On Contact Anosov Flows

By Liverani Carlangelo. 2003

189 citations recorded by the source corpus.

Exponential decay of correlations for $\Co^{(4)}$ Contact Anosov flows is established. This implies, in particular, exponential decay of correlations for all smooth geodesic flows in strictly negative curvature.

View Paper: https://www.semanticscholar.org/paper/f20f3d480af56b533817bc2dc4a79c531c18861d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

1. Ian Melbourne
2. S. Gouëzel
3. Luís Barreira
4. M. Tsujii
5. Yakov Pesin
6. P. Giulietti
7. C. Liverani
8. O. Sarig

Citations in the corpus

• Operator renewal theory and mixing rates for dynamical systems with infinite measure (2010, background): “Landmark results include the proof of exponential decay of correlations for certain classes of uniformly hyperbolic flows [8, 12, 30] and planar dispersing billiards [42].”
• Stability of mixing and rapid mixing for hyperbolic flows (2007, unknown): “More recently, Chernov [11], Dolgopyat [14] and Liverani [21] have obtained results on exponential rates of mixing for restricted classes of Anosov flows.”
• A review of linear response theory for general differentiable dynamical systems (2009, background): “correlation functions for hyperbolic flows (see in particular Chernov [13], Dolgopyat [14, 15], Liverani [27], Fields et al [18]).”
• Chapter 2 – Smooth Ergodic Theory and Nonuniformly Hyperbolic Dynamics (2006, methodology): “We end this section by mentioning the works of Pollicott [206] and Baladi & Vallée [19] for versions of Dolgopyat’s estimate for semi-flows over piecewise expanding maps of the interval, Dolgopyat’s study of exponential and rapid mixing for generic hyperbolic flows [73, 75], the paper by Stoyanov [2”
• Smooth approximation of stochastic differential equations (2014, methodology): “MR2113022 [28] Liverani, C., Saussol, B. and Vaienti, S. (1999).”
• Averaging and Invariant Measures (2005, unknown): “(1) Anosov diffeomorphisms [7]; (2) time one maps of Anosov flows [15], [34], [22]; (3) partially hyperbolic translations on homogeneous spaces [32]; (4) compact group extensions of Anosov diffeomorphisms [17]; (5) partially hyperbolic toral automorphisms [28]; (6) mostly contracting systems (see…”
• Decay of correlations in suspension semi-flows of angle-multiplying maps (2005, background): “More recently, Liverani [13] extended the result to Anosov flows preserving contact structures and, in particular, to all geodesic flows in strictly negative curvature.”
• Contact Anosov flows and the Fourier–Bros–Iagolnitzer transform (2011, methodology): “We refer to [2, 4, 9, 14, 16] and the references therein for works related to this paper.”
• Anosov flows and dynamical zeta functions (2012, background): “Then JH̃ has range in Cr(Bdu(0, δ),R) and ‖JH̃‖C̟∗(Bds+1(0,δ),Cr(Bdu (0,δ),R)) ≤ C#. 66 66 See [34] and referencess therein for the proofs of these statements.”
• Banach spaces adapted to Anosov systems (2004, background): “Indeed, the ideas in [7] have already been applied with success to some partially hyperbolic situations (flows) [19] and we expect them to be applicable to the study of dynamical zeta functions.”
• Differentiation of SRB states for hyperbolic flows (2004, background): “Since C is Hölder but not smooth in general, only [20] applies directly in the present situation.”
• Rapid Decay of Correlations for Nonuniformly Hyperbolic Flows (2006, background): “Liverani [23] extended this result to arbitrary dimensional geodesic flows in negative curvature and more generally to contact Anosov flows.”