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Transactions of the American Mathematical Society

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

 
 

 

Continued fractions with $ SL(2, Z)$-branches: combinatorics and entropy


Authors: Carlo Carminati, Stefano Isola and Giulio Tiozzo
Journal: Trans. Amer. Math. Soc. 370 (2018), 4927-4973
MSC (2010): Primary 11A55, 11K50, 37A10
DOI: https://doi.org/10.1090/tran/7109
Published electronically: February 21, 2018
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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
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
Email: tiozzo@math.utoronto.ca

DOI: https://doi.org/10.1090/tran/7109
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).
Article copyright: © Copyright 2018 American Mathematical Society

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