Hyperbolic graphs: Critical regularity and box dimension
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- by L. J. Díaz, K. Gelfert, M. Gröger and T. Jäger PDF
- Trans. Amer. Math. Soc. 371 (2019), 8535-8585 Request permission
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.References
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Additional Information
- 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
- MR Author ID: 687384
- Email: gelfert@im.ufrj.br
- M. Gröger
- Affiliation: Friedrich-Schiller-University Jena, Institute of Mathematics, Ernst-Abbe-Platz 2, 07743 Jena, Germany
- MR Author ID: 1016358
- 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
- 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. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 8535-8585
- MSC (2010): Primary 37C45, 37D20, 37D35, 37D30
- DOI: https://doi.org/10.1090/tran/7454
- MathSciNet review: 3955556