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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Hyperbolic graphs: Critical regularity and box dimension


Authors: L. J. Díaz, K. Gelfert, M. Gröger and T. Jäger
Journal: Trans. Amer. Math. Soc. 371 (2019), 8535-8585
MSC (2010): Primary 37C45, 37D20, 37D35, 37D30
DOI: https://doi.org/10.1090/tran/7454
Published electronically: February 27, 2019
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Abstract: We study fractal properties of invariant graphs of hyperbolic and partially hyperbolic skew product diffeomorphisms in dimension three. We describe the critical (either Lipschitz or at all scales Hölder continuous) regularity of such graphs. We provide a formula for their box dimension given in terms of appropriate pressure functions. We distinguish three scenarios according to the base dynamics: Anosov, one-dimensional attractor, or Cantor set. A key ingredient for the dimension arguments in the latter case will be the presence of a so-called fibered blender.


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L. J. Díaz
Affiliation: Departamento de Matemática PUC-Rio, Marquês de São Vicente 225, Gávea, Rio de Janeiro 22451-900, Brazil
Email: lodiaz@mat.puc-rio.br

K. Gelfert
Affiliation: Instituto de Matemática Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Cidade Universitária - Ilha do Fundão, Rio de Janeiro 21945-909, Brazil
Email: gelfert@im.ufrj.br

M. Gröger
Affiliation: Friedrich-Schiller-University Jena, Institute of Mathematics, Ernst-Abbe-Platz 2, 07743 Jena, Germany
Email: maik.groeger@uni-jena.de

T. Jäger
Affiliation: Friedrich-Schiller-University Jena, Institute of Mathematics, Ernst-Abbe-Platz 2, 07743 Jena, Germany
Email: tobias.jaeger@uni-jena.de

DOI: https://doi.org/10.1090/tran/7454
Keywords: Box dimension, fibered blender, invariant graph, hyperbolicity, skew product, topological pressure
Received by editor(s): February 21, 2017
Received by editor(s) in revised form: September 20, 2017, and November 7, 2017
Published electronically: February 27, 2019
Additional Notes: This research was supported, in part, by CNE-FaperjE/26/202.977/2015 and CNPq research grants 302879/2015-3 and 302880/2015-1, by Universal 474406/2013-0 and 474211/2013-4 (Brazil), by EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems FP7-PEOPLE-2012-IRSES 318999 BREUDS, and by DFG Emmy-Noether grant Ja 1721/2-1 and DFG Heisenberg grant Oe 538/6-1. This project was also part of the activities of the Scientific Network “Skew product dynamics and multifractal analysis” (DFG grant Oe 538/3-1).
The first and second authors thank ICERM (USA) and CMUP (Portugal) for their hospitality and financial support.
Article copyright: © Copyright 2019 American Mathematical Society