We show that each limiting semiclassical measure obtained from a sequence of eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on the entire cosphere bundle. The key new ingredient for the proof is the fractal uncertainty principle, first formulated by Dyatlov-Zahl and proved for porous sets in Bourgain-Dyatlov.

We show that for any amenable group \Gamma and any Z\Gamma-module M of type FL with vanishing Euler characteristic, the entropy of the natural \Gamma-action on the Pontryagin dual of M is equal to the L^2-torsion of M. As a particular case, the entropy of the principal algebraic action associated with the module Z\Gamma / Z\Gamma f is equal to the logarithm of the Fuglede-Kadison determinant of f whenever f is a non-zero-divisor in Z\Gamma. This confirms a conjecture of Deninger. As a key step in the proof we provide a general Szeg\H{o}-type approximation theorem for the Fuglede-Kadison determinant on the group von Neumann algebra of an amenable group. As a consequence of the equality between L^2-torsion and entropy, we show that the L^2-torsion of a non-trivial amenable group with finite classifying space vanishes. This was conjectured by Lueck. Finally, we establish a Milnor-Turaev formula for the L^2-torsion of a finite \Delta-acyclic chain complex.

In this paper, we study time-periodic perturbation of classical systems with two degrees of freedom. A transition chain is established, by passing through small neighborhood of double resonant point, to connect any two cohomology classes corresponding to resonant frequencies. Applying the result to nearly integrable Hamiltonian systems with three degrees of freedom, one obtains a transition chain along which one is able to construct diffusion orbits suggested by Arnold in [A66].

This paper studies a two-strain SIS epidemic model with a competing mechanism and a saturating incidence rate on complex networks. This type of incidence rate could be used to reflect the crowding effect of the infective individuals. We first obtain the associated reproduction numbers for each of the two strains which determine the existence of the boundary equilibria. Stability of the disease-free and boundary equilibria are further examined. Besides, we also show that the two competing strains can coexist under certain conditions. Interestingly, the saturating incidence rate can have specific effects on not only the stability of the boundary equilibria, but also the existence of the coexistence equilibrium. Numerical simulations are presented to support the theoretical results.

We show that for a class of C2 quasiperiodic potentials and for any fixed Diophantinefrequency, the Lyapunov exponent of the corresponding Schrödinger cocycles, as a function of energies, are uniformly positive and weakly Hölder continuous. As a corollary, we obtain that the corresponding integrated density of states is weakly Hölder continuous as well. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to more general SL(2, R)cocycles, which in turn can be applied to get uniform positivity and continuity of Lyapunovexponents around unique nondegenerate extremal points of any smooth potential, and to a certain class of C2 Szegő cocycles.