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.

We study the two main types of trajectories of the ABC ow in the near-integrable regime: spiral orbits and edge orbits. The former are helical orbits which are perturbations of similar orbits that exist in the integrable regime, while the latter exist only in the non-integrable regime. We prove existence of ballistic (i.e., linearly growing) spiral orbits by using the contraction mapping principle in the Hamiltonian formulation, and we also nd and analyze ballistic edge orbits. We discuss the relationship of existence of these orbits with questions concerning front propagation in the presence of ows, in particular, the question of linear (i.e., maximal possible) front speed enhancement rate for ABC ows.

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.

In this paper, we prove that the ODE system, $ \dot{x}=sinz+cosy$ $ \dot{y}=sinx+cosz$ $ \dot{z}=siny+cosx$ whose right-hand side is the Arnold-Beltrami-Childress (ABC) flow with parameters A = B = C = 1, has periodic orbits on $(2 \pi T)^3$ with rotation vectors parallel to (1, 0, 0), (0, 1, 0), and (0, 0, 1). An application of this result is that the well-known G-equation model for turbulent combustion with this ABC flow on $R^3$ has a linear (i.e., maximal possible) flame speed enhancement rate as the amplitude of the flow grows.

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].

Data-based detection and quantification of causation in complex, nonlinear dynamical systems is of paramount importance to science, engineering, and beyond. Inspired by the widely used methodology in recent years, the cross-map-based techniques, we develop a general framework to advance towards a comprehensive understanding of dynamical causal mechanisms, which is consistent with the natural interpretation of causality. In particular, instead of measuring the smoothness of the cross-map as conventionally implemented, we define causation through measuring the scaling law for the continuity of the investigated dynamical system directly. The uncovered scaling law enables accurate, reliable, and efficient detection of causation and assessment of its strength in general complex dynamical systems, outperforming those existing representative methods. The continuity scaling-based framework is rigorously established and demonstrated using datasets from model complex systems and the real world.

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.

The commonly used spatial entropy hr(U) of the multi-dimensional shift space U is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space Zd, d ≥ 2. This work studies spatial entropy hΩ(U) of shift space U on general expanding system Ω = {Ω(n)}∞ n=1 where Ω(n) is increasing finite sublattices and expands to Zd. Ω is called genuinely d-dimensional if Ω(n) contains no lower-dimensional part whose size is comparable to that of its d-dimensional part. We show that hr(U) is the supremum of hΩ(U) for all genuinely two-dimensional Ω. Furthermore, when Ω is genuinely d-dimensional and satisfies certain conditions, then hΩ(U) = hr(U). On the contrary, when Ω(n) contains a lower-dimensional part, then hr(U) < hΩ(U) for some U. Therefore, hr(U) is appropriate to be the d-dimensional spatial entropy.

For two-sided topological Markov chains, we show that the set of points for which the two-sided Birkhoff averages of a continuous function diverge is residual. We also show that the set of points for which the Birkhoff averages have a given set of accumulation points other than a singleton is residual. A nontrivial consequence of our results is that the set of points for which the local entropies of an invariant measure on a locally maximal hyperbolic set does not exist is residual. This strongly contrasts to the Shannon–McMillan–Breiman theorem in the context of ergodic theory, which says that local entropies exist on a full measure set.