Stochastic stability of non-uniformly hyperbolic diffeomorphisms

José F. Alves; Vítor Araújo; Carlos H. Vásquez

doi:10.1142/S0219493707002049

Stochastic stability of non-uniformly hyperbolic diffeomorphisms

José F. Alves, Vítor Araújo, Carlos H. Vásquez

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

We prove that the statistical properties of random perturbations of a diffeomorphism with dominated splitting having mostly contracting center-stable direction and non-uniformly expanding center-unstable direction are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic stability of such dynamical systems. We show that a certain C²-open class of non-uniformly hyperbolic diffeomorphisms introduced by Alves, Bonatti and Viana in [2] are stochastically stable.

Original language	English
Pages (from-to)	299-333
Number of pages	35
Journal	Stochastics and Dynamics
Volume	7
Issue number	3
DOIs	https://doi.org/10.1142/S0219493707002049
State	Published - Sep 2007
Externally published	Yes

Keywords

Dominated splitting
Non-uniform hyperbolicity
Random perturbation
SRB measure
Stochastic stability

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10.1142/S0219493707002049

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title = "Stochastic stability of non-uniformly hyperbolic diffeomorphisms",

abstract = "We prove that the statistical properties of random perturbations of a diffeomorphism with dominated splitting having mostly contracting center-stable direction and non-uniformly expanding center-unstable direction are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic stability of such dynamical systems. We show that a certain C2-open class of non-uniformly hyperbolic diffeomorphisms introduced by Alves, Bonatti and Viana in [2] are stochastically stable.",

keywords = "Dominated splitting, Non-uniform hyperbolicity, Random perturbation, SRB measure, Stochastic stability",

author = "Alves, {Jos{\'e} F.} and V{\'i}tor Ara{\'u}jo and V{\'a}squez, {Carlos H.}",

note = "Funding Information: J. F. A. and V. A. were partially supported by FCT through CMUP, POCI/ MAT/61237/2004, and grants of FCT. C. V. was partially supported by Fondecyt 11060497 and DGIP-UCN 2005 10301201 and V. A. was also partially supported by CNPq. Part of this work was done while V. A. enjoyed a leave at IMPA and IM-UFRJ and C. V enjoyed a leave at CMUP.",

year = "2007",

month = sep,

doi = "10.1142/S0219493707002049",

language = "English",

volume = "7",

pages = "299--333",

journal = "Stochastics and Dynamics",

issn = "0219-4937",

publisher = "World Scientific Publishing Co. Pte Ltd",

number = "3",

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TY - JOUR

T1 - Stochastic stability of non-uniformly hyperbolic diffeomorphisms

AU - Alves, José F.

AU - Araújo, Vítor

AU - Vásquez, Carlos H.

N1 - Funding Information: J. F. A. and V. A. were partially supported by FCT through CMUP, POCI/ MAT/61237/2004, and grants of FCT. C. V. was partially supported by Fondecyt 11060497 and DGIP-UCN 2005 10301201 and V. A. was also partially supported by CNPq. Part of this work was done while V. A. enjoyed a leave at IMPA and IM-UFRJ and C. V enjoyed a leave at CMUP.

PY - 2007/9

Y1 - 2007/9

N2 - We prove that the statistical properties of random perturbations of a diffeomorphism with dominated splitting having mostly contracting center-stable direction and non-uniformly expanding center-unstable direction are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic stability of such dynamical systems. We show that a certain C2-open class of non-uniformly hyperbolic diffeomorphisms introduced by Alves, Bonatti and Viana in [2] are stochastically stable.

AB - We prove that the statistical properties of random perturbations of a diffeomorphism with dominated splitting having mostly contracting center-stable direction and non-uniformly expanding center-unstable direction are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic stability of such dynamical systems. We show that a certain C2-open class of non-uniformly hyperbolic diffeomorphisms introduced by Alves, Bonatti and Viana in [2] are stochastically stable.

KW - Dominated splitting

KW - Non-uniform hyperbolicity

KW - Random perturbation

KW - SRB measure

KW - Stochastic stability

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U2 - 10.1142/S0219493707002049

DO - 10.1142/S0219493707002049

M3 - Article

AN - SCOPUS:34548302057

SN - 0219-4937

VL - 7

SP - 299

EP - 333

JO - Stochastics and Dynamics

JF - Stochastics and Dynamics

IS - 3

ER -

Stochastic stability of non-uniformly hyperbolic diffeomorphisms

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Keywords

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