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 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.

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.

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.

The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. This extends the result in Avila et al. (Ann Math 184:527–558, ADK16), where integrability was assumed on a larger set. In particular, it shows that (local) integrability near the boundary implies global integrability. One of the crucial ideas in the proof consists in analyzing Taylor expansion of the corresponding action-angle coordinates with respect to the eccentricity parameter, deriving and studying higher order conditions for the preservation of integrable rational caustics.

We study Schrödinger operators with a one-frequency analytic potential, focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials. From the dynamical point of view, the transition signals the emergence of non-uniform hyperbolicity, so the dependence of the Lyapunov exponent with respect to parameters plays a central role in the analysis. Though often ill-behaved by conventional measures, we show that the Lyapunov exponent is in fact remarkably regular in a “stratified sense” which we define: the irregularity comes from the matching of nice (analytic or smooth) functions along sets with complicated geometry. This result allows us to establish that the “critical set” for the transition lies within countably many codimension one subvarieties of the (infinite-dimensional) parameter space. A more refined renormalization-based analysis shows that the critical set is rather thin within those subvarieties, and allows us to conclude that a typical potential has no critical energies. Such acritical potentials also form an open set and have several interesting properties: only finitely many “phase transitions” may happen, but never at any specific point in the spectrum, and the Lyapunov exponent is minorated in the region of the spectrum where it is positive. On the other hand, we do show that the number of phase transitions can be arbitrarily large.

In this paper we show that for a generic strictly convex domain, one can recover the eigendata corresponding to Aubry–Mather periodic orbits of the induced billiard map from the (maximal) marked length spectrum of the domain.

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.

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].

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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.

In this paper, we propose a direct method to evaluate Hankel determinants for some generating functions satisfying a certain type of quadratic equations, which cover generating functions of Catalan numbers, Motzkin numbers and Schrder numbers. Additionally, four recent conjectures proposed by Cigler (2011) [3] are proved.

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.

We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products q_n d(q_n θ,Z) where the qn's are the denominators of the convergents associated with the real number θ by the ordinary continued fraction algorithm. Beside these two main results, we show that when d≥2, for almost all vectors θ∈R^d, lim inf_{n→∞} q_{n+k} d(q_n θ,Z^d)=0 for all positive integers k, where (q_n)_{n∈N} is the sequence of best approximation denominators of θ.

We study infinite iterated functions systems (IIFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focusing on IFSs that do not satisfy the open set condition. By assuming the logarithmic distortion property and some cardinality growth condition, we obtain a formula for the Hausdorff, box, and packing dimensions of the limit set in terms of certain topological pressure. By assuming, in addition, the weak separation condition, we show that these dimensions are equal to the growth dimension of the limit set.

By constructing an infinite graph-directed iterated iterated function system associated with a finite iterated function system, we develop a new approach for proving the differentiability of the $L^q$-spectrum and establishing the multifractal formalism of certain self-similar measures with overlaps, especially those defined by similitudes with different contraction ratios. We apply our technique to a well-known class of self-similar measures of generalized finite type.

We show that given $${n \geqslant 3}$$ , $${q \geqslant 1}$$ , and a finite set $${\{y_1, \ldots, y_q \}}$$ in $${\mathbb{R}^n}$$ there exists a quasiregular mapping $${\mathbb{R}^n\to \mathbb{R}^n}$$ omitting exactly points $${y_1, \ldots, y_q}$$ .

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Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp “deflection” estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.

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.

This work is concerned with zeta functions of two-dimensional shifts of finite type. A two-dimensional zeta function ζ0(s) which generalizes the Artin-Mazur zeta function was given by Lind for Z2-action φ. The n-th order zeta function ζn of φ on Zn×∞, n ≥ 1, is studied first. The trace operator Tn which is the transition matrix for x-periodic patterns of period n with height 2 is rotationally symmetric. The rotational symmetry of Tn induces the reduced trace operator τn and ζn = (det (I − snτn))−1. The zeta function ζ =∞ Qn=1 (det (I − snτn))−1 in the x-direction is now a reciprocal of an infinite product of polynomials. The zeta function can be presented in the y-direction and in the coordinates of any unimodular transformation in GL2(Z). Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function ζ0(s). The Taylor series at the origin for these zeta functions are equal with integer coefficients, yielding a family of identities which are of interest in number theory. The method applies to thermodynamic zeta functions for the Ising model with finite range interactions.

We study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds. Assuming that every non-trivial element acts ergodically, we prove that such actions are mixing of all orders. We further show exponential 2-mixing and 3-mixing. As an application we prove smooth cocycle rigidity for higher-rank abelian groups of nilmanifold automorphisms.

We study the effective fronts of first order front propagations in two dimensions (n=2) in the periodic setting. Using PDE-based approaches, we show that for every α∈(0,1), the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts for given front speeds in C^{1,α}(T^2,(0,∞)). This result can also be formulated in the language of stable norms corresponding to periodic metrics in T^2. Similar results were known long time ago when n≥3 for front speeds in C^∞(T^n,(0,∞)). Due to topological restrictions, the two dimensional case is much more subtle. In fact, the effective front is C^1, which cannot be a polygon, for given C^{1,1}(T^2,(0,∞)) front speeds. Our regularity requirements on front speeds are hence optimal. To the best of our knowledge, this is the first time that polygonal effective fronts have been constructed in two dimensions.

We develop a generic game platform that can be used to model various real-world systems with multiple intelligent cloud-computing pools and parallel-queues for resources-competing users. Inside the platform, the software structure is modelled as Blockchain. All the users are associated with Big Data arrival streams whose random dynamics is modelled by triply stochastic renewal reward processes (TSRRPs). Each user may be served simultaneously by multiple pools while each pool with parallel- servers may also serve multi-users at the same time via smart policies in the Blockchain, e.g. a Nash equilibrium point myopically at each fixed time to a game-theoretic scheduling problem. To illustrate the effectiveness of our game platform, we model the performance measures of its internal data flow dynamics (queue length and workload processes) as reflecting diffusion with regime-switchings (RDRSs) under our scheduling policies. By RDRS models, we can prove our myopic game-theoretic policy to be an asymptotic Pareto minimal-dual-cost Nash equilibrium one globally over the whole time horizon to a randomly evolving dynamic game problem. Iterative schemes for simulating our multi-dimensional RDRS models are also developed with the support of numerical comparisons.