DOI: 10.1017/s0143385700009615. Dimension, entropy and Lyapunov exponents

Dimension, entropy and Lyapunov exponents

10.1017/s0143385700009615

Crossref journal-article

Cambridge University Press (CUP)

Ergodic Theory and Dynamical Systems (56)

Abstract

AbstractWe consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure μ. Define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure. We prove a formula relating HD (μ) to the entropy and Lyapunov exponents of the map. Other classical notions of fractional dimension such as capacity and Rényi dimension are discussed. They are shown to be equal to Hausdorff dimension in the present context.

Bibliography

Young, L.-S. (1982). Dimension, entropy and Lyapunov exponents. Ergodic Theory and Dynamical Systems, 2(1), 109â124.

Authors 1

Lai-Sang Young (first)

References 22 Referenced 479

10.1017/S0143385700001309
10.1090/S0002-9904-1967-11798-1
10.1007/BF01208896
10.1007/BFb0081279
[18] Rényi A. . Dimension, entropy and information. Transactions of the second Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (1957) 545–556.
[8] Frederickson P. , Kaplan J. , Yorke E. & Yorke J. . The Liapunov dimension of strange attractors. Preprint.
{'key': 'S0143385700009615_ref011', 'first-page': '861', 'article-title': 'A new invariant for transitive dynamical systems', 'volume': '119', 'author': 'Kolmogorov', 'year': '1958', 'journal-title': 'Dokl. Akad. Nauk SSSR'} / Dokl. Akad. Nauk SSSR / A new invariant for transitive dynamical systems by Kolmogorov (1958)
10.1017/S0143385700001371
10.1093/qmath/os-20.1.31
10.1007/BF02684777
10.1007/BF01448030
10.1007/BF02684768
{'key': 'S0143385700009615_ref009', 'first-page': '228', 'volume-title': 'Chaotic Behavior of Multidimensional Difference Equations', 'author': 'Kaplan', 'year': '1979'} / Chaotic Behavior of Multidimensional Difference Equations by Kaplan (1979)
10.1007/BF02584795
[5] Brin M. & Katok A. . On local entropy. Preprint.
10.1017/S0143385700001188
10.1007/BF01762666
10.1070/RM1977v032n04ABEH001639
{'key': 'S0143385700009615_ref002', 'volume-title': 'Ergodic Theory and Information', 'author': 'Billingsley', 'year': '1965'} / Ergodic Theory and Information by Billingsley (1965)
{'key': 'S0143385700009615_ref006', 'first-page': '1135', 'article-title': 'Dimension de Hausdorff des attracteurs', 'volume': '24', 'author': 'Douady', 'year': '1980', 'journal-title': 'C.R. Acad. Sci.'} / C.R. Acad. Sci. / Dimension de Hausdorff des attracteurs by Douady (1980)
{'key': 'S0143385700009615_ref015', 'volume-title': 'Differentiable Dynamics', 'author': 'Nitecki', 'year': '1971'} / Differentiable Dynamics by Nitecki (1971)
{'key': 'S0143385700009615_ref016', 'first-page': '179', 'article-title': 'A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems', 'volume': '19', 'author': 'Oseledec', 'year': '1968', 'journal-title': 'Trudy Moskov. Mat. Obšč.'} / Trudy Moskov. Mat. Obšč. / A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems by Oseledec (1968)

Dates

Type	When
Created	16 years ago (Aug. 13, 2009, 5:45 a.m.)
Deposited	6 years, 3 months ago (May 24, 2019, 1:41 p.m.)
Indexed	2 weeks, 2 days ago (Aug. 12, 2025, 6:09 p.m.)
Issued	43 years, 5 months ago (March 1, 1982)
Published	43 years, 5 months ago (March 1, 1982)
Published Online	16 years ago (Aug. 13, 2009)
Published Print	43 years, 5 months ago (March 1, 1982)

Funders 0

None

BibTeX

@article{Young_1982, title={Dimension, entropy and Lyapunov exponents}, volume={2}, ISSN={1469-4417}, url={http://dx.doi.org/10.1017/s0143385700009615}, DOI={10.1017/s0143385700009615}, number={1}, journal={Ergodic Theory and Dynamical Systems}, publisher={Cambridge University Press (CUP)}, author={Young, Lai-Sang}, year={1982}, month=mar, pages={109–124} }

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