Continued fractions with $SL(2, Z)$-branches: combinatorics and entropy
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- by Carlo Carminati, Stefano Isola and Giulio Tiozzo PDF
- Trans. Amer. Math. Soc. 370 (2018), 4927-4973 Request permission
Abstract:
We study the dynamics of a family $K_\alpha$ of discontinuous interval maps whose (infinitely many) branches are Möbius transformations in $SL(2, \mathbb {Z})$ and which arise as the critical-line case of the family of $(a, b)$-continued fractions.
We provide an explicit construction of the bifurcation locus $\mathcal {E}_{KU}$ for this family, showing it is parametrized by Farey words and it has Hausdorff dimension zero. As a consequence, we prove that the metric entropy of $K_\alpha$ is analytic outside the bifurcation set but not differentiable at points of $\mathcal {E}_{KU}$ and that the entropy is monotone as a function of the parameter.
Finally, we prove that the bifurcation set is combinatorially isomorphic to the main cardioid in the Mandelbrot set, providing one more entry to the dictionary developed by the authors between continued fractions and complex dynamics.
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Additional Information
- Carlo Carminati
- Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
- MR Author ID: 348765
- Email: carlo.carminati@unipi.it
- Stefano Isola
- Affiliation: Scuola di Scienze e Tecnologie, Università di Camerino, via Madonna delle Carceri, 62032 Camerino, Italy
- Email: stefano.isola@unicam.it
- Giulio Tiozzo
- Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada
- MR Author ID: 907703
- Email: tiozzo@math.utoronto.ca
- Received by editor(s): February 21, 2014
- Received by editor(s) in revised form: October 14, 2016
- Published electronically: February 21, 2018
- Additional Notes: The authors acknowledge the support of the CRM “Ennio de Giorgi” of Pisa and the program “Dynamical Numbers” at the Max Planck Institute of Bonn.
The first author was partially supported by the GNAMPA group of the Istituto Nazionale di Alta Matematica (INdAM) and the MIUR project “Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite” (PRIN 2010JJ4KPA_008). - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 4927-4973
- MSC (2010): Primary 11A55, 11K50, 37A10
- DOI: https://doi.org/10.1090/tran/7109
- MathSciNet review: 3812101