Transience and multifractal analysis

Godofredo Iommi; Thomas Jordan; Michael John Todd

doi:10.1016/j.anihpc.2015.12.007

Transience and multifractal analysis

Godofredo Iommi, Thomas Jordan, Michael John Todd

Pure Mathematics

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

Abstract

We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit an example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.

Original language	English
Pages (from-to)	407-421
Journal	Annales de l'Institut Henri Poincare (C) Non Linear Analysis
Volume	34
Issue number	2
Early online date	11 Jan 2016
DOIs	https://doi.org/10.1016/j.anihpc.2015.12.007
Publication status	Published - Mar 2017

Keywords

Multifractal analysis
Ergodic theory
Lyapunov exponents

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10.1016/j.anihpc.2015.12.007

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© 2016, Publisher / the Author(s). This work is made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at www.sciencedirect.com / https://dx.doi.org/10.1016/j.anihpc.2015.12.007
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title = "Transience and multifractal analysis",

abstract = "We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit an example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.",

keywords = "Multifractal analysis, Ergodic theory, Lyapunov exponents",

author = "Godofredo Iommi and Thomas Jordan and Todd, {Michael John}",

note = "G.I. was partially supported by the Center of Dynamical Systems and Related Fields c{\'o}digo ACT1103 and by Proyecto Fondecyt 1150058. T.J. wishes to thank Proyecto Mecesup-0711 for funding his visit to PUC-Chile. M.T. is grateful for the support of Proyecto Fondecyt 1110040 for funding his visit to PUC-Chile and for partial support from NSF grant DMS 1109587.",

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doi = "10.1016/j.anihpc.2015.12.007",

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AU - Jordan, Thomas

AU - Todd, Michael John

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N2 - We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit an example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.

AB - We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit an example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.

KW - Multifractal analysis

KW - Ergodic theory

KW - Lyapunov exponents

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JF - Annales de l'Institut Henri Poincare (C) Non Linear Analysis

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ER -

Transience and multifractal analysis

Abstract

Keywords

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