Dynamics of continued fractions and kneading sequences of unimodal maps

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.Comment: 21 pages, 3 figures. New section added with additional results and applications. Figures and references added. Introduction rearrange

There is only one KAM curve

International audienceWe 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

Global and local minima of α\alpha-Brjuno functions

The aim of this article is to analyze some peculiar features of the global (and local) minima of $\alpha$-Brjuno functions $B_\alpha$ where $\alpha\in[\frac{1}{2},1].$ Our starting point is the result by Balazard--Martin (2020), who showed that the minimum of $B_1$ is attained at $g:=\frac{\sqrt 5 -1}{2}$; analyzing the scaling properties of $B_1$ near $g$ we shall deduce that all preimages of $g$ under the Gauss map are also local minima for $B_1$. Next we consider the problem of characterizing global and local minima of $B_\alpha$ for other values of $\alpha$: we show that for $\alpha\in (g,1)$ the global minimum is again attained at $g$, while for $\alpha=1/2$ the function $B_{1/2}$ attains its minimum at $\gamma:=\sqrt{2}-1$.Comment: 18 pages, 6 figure