Gibbs Measures in Ergodic Theory

Sinaĭ, Ya. G.

doi:10.1007/978-0-387-87870-6_12

Cited by 186 publications

(225 citation statements)

References 0 publications

Supporting

Mentioning

221

Contrasting

Unclassified

Order By: Relevance

“…It is a standard construction, see [174,Section 3] and [176,Lemma 1.6], to show that for any Gibbs measure on (Σ A , σ), which is defined naturally through two-sided k-cylinders, there is a Gibbs measure on (Σ + A , σ) with precisely the same ergodic properties. In particular, this can be exploited as in [147] to prove a version of Theorem 5.4.3 for two-sided cylinders.…”

Section: Two-sided Shiftsmentioning

confidence: 99%

Extremes and Recurrence in Dynamical Systems

Lucarini¹,

Faranda²,

Freitas³

et al. 2016

189

253

View full text Add to dashboard Cite

ii 4.1.2 The New Conditions 44 4.1.3 The Existence of EVL for General Stationary Stochastic Processes under Weaker Hypotheses 46 4.1.4 Proofs of Theorem 4.1.4 and Corollary 4.1.5 48 4.2 Extreme Values for Dynamically Defined Stochastic Processes 53 4.2.1 Observables and Corresponding Extreme Value Laws 55 4.2.2 Extreme Value Laws for Uniformly Expanding Systems 59 4.2.3 Example 4.2.1 revisited 61 4.2.4 Proof of the Dichotomy for Uniformly Expanding Maps 63 4.3 Point Processes of Rare Events 64 4.3.1 Absence of Clustering 64 4.3.2 Presence of Clustering 65 4.3.3 Dichotomy for Uniformly Expanding Systems for Point Processes 67 4.4 Conditions Д q (u n ), D 3 (u n ), D p (u n ) * and Decay of Correlations 68 4.5 Specific Dynamical Systems where the Dichotomy Applies 71 4.5.1 Rychlik Systems 72 4.5.2 Piecewise Expanding Maps in Higher Dimensions 73 4.6 Extreme Value Laws for Physical Observables 74

show abstract

Section: Two-sided Shiftsmentioning

confidence: 99%

Extremes and Recurrence in Dynamical Systems

Lucarini¹,

Faranda²,

Freitas³

et al. 2016

189

253

View full text Add to dashboard Cite

show abstract

“…One can then apply Fubini's theorem to conclude that a Gibbs state is indeed an SRB measure: its basin ( ) has full Lebesgue measure in the open neighborhood of Λ. These results make up most of [71,66,17].…”

Section: From Lorenz Back To Hadamard mentioning

confidence: 96%

“…In the early 1970s, Sinai, Ruelle and Bowen have discovered a fundamental concept to answer this question [71,66,17].…”

Section: From Lorenz Back To Hadamard mentioning

confidence: 99%

The Lorenz Attractor, a Paradigm for Chaos

Ghys

2013

Progress in Mathematical Physics

View full text Add to dashboard Cite

Abstract. It is very unusual for a mathematical or physical idea to disseminate into the society at large. An interesting example is chaos theory, popularized by Lorenz's butterfly effect: "does the flap of a butterfly's wings in Brazil set off a tornado in Texas?" A tiny cause can generate big consequences! Mathematicians (and non mathematicians) have known this fact for a long time! Can one adequately summarize chaos theory is such a simple minded way? In this review paper, I would like first of all to sketch some of the main steps in the historical development of the concept of chaos in dynamical systems, from the mathematical point of view. Then, I would like to present the present status of the Lorenz attractor in the panorama of the theory, as we see it Today.

show abstract

“…and it has both stable and unstable directions in the tangent bundle; the map f is not assumed to be expanding on Λ. For Anosov diffeomorphisms or for diffeomorphisms having a hyperbolic attractor, we have the existence of Sinai-Ruelle-Bowen (SRB) measures (see for instance [17], [4], [14], [1], [2], [3], [20], etc.) SRB measures exist also for smooth endomorphisms with hyperbolic attractors and are equal to the equilibrium measures of the unstable potentials, on inverse limit spaces (see [12]).…”

Section: Introductionmentioning

confidence: 99%

“…Examples and properties of endomorphisms with some hyperbolicity have been studied by many authors, for instance [4], [11], [16], [18], [6], [19], [7], [9], etc. Notice that if Λ is a repeller, then the local stable manifolds are contained in Λ. Hyperbolicity on Λ assures the existence of a unique equilibrium (Gibbs) measure µ φ for any given Hölder continuous potential φ on Λ; equilibrium measures are of great interest and have been studied intensively in the literature (for instance [17], [1], [14], [3], [5], [9], etc. )…”

Section: Introductionmentioning

confidence: 99%