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

Normal fault earthquakes or graviquakes

Earthquakes are dissipation of energy throughout elastic waves. Canonically is the elastic energy accumulated during the interseismic period. However, in crustal extensional settings, gravity is the main energy source for hangingwall fault collapsing. Gravitational potential is about 100 times larger than the observed magnitude, far more than enough to explain the earthquake. Therefore, normal faults have a different mechanism of energy accumulation and dissipation (graviquakes) with respect to other tectonic settings (strike-slip and contractional), where elastic energy allows motion even against gravity. The bigger the involved volume, the larger is their magnitude. The steeper the normal fault, the larger is the vertical displacement and the larger is the seismic energy released. Normal faults activate preferentially at about 60° but they can be shallower in low friction rocks. In low static friction rocks, the fault may partly creep dissipating gravitational energy without releasing great amount of seismic energy. The maximum volume involved by graviquakes is smaller than the other tectonic settings, being the activated fault at most about three times the hypocentre depth, explaining their higher b-value and the lower magnitude of the largest recorded events. Having different phenomenology, graviquakes show peculiar precursor

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

Tanaka-Ito α{\alpha}-continued fractions and matching

Two closely related families of ${\alpha}$-continued fractions were introduced in 1981: by Nakada on the one hand, by Tanaka and Ito on the other hand. The behavior of the entropy as a function of the parameter ${\alpha}$ has been studied extensively for Nakada's family, and several of the results have been obtained exploiting an algebraic feature called matching. In this article we show that matching occurs also for Tanaka-Ito ${\alpha}$-continued fractions, and that the parameter space is almost completely covered by matching intervals. Indeed, the set of parameters for which the matching condition does not hold, called bifurcation set, is a zero measure set (even if it has full Hausdorff dimension). This property is also shared by Nakada's ${\alpha}$-continued fractions, and yet there also are some substantial differences: not only does the bifurcation set for Tanaka-Ito continued fractions contain infinitely many rational values, it also contains numbers with unbounded partial quotients