CLASSIFYING C1+ STRUCTURES ON DYNAMICAL FRACTALS .1. THE MODULI SPACE OF SOLENOID FUNCTIONS FOR MARKOV MAPS ON TRAIN TRACKS

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-Holder 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 Holder continuous functions on the space {O,..., d - 1}(N) satisfying the matching condition.