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Classifying C1+ structures on dynamical fractals: 1. The moduli space of solenoid functions for Markov maps on train tracks

Published online by Cambridge University Press:  19 September 2008

A. A. Pinto  and
D. A. Rand
A. A. Pinto
Affiliation:
Faculdade de Ciencias, Universidade do Porto, 4000 Porto, Portugal
D. A. Rand
Affiliation:
Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
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Abstract

Sullivan's scaling function provides a complete description of the smooth conjugacy classes of cookie-cutters. However, for smooth conjugacy classes of Markov maps on a train track, such as expanding circle maps and train track mappings induced by pseudo-Anosov systems, the generalisation of the scaling function suffers from a deficiency. It is difficult to characterise the structure of the set of those scaling functions which correspond to smooth mappings. We introduce a new invariant for Markov maps called the solenoid function. We prove that for any prescribed topological structure, there is a one-to-one correspondence between smooth conjugacy classes of smooth Markov maps and pseudo-Hölder solenoid functions. This gives a characterisation of the moduli space for smooth conjugacy classes of smooth Markov maps. For smooth expanding maps of the circle with degree d this moduli space is the space of Hölder continuous functions on the space {0,…, d − 1}ℕ satisfying the matching condition.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 4 , August 1995 , pp. 697 - 734
Copyright
Copyright © Cambridge University Press 1995

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