Ergodic Flows Research Articles

Non-classifiability of ergodic flows up to time change

Non-classifiability of ergodic flows up to time change

On the ergodicity of the frame flow on even-dimensional manifolds

It is known that the frame flow on a closed n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n$\\end{document}-dimensional Riemannian manifold with negative sectional curvature is ergodic if n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n$\\end{document} is odd and n≠7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n \ eq 7$\\end{document}. In this paper we study its ergodicity in the remaining cases. For n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n$\\end{document} even and n≠8,134\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n \ eq 8, 134$\\end{document}, we show that: if n≡2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n \\equiv 2$\\end{document} mod 4 or n=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n=4$\\end{document}, the frame flow is ergodic if the manifold is ∼0.3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\sim 0.3$\\end{document}-pinched,if n≡0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n \\equiv 0$\\end{document} mod 4, it is ergodic if the manifold is ∼0.6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\sim 0.6$\\end{document}-pinched. In the three dimensions n=7,8,134\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n=7,8,134$\\end{document}, the respective pinching bounds that we need in order to prove ergodicity are 0.4962..., 0.6212..., and 0.5788.... This is a significant improvement over the previously known results and a step forward towards solving a long-standing conjecture of Brin asserting that 0.25-pinched even-dimensional manifolds have an ergodic frame flow.

Zeta functions and topology of Heisenberg cycles for linear ergodic flows

Placing a Dirac–Schrödinger operator along the orbit of a flow on a compact manifold M defines an {\mathbb{R}} -equivariant spectral triple over the algebra of smooth functions on M . We study some of the properties of these triples, with special attention to their zeta functions. These zeta functions are defined for \operatorname{Re}(s)>1 by \operatorname{Trace}(f_{p}H^{-s}) , where f_{p} is the uniformly continuous function on the real line obtained by restricting the continuous or smooth function f on M to the orbit of a point p\in M , and H = -\frac{\partial^{2}}{\partial x^{2}}+ x^{2} is the harmonic oscillator. The meromorphic continuation property and pole structure of these zeta functions are related to ergodic time averages in dynamics. In the case of the periodic flow on the circle, one obtains a spectral triple over the smooth irrational torus A_{\hslash}^{\infty} \subset A_{\hslash} already studied by Lesch and Moscovici. We strengthen a result of these authors, showing that the zeta function \operatorname{Trace}(aH^{-s}) extends meromorphically to \mathbb{C} for any element a of the C^{*} -algebra A_{\hslash} . Another variant of our construction yields a spectral cycle for A_{\hslash}\otimes A_{1/\hslash} and a spectral triple over a suitable subalgebra with the meromorphic continuation property if \hslash satisfies a Diophantine condition. The class of this cycle defines a fundamental class in the sense that it determines a KK-duality between A_{\hslash} and A_{1/\hslash} . We employ the local index theorem of Connes and Moscovici in order to elaborate an index theorem of Connes for certain classes of differential operators on the line and compute the intersection form on K-theory induced by the fundamental class.

Genericity of trivial Lyapunov spectrum for Lp-cocycles derived from second order linear homogeneous differential equations

We consider a probability space M on which an ergodic flow φt:M→M is defined. We study a family of continuous-time linear cocycles, referred to as kinetic, that are associated with solutions of the second-order linear homogeneous differential equation x¨+α(φt(ω))x˙+β(φt(ω))x=0. Here, the parameters α and β evolve along the φt-orbit of ω∈M. Our main result states that for a generic subset of kinetic continuous-time linear cocycles, where generic means a Baire second category with respect to an Lp-like topology on the infinitesimal generator, the Lyapunov spectrum is trivial.

Integrating eco‐evolutionary dynamics into matrix population models for structured populations: Discrete and continuous frameworks

AbstractState‐structured populations are ubiquitous in biology, from the age‐structure of animal societies to the life cycles of parasitic species. Understanding how this structure contributes to eco‐evolutionary dynamics is critical not only for fundamental understanding but also for conservation and treatment purposes. Although some methods have been developed in the literature for modelling eco‐evolutionary dynamics in structured population, such methods are wholly lacking in the function evolutionary game theoretic framework.In this paper, we integrate standard matrix population modelling into the function framework to create a theoretical framework to probe eco‐evolutionary dynamics in structured populations. This framework encompasses age‐ and stage‐structured matrix models with basic density‐ and frequency‐dependent transition rates and probabilities.For both discrete and continuous time models, we define and characterize asymptotic properties of the system such as eco‐evolutionary equilibria (including ESSs) and the convergence stability of these equilibria. For multistate structured populations, we introduce an ergodic flow preserving folding method for analysing such models.The methods developed in this paper for state‐structured populations and their extensions to multistate‐structured populations provide a simple way to create, analyse and simulate eco‐evolutionary dynamics in structured populations. Furthermore, their generality allows these techniques to be applied to a variety of problems in ecology and evolution.

Plenty of hyperbolicity on a class of linear homogeneous jerk differential equations

We consider 3times 3 partially hyperbolic linear differential systems over an ergodic flow X^t and derived from the linear homogeneous differential equation dddot{x}(t)+beta (X^t(t))dot{x}(X^t(t))+gamma (t) x(t)=0. Assuming that the partial hyperbolic decomposition E^soplus E^coplus E^u is proper and displays a zero Lyapunov exponent along the central direction E^c we prove that some C^0 perturbation of the parameters beta (t) and gamma (t) can be done in order to obtain non-zero Lyapunov exponents and so a chaotic behaviour of the solution.

Riemann moduli spaces are quantum ergodic

In this note we show that the Riemann moduli spaces $M_{\gamma,n}$ equipped with the Weil–Petersson metric are quantum ergodic for $3 \gamma + n \geq 4$. We also provide other examples of singular spaces with ergodic geodesic flow for which quantum ergodicity holds.

BERNOULLI ACTIONS OF TYPE III WITH PRESCRIBED ASSOCIATED FLOW

AbstractWe prove that many, but not all, injective factors arise as crossed products by nonsingular Bernoulli actions of the group $\mathbb {Z}$ . We obtain this result by proving a completely general result on the ergodicity, type and Krieger’s associated flow for Bernoulli shifts with arbitrary base spaces. We prove that the associated flow must satisfy a structural property of infinite divisibility. Conversely, we prove that all almost periodic flows, as well as many other ergodic flows, do arise as associated flow of a weakly mixing Bernoulli action of any infinite amenable group. As a byproduct, we prove that all injective factors with almost periodic flow of weights are infinite tensor products of $2 \times 2$ matrices. Finally, we construct Poisson suspension actions with prescribed associated flow for any locally compact second countable group that does not have property (T).

Quantum ergodicity for pseudo-Laplacians

We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on surfaces with hyperbolic cusps and ergodic geodesic flows.

Reconstruction of Diffusion Coefficients and Power Exponents from Single Lagrangian Trajectories

Using discrete wavelets, a novel technique is developed to estimate turbulent diffusion coefficients and power exponents from single Lagrangian particle trajectories. The technique differs from the classical approach (Davis (1991)’s technique) because averaging over a statistical ensemble of the mean square displacement (<X2>) is replaced by averaging along a single Lagrangian trajectory X(t) = {X(t), Y(t)}. Metzler et al. (2014) have demonstrated that for an ergodic (for example, normal diffusion) flow, the mean square displacement is <X2> = limT→∞τX2(T,s), where τX2 (T, s) = 1/(T − s) ∫0T−s(X(t+Δt) − X(t))2 dt, T and s are observational and lag times but for weak non-ergodic (such as super-diffusion and sub-diffusion) flows <X2> = limT→∞≪τX2(T,s)≫, where ≪…≫ is some additional averaging. Numerical calculations for surface drifters in the Black Sea and isobaric RAFOS floats deployed at mid depths in the California Current system demonstrated that the reconstructed diffusion coefficients were smaller than those calculated by Davis (1991)’s technique. This difference is caused by the choice of the Lagrangian mean. The technique proposed here is applied to the analysis of Lagrangian motions in the Black Sea (horizontal diffusion coefficients varied from 105 to 106 cm2/s) and for the sub-diffusion of two RAFOS floats in the California Current system where power exponents varied from 0.65 to 0.72. RAFOS float motions were found to be strongly non-ergodic and non-Gaussian.

Noncommutative weighted individual ergodic theorems with continuous time

We show that ergodic flows in the noncommutative [Formula: see text]-space (associated with a semifinite von Neumann algebra) generated by continuous semigroups of positive Dunford–Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly. The corresponding local ergodic theorem is also proved. We then extend these results to arbitrary noncommutative fully symmetric spaces and present applications to noncommutative Orlicz (in particular, noncommutative [Formula: see text]-spaces), Lorentz, and Marcinkiewicz spaces. The commutative counterparts of the results are derived.

On ergodic flows with simple Lebesgue spectrum

We prove the existence of ergodic flows with invariant probability measure having a Lebesgue spectrum of multiplicity . Bibliography: 15 titles.

On ergodic flows with simple Lebesgue spectrum

В работе доказывается существование эргодических потоков с инвариантной вероятностной мерой, обладающих лебеговским спектром кратности $1$. Библиография: 15 названий.

The Lyapunov exponents of generic skew-product compact semiflows

Let $${\mathscr {F}}_K$$ denote the set of infinite-dimensional cocycles over a $$\mu $$ -ergodic flow $$\varphi ^t:M\rightarrow M$$ and with fiber dynamics given by a compact semiflow on a Hilbert space. We prove that there exists a residual subset $${\mathscr {R}}$$ of $${\mathscr {F}}_K$$ such that for $$\Phi \in {\mathscr {R}}$$ and for $$\mu $$ -almost every $$x\in M$$ , either:

Inner radius of nodal domains of quantum ergodic eigenfunctions

In this short note we show that the lower bounds of Mangoubi on the inner radius of nodal domains can be improved for quantum ergodic sequences of eigenfunctions, according to a certain power of the radius of shrinking balls on which the eigenfunctions equidistribute. We prove such improvements using a quick application of our recent results (arXiv:1606.02057), which give modified growth estimates for eigenfunctions that equidistribute on small balls. Since by the results of Han and Hezari-Riviere small scale QE holds for negatively curved manifolds on logarithmically shrinking balls, we get logarithmic improvements on the inner radius of such manifolds. We also get improvements for manifolds with ergodic geodesic flows. In addition using the small scale equidistribution results of Lester-Rudnick, one gets polynomial betterments of for toral eigenfunctions in dimensions $n \geq 3$. The results work only for a full density subsequence of eigenfunctions.

On the Lower Bound of the Inner Radius of Nodal Domains

We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions varphi _{lambda } on a closed Riemannian manifold (M, g) . In the real-analytic case, we present an improvement of the currently best-known bounds, due to Mangoubi (Commun Partial Differ Equ 33:1611–1621, 2008; Can Math Bull 51(2):249–260, 2008). Furthermore, using recent results of Hezari (P Am Math Soc, 2016, https://doi.org/10.1090/proc/13766; Anal PDE 11(4):855–871, 2018), we obtain log -type improvements in the case of negative curvature and improved bounds for (M, g) possessing an ergodic geodesic flow.

Carlson's theorem for different measures

We use an observation of Bohr connecting Dirichlet series in the right half plane C+ to power series on the polydisk to interpret Carlson's theorem about integrals in the mean as a special case of the ergodic theorem by considering any vertical line in the half plane as an ergodic flow on the polytorus. Of particular interest is the imaginary axis because Carlson's theorem for Lebesgue measure does not hold there. In this note, we construct measures for which Carlson's theorem does hold on the imaginary axis for functions in the Dirichlet series analog of the disk algebra A(C+).

Global and cocycle attractors for non-autonomous reaction-diffusion equations. The case of null upper Lyapunov exponent

In this paper we obtain a detailed description of the global and cocycle attractors for the skew-product semiflows induced by the mild solutions of a family of scalar linear-dissipative parabolic problems over a minimal and uniquely ergodic flow. We consider the case of null upper Lyapunov exponent for the linear part of the problem. Then, two different types of attractors can appear, depending on whether the linear equations have a bounded or an unbounded associated real cocycle. In the first case (e.g. in periodic equations), the structure of the attractor is simple, whereas in the second case (which occurs in aperiodic equations), the attractor is a pinched set with a complicated structure. We describe situations when the attractor is chaotic in measure in the sense of Li–Yorke. Besides, we obtain a non-autonomous discontinuous pitchfork bifurcation scenario for concave equations, applicable for instance to a linear-dissipative version of the Chafee–Infante equation.

On isomorphism problem for von Neumann flows with one discontinuity

A von Neumann flow is a special flow over an irrational rotation of the circle and under a piecewise C1 roof function with a non-zero sum of jumps. We prove that the absolute value of the jump is a (measure theoretic) invariant in the class of von Neumann special flows with one discontinuity, i.e., two ergodic von Neumann flows with one discontinuity are not isomorphic if the jumps of the roof functions have different absolute values, regardless of the irrational rotation in the base.

Eddy diffusivity of quasi-neutrally-buoyant inertial particles

We investigate the large-scale transport properties of quasi-neutrally-buoyant inertial particles carried by incompressible zero-mean periodic or steady ergodic flows. We show how to compute large-scale indicators such as the inertial-particle terminal velocity and eddy diffusivity from first principles in a perturbative expansion around the limit of added-mass factor close to unity. Physically, this limit corresponds to the case where the mass density of the particles is constant and close in value to the mass density of the fluid which is also constant. Our approach differs from the usual over-damped expansion inasmuch we do not assume a separation of time scales between thermalization and small-scale convection effects. For general incompressible flows, we derive closed-form cell equations for the auxiliary quantities determining the terminal velocity and effective diffusivity. In the special case of parallel flows these equations admit explicit analytic solution. We use parallel flows to show that our approach enables to shed light onto the behavior of terminal velocity and effective diffusivity for Stokes numbers of the order of unity.