Given a dynamical system, we prove that the shortest distance between two n-orbits scales like n to a power even when the system has slow mixing properties, thus building and improving on results of Barros, Liao and the first author [On the shortest distance between orbits and the longest common substring problem. Adv. Math. 344 (2019), 311–339]. We also extend these results to flows. Finally, we give an example for which the shortest distance between two orbits has no scaling limit.

We prove a concentration inequality for sequential dynamical systems of the unit interval enjoying an exponential loss of memory in the BVnorm and we investigate several of its consequences. In particular, this covers compositions of $\unicode[STIX]{x1D6FD}$ -transformations, with all $\unicode[STIX]{x1D6FD}$ lying in a neighborhood of a fixed $\unicode[STIX]{x1D6FD}_{\star }>1$ , and systems satisfying a covering-type assumption.

We consider toral extensions of hyperbolic dynamical systems. We prove that its quantitative recurrence (also with respect to given observables) and hitting time scale behavior depend on the arithmetical properties of the extension. By this we show that those systems have a polynomial decay of correlations with respect to $C^{r}$ observables, and give estimations for its exponent, which depend on $r$ and on the arithmetical properties of the system. We also show examples of systems of this kind having no shrinking target property, and having a trivial limit distribution of return time statistics.

A SiC-based ceramic foam applied in solar thermal processes was characterized in detail in terms of its textural parameters and its radiative properties. Scanning electron microscopy and x-ray µ-tomography were first performed to investigate the 3D texture of the sample at several length scales. Infrared reflectance microscopy was also applied to probe the local optical responses on the struts constituting the foam. Based on the whole set of experimental data, a numerical tool (C++) was implemented to reconstruct virtual SiC foams. A Monte Carlo Ray Tracing code (iMorphRad, C++) was then used to compute the normal spectral emittance for the real SiC foam and for another reconstructed SiC foam with similar textural features. The two numerically determined emittances were then compared with previous infrared spectroscopy experimental measurements. This numerical procedure enables us to propose a methodology for the design of SiC foams with prescribed radiative properties.

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation with the weight function. Firstly, we introduce local Carleman estimates for elliptic operators and deduce unique continuation properties as well as interpolation inequalities. These latter inequalities yield a remarkable spectral inequality and the null controllability of the heat equation. Secondly, we prove Carleman estimates for parabolic operators. We state them locally in space at first, and patch them together to obtain a global estimate. This second approach also yields the null controllability of the heat equation.

We study Poincaré recurrence for flows and observations of flows. For Anosov flow, we prove that the recurrence rates are linked to the local dimension of the invariant measure. More generally, we give for the recurrence rates for the observations an upper bound depending on the push-forward measure. When the flow is metrically isomorphic to a suspension flow for which the dynamic on the base is rapidly mixing, we prove the existence of a lower bound for the recurrence rates for the observations. We apply these results to the geodesic flow and we compute the recurrence rates for a particular observation of the geodesic flow, i.e. the projection on the manifold.