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

Carminati, Carlo; Isola, Stefano; Tiozzo, Giulio

doi:10.1090/tran/7109

Continued fractions with $SL(2, Z)$-branches: combinatorics and entropy
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by Carlo Carminati, Stefano Isola and Giulio Tiozzo PDF

Trans. Amer. Math. Soc. 370 (2018), 4927-4973 Request permission

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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