Unique Ergodicity Research Articles

Subconvexity implies effective quantum unique ergodicity for Hecke–Maaß cusp forms on SL2(ℤ)∖SL2(ℝ)

Subconvexity implies effective quantum unique ergodicity for Hecke–Maaß cusp forms on SL2(ℤ)∖SL2(ℝ)

The ergodicity and uniform large deviations for the 1D stochastic Landau-Lifshitz-Bloch equation

In this article, we consider two aspects asymptotic behaviors for the 1D stochastic Landau-Lifshitz-Bloch equation driven by additive noise including the unique ergodicity and the uniform large deviations. Using a deterministic argument relying on the exponential moment and exponential stability estimates, we first establish the unique ergodicity of invariant measure in H 1. Then, the uniform large deviations is established in C ( [ 0 , T ] ; H 1 ) by the uniform contraction principle.

Unique ergodicity in stochastic electroconvection

We consider a stochastic electroconvection model describing the nonlinear evolution of a surface charge density in a two-dimensional fluid with additive stochastic forcing. We prove the existence and uniqueness of solutions, we define the corresponding Markov semigroup, and we study its Feller properties. When the noise forces enough modes in phase space, we obtain the uniqueness of the smooth invariant measure for the Markov transition kernels associated with the model.

Optimal lower bound on eigenvector overlaps for non-Hermitian random matrices

We consider large non-Hermitian N×N matrices with an additive independent, identically distributed (i.i.d.) noise for each matrix elements. We show that already a small noise of variance 1/N completely thermalises the bulk singular vectors, in particular they satisfy the strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence. In physics terms, we thus extend the Eigenstate Thermalisation Hypothesis, formulated originally by Deutsch [34] and proven for Wigner matrices in [23], to arbitrary non-Hermitian matrices with an i.i.d. noise. As a consequence we obtain an optimal lower bound on the diagonal overlaps of the corresponding non-Hermitian eigenvectors. This quantity, also known as the (square of the) eigenvalue condition number measuring the sensitivity of the eigenvalue to small perturbations, has notoriously escaped rigorous treatment beyond the explicitly computable Ginibre ensemble apart from the very recent upper bounds given in [7] and [45]. As a key tool, we develop a new systematic decomposition of general observables in random matrix theory that governs the size of products of resolvents with deterministic matrices in between.

Minimality and unique ergodicity of Veech 1969 type interval exchange transformations

Minimality and unique ergodicity of Veech 1969 type interval exchange transformations

Bulk universality and quantum unique ergodicity for random band matrices in high dimensions

Bulk universality and quantum unique ergodicity for random band matrices in high dimensions

Spreading speeds for time heterogeneous prey-predator systems with nonlocal diffusion on a lattice

We investigate the spreading behaviour for the solutions of a non-autonomous prey-predator system on a discrete lattice. These time variations are assumed to enjoy an averaging property. This includes periodicity, almost periodicity and unique ergodicity as special cases. The spatial motion of individuals from one site to another is modelled by a discrete convolution operator. In order to take into account external fluctuations such as seasonality, daily variations and so on, the convolution kernels and reaction terms may vary with time. Our analysis of the spreading speeds of invasion of the species is based on the careful and detailed study of the hair-trigger effect and spreading speed for a non-autonomous scalar Fisher-KPP equation on a lattice. Then, we are able to compare the solutions of the prey-predator system with those of a suitable scalar Fisher-KPP equation and derive the invasion speeds of the prey and of the predator.

Support set and unique ergodicity of stochastic KdV equation

This paper presents the support set and unique ergodicity of invariant measure given in Ekren (2018) of stochastic KdV equation, which also offers an answer to the question about support set raised in Wu (2023). To begin with, we investigate the exponential decay of the H2 norm of the solution. Subsequently, we employ the truncated method in conjunction with the dominated convergence theorem to prove the support set and regularity of the invariant measure. Finally, we establish the unique ergodicity with the help of the Birkhoff ergodic theorem, primarily relying on precise estimates of ‖ux‖L∞.

Contributions to the ergodic theory of hyperbolic flows: unique ergodicity for quasi-invariant measures and equilibrium states for the time-one map

Contributions to the ergodic theory of hyperbolic flows: unique ergodicity for quasi-invariant measures and equilibrium states for the time-one map

On the unique ergodicity for a class of 2 dimensional stochastic wave equations

We study the global-in-time dynamics for a stochastic semilinear wave equation with cubic defocusing nonlinearity and additive noise, posed on the 2 2 -dimensional torus. The noise is taken to be slightly more regular than space-time white noise. In this setting, we show existence and uniqueness of an invariant measure for the Markov semigroup generated by the flow over an appropriately chosen Banach space. This extends a result of the second author [Comm. Math. Phys. 377 (2020), pp. 1311–1347] to a situation where the invariant measure is not explicitly known.

Quantum Unique Ergodicity for Cayley Graphs of Quasirandom Groups

A finite group G is called C-quasirandom (by Gowers) if all non-trivial irreducible complex representations of G have dimension at least C. For any unit ell ^{2} function on a finite group we associate the quantum probability measure on the group given by the absolute value squared of the function. We show that if a group is highly quasirandom, in the above sense, then any Cayley graph of this group has an orthonormal eigenbasis of the adjacency operator such that the quantum probability measures of the eigenfunctions put close to the correct proportion of their mass on suitably selected subsets of the group that are not too small.

Spreading speeds for time heterogeneous prey–predator systems with diffusion

We investigate the large time behavior for two components reaction–diffusion systems of prey–predator type in a time varying environment. Here we assume that these variations in time exhibit an averaging property, which will be called mean value in this work. This framework includes in particular time periodicity, almost periodicity and unique ergodicity. We describe the spreading behavior of the prey and the predator, wherein the two populations are able to co-invade the empty space. Our analysis is based the parabolic strong maximum principle for scalar equation and on the derivation of local pointwise estimates that are used to compare the solutions of the prey–predator problem with those of a KPP scalar equation on suitable spatio-temporal domains.

Generalizations of Sturmian sequences associated with N-continued fraction algorithms

Given a positive integer N and x∈[0,1]∖Q, an N-continued fraction expansion of x is defined analogously to the classical continued fraction expansion, but with the numerators being all equal to N. Inspired by Sturmian sequences, we introduce the N-continued fraction sequencesω(x,N) and ωˆ(x,N), which are related to the N-continued fraction expansion of x. They are infinite words over a two letter alphabet obtained as the limit of a directive sequence of certain substitutions, hence they are S-adic sequences. When N=1, we are in the case of the classical continued fraction algorithm, and obtain the well-known Sturmian sequences. We show that ω(x,N) and ωˆ(x,N) are C-balanced for some explicit values of C and compute their factor complexity function. We also obtain uniform word frequencies and deduce unique ergodicity of the associated subshifts. Finally, we provide a Farey-like map for N-continued fraction expansions, which provides an additive version of N-continued fractions, for which we prove ergodicity and give the invariant measure explicitly.

On the ergodic control of ensembles in the presence of non-linear filters

In many sharing-economy applications, as well as in other, conventional applications, one wishes to regulate the behaviour of an ensemble of agents with guarantees on both the regulation of the ensemble in aggregate and the revenue or quality of service associated with each agent. Previous work [Automatica, Volume 108, 108483] has developed guarantees of unique ergodicity when there are linear filters. Here, we extend the guarantees to systems including non-linear elements, such as non-linear filters.

Random Splitting of Fluid Models: Unique Ergodicity and Convergence

Random Splitting of Fluid Models: Unique Ergodicity and Convergence

Invariant probability measures from pseudoholomorphic curves Ⅰ

We introduce a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds using pseudoholomorphic curve techniques from symplectic geometry. These flows include any non-singular volume preserving flow in dimension three, and autonomous Hamiltonian flows on closed, regular energy levels in symplectic manifolds of any dimension. As an application, we use our method to prove the existence of obstructions to unique ergodicity for this class of flows, generalizing results of Taubes and Ginzburg–Niche.

Unique ergodicity of the horocycle flow of a higher genus compact surface with no conjugate points and continuous Green bundles

Unique ergodicity of the horocycle flow of a higher genus compact surface with no conjugate points and continuous Green bundles

Invariant Conditional Expectations and Unique Ergodicity for Anzai Skew-Products

Anzai skew-products are shown to be uniquely ergodic with respect to the fixed-point subalgebra if and only if there is a unique conditional expectation onto such a subalgebra which is invariant under the dynamics. For the particular case of skew-products, this solves a question raised by B. Abadie and K. Dykema in the wider context of $$C^*$$ -dynamical systems.

Cold dynamics in cellular automata: a tutorial

This tutorial is about cellular automata that exhibit 'cold dynamics'. By this we mean zero entropy, stabilization of all orbits, trivial asymptotic dynamics, etc. These are purely transient irreversible dynamics, but they capture many examples from the literature. A rich zoo of properties is presented and discussed: nilpotency and asymptotic, generic or mu-variants, unique ergodicity, convergence, bounded-changeness, freezingness. They all correspond to the 'cold dynamics' paradigm in some way, and we study their links and differences by various examples and results from the literature. Besides dynamical considerations, we also focus on computational aspects: we show how such 'cold cellular automata' can still compute under their dynamical constraint, and what are their computational limitation.

Effective Quantum Unique Ergodicity for Hecke–Maass Newforms and Landau–Siegel Zeros

Abstract We show that Landau–Siegel zeros for Dirichlet L-functions do not exist or that quantum unique ergodicity for $\mathrm{GL}_2$ Hecke–Maass newforms holds with an effective rate of convergence. This follows from a more general result: Landau–Siegel zeros of Dirichlet L-functions repel the zeros of all other automorphic L-functions from the line $\mathrm{Re}(s)=1$.