The bifurcation locus for numbers of bounded type

Abstract

We define a family B(t) of compact subsets of the unit interval which generalizes the sets of numbers whose continued fraction expansion has bounded digits. We study how the set B(t) changes as one moves the parameter t, and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behavior as the usual family of quadratic polynomials. The set E of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension 1. We also show that the Hausdorff dimension of B(t) varies continuously with the parameter, and the dimension of each individual set equals the dimension of a corresponding section of the bifurcation set E.

32 pages, 2 figures. Accepted version in Ergodic Th. Dynam. Systems

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Cited by in corpus (6)

• Hausdorff dimensions of perturbations of a conformal iterated function system via thermodynamic formalism
• Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set
• Tanaka-Ito $\alpha$-continued fractions and matching
• On the bifurcation set of unique expansions
• On a class of self-similar sets which contain finitely many common points
• Density spectrum of Cantor measure