Train tracks with C¹⁺ self-renormalizable structures

Abstract

We prove a one-to-one correspondence between C ¹⁺  conjugacy classes of diffeomorphisms with hyperbolic sets contained in surfaces and stable and unstable pairs of one-dimensional C ¹⁺  self-renormalizable structures.

Keywords:

• hyperbolic dynamics
• renormalization
• Markov maps
• minimal sets

Acknowledgements

We are very grateful to Welington de Melo and Stefano Luzzatto for very useful discussions on this work. We also thank the referees for useful comments. We thank IHES, CUNY, IMPA, Stony Brook and University of Warwick for their hospitality. We thank Calouste Gulbenkian Foundation, PRODYN-ESF, POCTI and POCI by FCT and Ministério da Ciência e da Tecnologia and Centros de Matemática da Universidade do Minho e da Universidade do Porto for their financial support of A.A. Pinto, and the Wolfson Foundation and the UK Engineering and Physical Sciences Research Council for their financial support of D.A. Rand.

Notes

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Additional information

Notes on contributors

D.A. Rand

1