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The bifurcation locus for numbers of bounded type

Published online by Cambridge University Press:  08 April 2021

Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, PisaI-56127, Italy (e-mail:
Department of Mathematics, University of Toronto, 40 St George St, TorontoON, Canada


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 $\mathcal {E}$ .

Original Article
© The Author(s), 2021. Published by Cambridge University Press

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