Entropy, volume growth and SRB measures for Banach space mappings

Alex Blumenthal, Lai-Sang Young

Research output: Contribution to journalArticle

Abstract

We consider (Formula presented.) Fréchet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces. SRB measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds, equivalently the sum of positive Lyapunov exponents of the map. In addition to numerous difficulties incurred by our infinite-dimensional setting, a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions. The results here generalize previously known results for diffeomorphisms of finite dimensional Riemannian manifolds, and are applicable to dynamical systems defined by large classes of dissipative parabolic PDEs.

Original languageEnglish (US)
Pages (from-to)1-61
Number of pages61
JournalInventiones Mathematicae
DOIs
StateAccepted/In press - Jul 21 2016

Fingerprint

SRB Measure
Volume Growth
Unstable Manifold
Entropy
Banach space
Parabolic PDEs
Borel Measure
Diffeomorphisms
Lyapunov Exponent
Probability Measure
Differentiable
Riemannian Manifold
Determinant
Dynamical system
Subspace
Generalise
Invariant
Class

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Entropy, volume growth and SRB measures for Banach space mappings. / Blumenthal, Alex; Young, Lai-Sang.

In: Inventiones Mathematicae, 21.07.2016, p. 1-61.

Research output: Contribution to journalArticle

@article{6abeb94e8ad44f6ab08e0f1162a2e527,
title = "Entropy, volume growth and SRB measures for Banach space mappings",
abstract = "We consider (Formula presented.) Fr{\'e}chet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces. SRB measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds, equivalently the sum of positive Lyapunov exponents of the map. In addition to numerous difficulties incurred by our infinite-dimensional setting, a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions. The results here generalize previously known results for diffeomorphisms of finite dimensional Riemannian manifolds, and are applicable to dynamical systems defined by large classes of dissipative parabolic PDEs.",
author = "Alex Blumenthal and Lai-Sang Young",
year = "2016",
month = "7",
day = "21",
doi = "10.1007/s00222-016-0678-0",
language = "English (US)",
pages = "1--61",
journal = "Inventiones Mathematicae",
issn = "0020-9910",
publisher = "Springer New York",

}

TY - JOUR

T1 - Entropy, volume growth and SRB measures for Banach space mappings

AU - Blumenthal, Alex

AU - Young, Lai-Sang

PY - 2016/7/21

Y1 - 2016/7/21

N2 - We consider (Formula presented.) Fréchet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces. SRB measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds, equivalently the sum of positive Lyapunov exponents of the map. In addition to numerous difficulties incurred by our infinite-dimensional setting, a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions. The results here generalize previously known results for diffeomorphisms of finite dimensional Riemannian manifolds, and are applicable to dynamical systems defined by large classes of dissipative parabolic PDEs.

AB - We consider (Formula presented.) Fréchet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces. SRB measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds, equivalently the sum of positive Lyapunov exponents of the map. In addition to numerous difficulties incurred by our infinite-dimensional setting, a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions. The results here generalize previously known results for diffeomorphisms of finite dimensional Riemannian manifolds, and are applicable to dynamical systems defined by large classes of dissipative parabolic PDEs.

UR - http://www.scopus.com/inward/record.url?scp=84979302740&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84979302740&partnerID=8YFLogxK

U2 - 10.1007/s00222-016-0678-0

DO - 10.1007/s00222-016-0678-0

M3 - Article

SP - 1

EP - 61

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

ER -