On differentiability of SRB states for partially hyperbolic systems

By Dmitry Dolgopyat. 2004

179 citations recorded by the source corpus.

Subfields: Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0baddaa20770bf6997aa51bd4a4c177d9ae3510e

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In Condensed Matter Physics

In Mathematical Physics

Citations in the corpus

On the ergodicity of partially hyperbolic systems (2005, result): “results in [ D ] with an argument of Ma~ n e | see [BPSW], p. 352).”
Stable ergodicity for partially hyperbolic attractors with negative central exponents (2007, background): “It is shown in [11] that if f is either a T(1) extension of an Anosov diffeomorphism or the time-1 map of an Anosov flow and g is close to id, then for typical g and any sufficiently large n, either Fn or F −1 n has negative central exponent with respect to any u-measure.”
Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows (2011, result): “Similar results hold for perturbations of the time-one map of volume-preserving Anosov flows: there exist open sets of perturbations with nonvanishing center exponents (Dolgopyat [19]), and the results in [49] also imply that volume must have atomic disintegration for these examples.”
Banach spaces adapted to Anosov systems (2004, background): “In the last few years the emphasis has been on strong stability properties with respect to various types of perturbation [4], dynamical zeta functions and the related smoothness issue (see [9, 14, 26]).”
Averaging and Invariant Measures (2005, unknown): “In this note we review recent results about statistical properties and stability of SRB measures [18], [19] and then show how the methods of the these two papers can be combined to obtain new results in this area.”
Linear response, or else (2014, background): “1, while Dolgopyat’s result [20] in a (not necessarily structurally stable) partially hyperbolic case is stated in § 3.”
SRB measures for partially hyperbolic systems whose central direction is weakly expanding (2014, unknown): “Additional results include conditions which imply uniqueness of the SRB measure for a diffeomor-phism and its perturbations [18, 19, 48], mixing properties of this measure [20, 21, 25], and even differentiability with respect to perturbations [26].”
Smooth Anosov flows: Correlation spectra and stability (2007, background): “Subsequently, D.Dolgopyat has extended such results to a large class of partially hyperbolic systems [ 8 ].”
Differentiation of SRB states for hyperbolic flows (2004, unknown): “(For other differentiability results see [14]).”
Physical measures and absolute continuity for one-dimensional center direction (2013, background): “Short afterwards, Castro [30] and Dolgopyat [31] proved exponential decay of correlations and other ergodic properties for certain diffeomorphisms with mostly contracting center.”
A simple framework to justify linear response theory (2009, result): “nessdoesn’tneedtohold for a6= a0, our result implies differentiability of the map a→ hϕ,µai at a= a0 for any choice of a→ µa. This is analogous to the result shown for partially hyperbolic systems in [Dol04]. Remark 2.7 The classical perturbation results found for example in [Kat76] do not seem to b”