Carlo Carminati

Tuning and plateaux for the entropy of α-continued fractions

The entropy h(Tα) of α-continued fraction transformations is known to be locally monotone outside a closed, totally disconnected set . We will exploit the explicit description of the fractal structure of 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.

The entropy of α-continued fractions: numerical results

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.

Chaotic dynamics for perturbations of infinite-dimensional Hamiltonian systems ☆

where w(t; z)∈R is the transverse de9ection of the axis of the beam; w(t; 0)=w(t; 1)= wzz(t; 0)=wzz(t; 1)=0, is an external load, ? 0 is a ratio indicating the extensional rigidity and is the damping. The #rst result on the existence of a chaotic dynamics for system (H ) has been given by Holmes and Marsden in [10] for a speci#c periodic forcing perturbation of the type P(t; w(t; z))=f(z) cos(!t). They use the theory of invariant manifolds and of non-linear semigroups in order to extend the classical Melnikov approach for planar ordinary di=erential equations to system (H ).

A canonical thickening of Q and the entropy of a-continued fraction transformations

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.

There is only one KAM curve

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.