otherwise, if f vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension (Φ 0 t ) t∈R . The proof of this result exploits the reduction of (Φ f t ) t∈R to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of (φ t ) t∈R on which f does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.

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“…The cocycles that arise in this setting are piecewise constant functions with values in Z d . First results in these geometric settings were only recently proved in [6,14,15,17,18].…”

Section: T ϕ (X Y) = (T X Y + ϕ(X))mentioning

confidence: 99%

Section: Rigidity Sets With Large Oscillations Of Birkhoff Sumsmentioning

confidence: 99%

Ergodic properties of infinite extensions of area-preserving flows

Frączek

Ulcigrai

2011

Math. Ann.

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show abstract

“…We give below a short description of this model (cf. [CoGu12]) and show that for special directions the previous results on ergodic sums over rotations apply.…”

Section: Description Of the Modelmentioning

confidence: 70%

“…One can reduce the model to the case of a direction of flow with angle η = π/4 and with the small obstacles condition a + b ≤ 1 (see [CoGu12]). Without loss of generality, we will consider this case.…”

Section: Rational Directions and Small Obstaclesmentioning

confidence: 99%