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Connecting ergodicity and dimension in dynamical systems

Published online by Cambridge University Press:  19 September 2008

C. D. Cutler
C. D. Cutler
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, CanadaN2L 3G1
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Abstract

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In this paper we make precise the relationship between local or pointwise dimension and the dimension structure of Borel probability measures on metric spaces. Sufficient conditions for exact-dimensionality of the stationary ergodic distributions associated with a dynamical system are obtained. A counterexample is provided to show that ergodicity alone is not sufficient to guarantee exactdimensionality even in the case of continuous maps or flows.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 3 , September 1990 , pp. 451 - 462
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

Billingsley, P., (1960), Hausdorff dimension in probability theory. Illinois J. Math. 4 187–209.Google Scholar
Billingsley, P., (1961), Hausdorff dimension in probability theory II. Illinois J. Math. 5 291–298.CrossRefGoogle Scholar
Billingsley, P., (1978), Ergodic Theory and Information. Krieger.Google Scholar
Cutler, C. D., (1986), The Hausdorff dimension distribution of finite measures in Euclidean space. Canad. J. Math. 38 (6) 1459–1484.Google Scholar
Cutler, C. D., and Dawson, D. A., (1989), Estimation of dimension for spatially-distributed data and related limit theorems. J. Multivariate Anal. 28 115–198CrossRefGoogle Scholar
Cutler, C. D., (1988), A note on equivalent interval covering systems for Hausdorff dimension on R. Int. J. Math. Math. Sci. 11 (4) 643–650.CrossRefGoogle Scholar
Eckmann, J.-P., and Ruelle, D., (1985), Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57 (3) Part 1, 617–656.Google Scholar
Falconer, K. J., (1985), The Geometry of Fractal Sets. Cambridge University Press.CrossRefGoogle Scholar
Farmer, J. D., Ott, E. and Yorke, J. A. (1983), The dimension of chaotic attractors. Physica 7D 153–180.Google Scholar
Guckenheimer, J. (1984), Dimension estimates for attractors. Contemp. Math. 28 357–367.Google Scholar
Rogers, C. A. (1970), Hausdorff Measures. Cambridge University Press.Google Scholar
Young, L.-S. (1982), Dimension, entropy, and Lyapunov exponents. Ergod. Th. & Dyn. Sys. 2 109–124.CrossRefGoogle Scholar