In this paper we construct a correspondence between the parameter spaces of two families of one-dimensional dynamical systems, the $\alpha$-continued fraction transformations $T_\alpha$ and unimodal maps. This correspondence identifies bifurcation parameters in the two families, and allows one to transfer topological and metric properties from one setting to the other. As an application, we recover results about the real slice of the Mandelbrot set, and the set of univoque numbers

We construct a countable family of open intervals contained in (0,1] whose endpoints are quadratic surds and such that their union is a full measure set. We then show that these intervals are precisely the monotonicity intervals of the entropy of α-continued fractions, thus proving a conjecture of Nakada and Natsui.

We study the dynamics of a family Kα of discontinuous interval maps whose (infinitely many) branches are Möbius transformations in SL(2, 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 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 Kα is analytic outside the bifurcation set but not differentiable at points of 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.

Abstract. The entropy h(Tα) of α-continued fraction transformations is known to be locally monotone outside a closed, totally disconnected set E. We will exploit the explicit description of the fractal structure of E to investigate the self-similarities displayed by the graph of the function α → h(Tα). Finally, we completely characterize the plateaux occurring in this graph, and classify the local monotonic behaviour.

No abstract

We consider the standard family of area-preserving twist maps of the annulus and the corresponding KAM curves. Addressing a question raised by Kolmogorov, we show that, instead of viewing these invariant curves as separate objects, each of which having its own Diophantine frequency, one can encode them in a single function of the frequency which is naturally defined in a complex domain containing the real Diophantine frequencies and which is monogenic in the sense of Borel; this implies a remarkable property of quasianalyticity, a form of uniqueness of the monogenic continuation, although real frequencies constitute a natural boundary for the analytic continuation from the Weierstraß point of view because of the density of the resonances.

We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as α-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.

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