The bifurcation locus for numbers of bounded type

Details The bifurcation locus for numbers of bounded type The bifurcation locus for numbers of bounded type

2011-09-02

arxiv.org/abs/1109.0516v1

Mathematics

Dynamical Systems

26 pages, 2 figures

Scientific paper

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. 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.

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