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

• Type: Article in International Scientific Journal
• Date: 1995

Authors

Alberto A. Pinto (Author) FCUP; Rand, DA (Author) Other

Journal

Title: Ergodic Theory and Dynamical Systems

Vol. 15

Pages: 697-734

ISSN: 0143-3857

Publisher: Cambridge University Press

Indexing

Scopus - 16 Citations

Other information

Authenticus ID: P-00K-JYX

DOI: 10.1017/s0143385700008622

Abstract (EN): 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} satisfying the matching condition.

Language: English

Type (Professor's evaluation): Scientific