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AMBIGUITY IN THE DETERMINATION OF THE FREE ENERGY ASSOCIATED WITH THE CRITICAL CIRCLE MAP

Part of: Infinite-dimensional dissipative dynamical systems Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  01 October 2008

BRIAN G. KENNY  and
TONY W. DIXON
BRIAN G. KENNY*
Affiliation:
Department of Theoretical Physics, Research School of Physical Science and Engineering, Australian National University, Canberra, ACT 0200, Australia (email: brianikenny@anu.edu.au)
TONY W. DIXON
Affiliation:
Formerly at School of Mathematics & Statistics, Curtin University of Technology, GPO Box U1987, Perth, WA 6845, Australia (email: tony.w.dixon@gmail.com)
*
✉For correspondence; e-mail: brianikenny@anu.edu.au
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Abstract

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We consider a simple model to describe the widths of the mode-locked intervals for the critical circle map. By using two different partitions of the rational numbers based on Farey series and Farey tree levels, respectively, we calculate the free energy analytically at selected points for each partition. It emerges that the result of the calculation depends on the method of partition. An implication of this finding is that the generalized dimensions Dq are different for the two types of partition except when q=0; that is, only the Hausdorff dimension is the same in both cases.

Keywords

circle mapsthermodynamic formalismFarey treesFarey series

MSC classification

Secondary: 37D35: Thermodynamic formalism, variational principles, equilibrium states 37L30: Attractors and their dimensions, Lyapunov exponents
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 2 , October 2008 , pp. 177 - 184
Copyright
Copyright © Australian Mathematical Society 2009

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