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The singularity spectrum f(α) for cookie-cutters

Published online by Cambridge University Press:  19 September 2008

D. A. Rand
D. A. Rand
Affiliation:
Nonlinear Systems Laboratory, Mathematics Institute, Warwick University, Coventry CV4 7AL, UK
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Abstract

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I use a thermodynamic formalism to study the spectrum f(α) which characterises the large fluctuations of pointwise dimension in a Gibbs state supported on a hyperbolic cookie-cutter. Amongst other things, it is proved thatf(α) is the Hausdorff dimension of the set of points with pointwise dimension α, that f(α) is real-analytic and that its Legendre transform τ(q) is related to the Renyi dimension Dq of the Gibbs state by the formula (1 − q)Dq = τ(q).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 3 , September 1989 , pp. 527 - 541
Copyright
Copyright © Cambridge University Press 1989

References

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