On contact Anosov flows

Abstract

Exponential decay of correlations for C 4 contact Anosov flows is established. This implies, in particular, exponential decay of correlations for all smooth geodesic flows in strictly negative curvature.

Exponential Decay of Correlations for Piecewise Cone Hyperbolic Contact Flows

“…Just like in [30], we do not claim that the transfer operator L 1 associated to the time-one map T 1 has a spectral gap. However, the resolvent method we adapt from [30] gives us a precise description of the spectrum of the generator X of the semigroup of operators L t in a half-plane large enough to deduce exponential decay of correlations (see Corollaries 3.6 and 3.10). The spaces H r,s,q p that we shall use are a modification (see Subsection 2.2) of the spaces H r,s p (s < 0 < r and 1 < p < ∞) of [6] for piecewise hyperbolic maps (the spaces in [6] generalise earlier constructions in [5] and [3], more directly related to standard Triebel spaces).…”

“…It follows that the "bounded" term in our Lasota-Yorke bound (Lemma 3.1) is not really bounded. The "compact" term in the Lasota-Yorke bound is not compact either, due to a loss of regularity in the flow direction of a more elementary origin (see Lemma 4.1), which also played a part in Liverani's [30] proof. Like in [30], we may overcome these problems because we work with the resolvent R(z) = ∞ 0 e −zt L t dt which involves integration along the time direction.…”

“…(By definition, L * t preserves dx.) The strategy, following [30], is to study L t as an operator on a suitable Banach space H of anisotropic distributions, and to prove Dolgopyat-like estimates. Just like in [30], we do not claim that the transfer operator L 1 associated to the time-one map T 1 has a spectral gap.…”

“…The strategy, following [30], is to study L t as an operator on a suitable Banach space H of anisotropic distributions, and to prove Dolgopyat-like estimates. Just like in [30], we do not claim that the transfer operator L 1 associated to the time-one map T 1 has a spectral gap. However, the resolvent method we adapt from [30] gives us a precise description of the spectrum of the generator X of the semigroup of operators L t in a half-plane large enough to deduce exponential decay of correlations (see Corollaries 3.6 and 3.10).…”

“…Using these spaces, we then adapt Liverani's [30] version of the Dolgopyat estimate for Anosov contact flows to obtain the first result of exponential decay of correlations for hyperbolic 1 An important exception is given by the "coupling" methods introduced in this context by L-S.Young in [48] and greatly generalised by D.Dolgopyat in [21] giving rise to the "coupling of standard pairs" method. See Chernov and others [16] for a review on how to obtain exponential mixing for the discrete-time Sinai billiard via coupling of standard pairs.…”