Unique Ergodicity Research Articles

Ergodic Theorems for Laminations and Foliations: Recent Results and Perspectives

This report discusses recent results as well as new perspectives in the ergodic theory for Riemann surface laminations, with an emphasis on singular holomorphic foliations by curves. The central notions of these developments are leafwise Poincaré metric, directed positive harmonic currents, multiplicative cocycles, and Lyapunov exponents. We deal with various ergodic theorems for such laminations: random and operator ergodic theorems, (geometric) Birkhoff ergodic theorems, Oseledec multiplicative ergodic theorem, and unique ergodicity theorems. Applications of these theorems are also given. In particular, we define and study the canonical Lyapunov exponents for a large family of singular holomorphic foliations on compact projective surfaces. Topological and algebro-geometric interpretations of these characteristic numbers are also treated. These results highlight the strong similarity as well as the fundamental differences between the ergodic theory of maps and that of Riemann surface laminations. Most of the results reported here are known. However, sufficient conditions for abstract heat diffusions to coincide with the leafwise heat diffusions (Section 5.2) are new ones.

A context in which finite or unique ergodicity is generic

AbstractWe show that for good measures, the set of homeomorphisms of Cantor space which preserve that measure and which have no invariant clopen sets contains a residual set of homeomorphisms which are uniquely ergodic. Additionally, we show that for refinable Bernoulli trial measures, the same set of homeomorphisms contains a residual set of homeomorphisms which admit only finitely many ergodic measures.

Positive Entropy Using Hecke Operators at a Single Place

AbstractWe prove the following statement: let $X=\textrm{SL}_n({{\mathbb{Z}}})\backslash \textrm{SL}_n({{\mathbb{R}}})$ and consider the standard action of the diagonal group $A<\textrm{SL}_n({{\mathbb{R}}})$ on it. Let $\mu $ be an $A$-invariant probability measure on $X$, which is a limit $$\begin{equation*} \mu=\lambda\lim_i|\phi_i|^2dx, \end{equation*}$$where $\phi _i$ are normalized eigenfunctions of the Hecke algebra at some fixed place $p$ and $\lambda>0$ is some positive constant. Then any regular element $a\in A$ acts on $\mu $ with positive entropy on almost every ergodic component. We also prove a similar result for lattices coming from division algebras over ${{\mathbb{Q}}}$ and derive a quantum unique ergodicity result for the associated locally symmetric spaces. This generalizes a result of Brooks and Lindenstrauss [2].

Stationary distribution and ergodicity of a stochastic cholera model with multiple pathways of transmission

A stochastic compartmental model for cholera is investigated, which incorporates the effect of two transmission pathways on the diseases via contaminated water. Multiple stages of infection and multiple states of pathogen and environmental white noises are considered, which include or extend the cholera models in some existing articles. The existence of a unique stationary distribution and ergodicity of the model are obtained by using a suitable Lyapunov function, which determines a critical value R0* corresponding to a basic reproduction number R0 of the ordinary differential equation. The results mean if R0*>1, the system still retains the stability and all individuals are persistent in a long term. In addition, the sufficient conditions for extinction of diseases are derived. What’s more, we provide a new method in constructing a Lyapunov function, which can be successfully applied to other complex high-dimensional systems.

On the Uniform Convergence of Ergodic Averages for $$C^*$$-Dynamical Systems

We investigate some ergodic and spectral properties of general (discrete) $$C^*$$ -dynamical systems $$({\mathfrak A},\Phi )$$ made of a unital $$C^*$$ -algebra and a multiplicative, identity-preserving $$*$$ -map $$\Phi :{\mathfrak A}\rightarrow {\mathfrak A}$$ , particularising the situation when $$({\mathfrak A},\Phi )$$ enjoys the property of unique ergodicity with respect to the fixed-point subalgebra. For $$C^*$$ -dynamical systems enjoying or not the strong ergodic property mentioned above, we provide conditions on $$\lambda $$ in the unit circle $$\{z\in {\mathbb {C}}\mid |z|=1\}$$ and the corresponding eigenspace $${\mathfrak A}_\lambda \subset {\mathfrak A}$$ for which the sequence of Cesaro averages $$\left( \frac{1}{n}\sum _{k=0}^{n-1}\lambda ^{-k}\Phi ^k\right) _{n>0}$$ , converges point-wise in norm. We also describe some pivotal examples coming from quantum probability, to which the obtained results can be applied.

Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise

In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the d-dimensional torus. This class includes the wave equation for d=1 and the beam equation for dle 3. We show that the Gibbs measure is the unique invariant measure for this system. Since the flow does not satisfy the strong Feller property, we introduce a new technique for showing unique ergodicity. This approach may be also useful in situations in which finite-time blowup is possible.

Unique Ergodicity of Deterministic Zero-Sum Differential Games

We study the ergodicity of deterministic two-person zero-sum differential games. This property is defined by the uniform convergence to a constant of either the infinite-horizon discounted value as the discount factor tends to zero, or equivalently, the averaged finite-horizon value as the time goes to infinity. We provide necessary and sufficient conditions for the unique ergodicity of a game. This notion extends the classical one for dynamical systems, namely when ergodicity holds with any (suitable) perturbation of the running payoff function. Our main condition is symmetric between the two players and involve dominions, i.e., subsets of states that one player can make approximately invariant.

Random Band Matrices in the Delocalized Phase I: Quantum Unique Ergodicity and Universality

Consider $N\times N$ symmetric one-dimensional random band matrices with general distribution of the entries and band width $W \geq N^{3/4+\varepsilon}$ for any $\varepsilon>0$. In the bulk of the spectrum and in the large $N$ limit, we obtain the following results. (i) The semicircle law holds up to the scale $N^{-1+\varepsilon}$ for any $\varepsilon>0$. (ii) The eigenvalues locally converge to the point process given by the Gaussian orthogonal ensemble at any fixed energy. (iii) All eigenvectors are delocalized, meaning their ${\rm L}^\infty$ norms are all simultaneously bounded by $N^{-\frac{1}{2}+\varepsilon}$ (after normalization in ${\rm L}^2$) with overwhelming probability, for any $\varepsilon>0$. (iv )Quantum unique ergodicity holds, in the sense that the local ${\rm L}^2$ mass of eigenvectors becomes equidistributed with overwhelming probability. We extend the mean-field reduction method \cite{BouErdYauYin2017}, which required $W=\Omega(N)$, to the current setting $W \ge N^{3/4+\varepsilon}$. Two new ideas are: (1) A new estimate on the generalized resolvent of band matrices when $W \geq N^{3/4+\varepsilon}$. Its proof, along with an improved fluctuation average estimate, will be presented in parts 2 and 3 of this series \cite {BouYanYauYin2018,YanYin2018}. (2) A strong (high probability) version of the quantum unique ergodicity property of random matrices. For its proof, we construct perfect matching observables of eigenvector overlaps and show they satisfying the eigenvector moment flow equation \cite{BouYau2017} under the matrix Brownian motions.

A convexity criterion for unique ergodicity of interval exchange transformations

A criterion for unique ergodicity for points of a curve in the space of interval exchange transformation is given.

Stability Estimates of Markov Semigroups on Abstract States Spaces

In this paper, the stability and perturbation estimates for Markov $$C_0$$-semigroups on abstract state spaces are explored using the Dobrushin ergodicity coefficient. Consequently, a linear relation between the stability of the semigroup and the sensitivity of its fixed point is obtained with respect to the perturbations of the semigroup. This investigation has led to the discovery of perturbation estimates for the time averages of uniform asymptotically stable semigroups. This work also proves the equivalence of uniform and weak ergodicities of time averages of $$C_0$$-Markov semigroups in terms of the ergodicity coefficient, which shines new light on this topic. Finally, in terms of weighted averages, the unique ergodicity of semigroups is also studied. Emphasis is laid on the newly obtained results, which are new discoveries in the classical and non-commutative settings.

Quantum unique ergodicity for half-integral weight automorphic forms

We investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture for half-integral weight automorphic forms. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for both half-integral weight holomorphic Hecke cusp forms for $\Gamma_0(4)$ lying in Kohnen's plus subspace and for half-integral weight Hecke Maa{\ss} cusp forms for $\Gamma_0(4)$ lying in Kohnen's plus subspace. By combining the former result along with an argument of Rudnick, it follows that under GRH the zeros of these holomorphic Hecke cusp equidistribute with respect to hyperbolic measure on $\Gamma_0(4)\backslash \mathbb H$ as the weight tends to infinity.

Some analytic aspects of automorphic forms on GL(2) of minimal type

Let \pi be a cuspidal automorphic representation of PGL _2(\mathbb A_\mathbb Q) of arithmetic conductor C and archimedean parameter T , and let \phi be an L^2 -normalized automorphic form in the space of \pi . The sup-norm problem asks for bounds on \| \phi \|_\infty in terms of C and T . The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the L^2 -mass |\phi|^2 (g) \, d g of \phi . Previous work on these problems in the conductor-aspect has focused on the case that \phi is a newform. In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor p of C , the local component \pi_p is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms \phi for which the local components \phi_p \in \pi_p are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of PGL _2(\mathbb{R}) . For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if \pi_\infty is a holomorphic discrete series of lowest weight k , we obtain the optimal bound C^{1/8 -\epsilon} k^{1/4 - \epsilon} \ll_{\epsilon} |\phi|_\infty \ll_{\epsilon} C^{1/8 + \epsilon} k^{1/4+\epsilon} . We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence [31] for newforms of powerful level.

Teichmüller dynamics and unique ergodicity via currents and Hodge theory

Abstract We present a cohomological proof that recurrence of suitable Teichmüller geodesics implies unique ergodicity of their terminal foliations. This approach also yields concrete estimates for periodic foliations and new results for polygonal billiards.

Measure Rigidity for Horospherical Subgroups of Groups Acting on Trees

Abstract We prove analogues of some of the classical results in homogeneous dynamics in nonlinear setting. Let $G$ be a closed subgroup of the group of automorphisms of a biregular tree and $\Gamma \leq G$ a discrete subgroup. For a large class of groups $G$, we give a classification of the probability measures on $G/\Gamma $ invariant under horospherical subgroups. When $\Gamma $ is a cocompact lattice, we show the unique ergodicity of the horospherical action. We prove Hedlund’s theorem for geometrically finite quotients. Finally, we show equidistribution of large compact orbits.

Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces

Let D be an indefinite quaternion division algebra over {{mathbb {Q}}}. We approach the problem of bounding the sup-norms of automorphic forms phi on D^times ({{mathbb {A}}}) that belong to irreducible automorphic representations and transform via characters of unit groups of orders of D. We obtain a non-trivial upper bound for Vert phi Vert _infty in the level aspect that is valid for arbitrary orders. This generalizes and strengthens previously known upper bounds for Vert phi Vert _infty in the setting of newforms for Eichler orders. In the special case when the index of the order in a maximal order is a squarefull integer N, our result specializes to Vert phi Vert _infty ll _{pi _infty , epsilon } N^{1/3 + epsilon } Vert phi Vert _2. A key application of our result is to automorphic forms phi which correspond at the ramified primes to either minimal vectors, in the sense of Hu et al. (Commun Math Helv, to appear) or p-adic microlocal lifts, in the sense of Nelson in “Microlocal lifts and and quantum unique ergodicity on mathrm{GL}_2({{mathbb {Q}}}_{p})” (Algebra Number Theory 12(9):2033–2064, 2018). For such forms, our bound specializes to Vert phi Vert _infty ll _{ epsilon } C^{frac{1}{6} + epsilon }Vert phi Vert _2 where C is the conductor of the representation pi generated by phi . This improves upon the previously known local boundVert phi Vert _infty ll _{lambda , epsilon } C^{frac{1}{4} + epsilon }Vert phi Vert _2 in these cases.

Exact number of ergodic invariant measures for Bratteli diagrams

For a Bratteli diagram B, we study the simplex M1(B) of probability measures on the path space of B which are invariant with respect to the tail equivalence relation. Equivalently, M1(B) is formed by probability measures invariant with respect to a homeomorphism of a Cantor set. We study relations between the number of ergodic measures from M1(B) and the structure and properties of the diagram B. We prove a criterion and find sufficient conditions of unique ergodicity of a Bratteli diagram, in which case the simplex M1(B) is a singleton. For a finite rank k Bratteli diagram B having exactly l≤k ergodic invariant measures, we explicitly describe the structure of the diagram and find the subdiagrams which support these measures. We find sufficient conditions under which: (i) a Bratteli diagram has a prescribed number (finite or infinite) of ergodic invariant measures, and (ii) the extension of a measure from a uniquely ergodic subdiagram gives a finite ergodic invariant measure. Several examples, including stationary Bratteli diagrams, Pascal-Bratteli diagrams, and Toeplitz flows, are considered.

Subconvex equidistribution of cusp forms: Reduction to Eisenstein observables

Let $\pi$ traverse a sequence of cuspidal automorphic representations of GL(2) with large prime level, unramified central character and bounded infinity type. For G either of the groups GL(1) or PGL(2), let H(G) denote the assertion that subconvexity holds for G-twists of the adjoint $L$-function of $\pi$, with polynomial dependence upon the conductor of the twist. We show that H(GL(1)) implies H(PGL(2)). In geometric terms, H(PGL(2)) corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, H(GL(1)) to the special case in which the relevant sequence of measures is tested against an Eisenstein series.

Quantum Unique Ergodicity of Degenerate Eisenstein Series on GL(n)

We prove quantum unique ergodicity for a subspace of the continuous spectrum spanned by the degenerate Eisenstein Series on GL(n).

Quantitative shrinking target properties for rotations and interval exchanges

This paper presents quantitative shrinking target results for rotations and interval exchange transformations. To do this a quantitative version of a unique ergodicity criterion of Boshernitzan is established.

Ergodic Approximation to Chemical Reaction System with Delay

In order to inherit numerically the ergodicity of the chemical reaction system with delays, we propose and study an Euler-type numerical method from the point of view of stochastic delay differential equations. We not only prove the unique exponential ergodicity of the numerical solution of the approximation, but also present error estimation on invariant measures, which gives order 1 under certain hypotheses. Numerical experiments are provided to illustrate the results.