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The bifurcation locus for numbers of bounded type

Part of: Low-dimensional dynamical systems Elementary number theory Diophantine approximation, transcendental number theory Smooth dynamical systems: general theory

Published online by Cambridge University Press:  08 April 2021

CARLO CARMINATI Open the ORCID record for CARLO CARMINATI[Opens in a new window]  and
GIULIO TIOZZO Open the ORCID record for GIULIO TIOZZO[Opens in a new window]
CARLO CARMINATI
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, PisaI-56127, Italy (e-mail: carminat@dm.unipi.it)
GIULIO TIOZZO*
Affiliation:
Department of Mathematics, University of Toronto, 40 St George St, TorontoON, Canada
*
e-mail: tiozzo@math.utoronto.ca
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Abstract

We define a family $\mathcal {B}(t)$ of compact subsets of the unit interval which provides a filtration of the set of numbers whose continued fraction expansion has bounded digits. We study how the set $\mathcal {B}(t)$ changes as the parameter t ranges in $[0,1]$ , and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behaviour as the family of real quadratic polynomials. The set $\mathcal {E}$ of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension $1$ . The Hausdorff dimension of $\mathcal {B}(t)$ varies continuously with the parameter, and we show that the dimension of each individual set equals the dimension of the corresponding section of the bifurcation set $\mathcal {E}$ .

Keywords

continued fractionsopen dynamical systemsDiophantine numbersHausdorff dimension

MSC classification

Primary: 11A55: Continued fractions 37C45: Dimension theory of dynamical systems
Secondary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 11J06: Markov and Lagrange spectra and generalizations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 7 , July 2022 , pp. 2239 - 2269
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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