A proof of the estimation from below in Pesin's entropy formula

Ledrappier, François; Strelcyn, Jean-Marie

doi:10.1017/s0143385700001528

Cited by 170 publications

(147 citation statements)

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“…More precisely what is exactly needed in [LS] is a family of local unstable manifolds V loc satisfying the conclusions of [LS,Proposition 3.3]: items (3.3.1) to (3.3.6), except (3.3.5), assert that the family of manifolds V loc has controlled geometry on a set of large µ measure, and (3.3.5) means that points in the same local leaf become exponentially close under backwards iteration, uniformly on sets of large measure. The reader will easily check these properties are true for the unstable disks constructed above, that is, the set of disks subordinate to T − .…”

Section: Fix ε > 0 and Consider A Collectionmentioning

confidence: 99%

Laminar currents and birational dynamics

Dujardin

2006

Duke Math. J.

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Abstract. We study the dynamics of a bimeromorphic map X → X, where X is a compact complex Kähler surface. Under a natural geometric hypothesis, we construct an invariant probability measure, which is mixing, hyperbolic and of maximal entropy. The proof relies heavily on the theory of laminar currents and is new even in the case of polynomial automorphisms of C 2 . This extends recent results by E. Bedford and J. Diller.

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Section: Fix ε > 0 and Consider A Collectionmentioning

confidence: 99%

Laminar currents and birational dynamics

Dujardin

2006

Duke Math. J.

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“…It guarantees the applicability of the theory elaborated in [5] (see [6]) to billiards in strictly convex bounded regions in Rn with boundary of class Ck, k > 3, in particular Pesin's entropy formula [8,9,7] holds for such billiards.…”

Section: Introductionmentioning

confidence: 82%

Smoothness of the billiard ball map for strictly convex domains near the boundary

Kovachev¹

1988

Proc. Amer. Math. Soc.

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“…This inequality, which is known as the Pesin inequality, turns into an equality for sufficiently chaotic systems, such as Anosov systems (Ledrappier and Strelcyn 1982;Castiglione et al 2008). This means that the production of information (that is, the gain in information about initial conditions from observing the trajectory for one more step) is only provided by unstable directions, so h KS 6 γ + i .…”

Section: Metric Entropymentioning

confidence: 99%

Shannon entropy: a rigorous notion at the crossroads between probability, information theory, dynamical systems and statistical physics

Lesne¹

2014

Math. Struct. Comp. Sci.

163

113

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Statistical entropy was introduced by Shannon as a basic concept in information theory measuring the average missing information in a random source. Extended into an entropy rate, it gives bounds in coding and compression theorems. In this paper, I describe how statistical entropy and entropy rate relate to other notions of entropy that are relevant to probability theory (entropy of a discrete probability distribution measuring its unevenness), computer sciences (algorithmic complexity), the ergodic theory of dynamical systems (Kolmogorov-Sinai or metric entropy) and statistical physics (Boltzmann entropy). Their mathematical foundations and correlates (the entropy concentration, Sanov, Shannon-McMillan-Breiman, Lempel-Ziv and Pesin theorems) clarify their interpretation and offer a rigorous basis for maximum entropy principles. Although often ignored, these mathematical perspectives give a central position to entropy and relative entropy in statistical laws describing generic collective behaviours, and provide insights into the notions of randomness, typicality and disorder. The relevance of entropy beyond the realm of physics, in particular for living systems and ecosystems, is yet to be demonstrated. IntroductionHistorically, many notions of entropy have been proposed. The first use of the word entropy dates back to Clausius (Clausius 1865), who coined this term from the Greek tropos, meaning transformation, and the prefix en-to recall its inseparable (in his work) relation to the notion of energy (Jaynes 1980). A statistical concept of entropy was introduced by Shannon in the theory of communication and transmission of information (Shannon 1948). † Part of this paper was presented during the 'Séminaire Philosophie et Mathématiques de l'École normale supérieure' in 2010 and the 'Séminaire MaMuPhy' at Ircam in 2011.

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A proof of the estimation from below in Pesin's entropy formula

Abstract: We give a proof of Pesin entropy formula in a very general setting.

Cited by 170 publications

References 15 publications

Laminar currents and birational dynamics

Laminar currents and birational dynamics

Smoothness of the billiard ball map for strictly convex domains near the boundary

Shannon entropy: a rigorous notion at the crossroads between probability, information theory, dynamical systems and statistical physics

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