In this paper we study persistence features of topological entropy and periodic orbit growth of Hamiltonian diffeomorphisms on surfaces with respect to Hofer's metric. We exhibit stability of these dynamical quantities in a rather strong sense for a specific family of maps studied by Polterovich and Shelukhin. A crucial ingredient comes from enhancement of lower bounds for the topological entropy and orbit growth forced by a periodic point, formulated in terms of the geometric self-intersection number and a variant of Turaev's cobracket of the free homotopy class that it induces. Those bounds are obtained within the framework of Le Calvez and Tal's forcing theory.

We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schrödinger cocycle over a skew-shift base with a cosine potential and the golden ratio as frequency. For coupling below 1, which is the threshold for Herman’s subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large-deviation estimates at that scale. Our main results demonstrate that these finite-size conditions imply the positivity of the infinite-volume Lyapunov exponent. This paper shows that it is possible to make the techniques developed for the study of Schrödinger operators with deterministic potentials, based on large-deviation estimates and the avalanche principle, effective.

In the realm of Delone sets in locally compact, second countable Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty–Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

The parareal in time algorithm allows for efficient parallel numerical simulations oftime-dependent problems. It is based on a decomposition of the time interval intosubintervals, and on a predictor-corrector strategy, where the propagations over eachsubinterval for the corrector stage are concurrently performed on the different processorsthat are available. In this article, we are concerned with the long time integration ofHamiltonian systems. Geometric, structure-preserving integrators are preferably employedfor such systems because they show interesting numerical properties, in particularexcellent preservation of the total energy of the system. Using a symmetrization procedureand/or a (possibly also symmetric) projection step, we introduce here several variants ofthe original plain parareal in time algorithm [L. Baffico, et al. Phys. Rev. E66 (2002) 057701; G. Bal and Y. Maday, A parareal timediscretization for nonlinear PDE’s with application to the pricing of an American put, inRecent developments in domain decomposition methods, Lect.Notes Comput. Sci. Eng. 23 (2002) 189–202; J.-L. Lions, Y. Madayand G. Turinici, C. R. Acad. Sci. Paris, Série I 332 (2001)661–668.] that are better adapted to the Hamiltonian context. These variants arecompatible with the geometric structure of the exact dynamics, and are easy to implement.Numerical tests on several model systems illustrate the remarkable properties of theproposed parareal integrators over long integration times. Some formal elements ofunderstanding are also provided.

We study Hamiltonian systems which generate extremal flows of regularvariational problems on smooth manifolds and demonstrate thatnegativity of the generalized curvature of such a system impliesthe existence of a global smooth optimal synthesis for the infinitehorizon problem.We also show that in the Euclidean case negativity of the generalized curvature is a consequence ofthe convexity of the Lagrangian with respect to the pair of arguments. Finally, we give a generic classification for 1-dimensional problems.