Positive Entropy Using Hecke Operators at a Single Place

We 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].

Deterministic and stochastic perturbations of area preserving flows on a two-dimensional torus

We study deterministic and stochastic perturbations of incompressible flows on a two-dimensional torus. Even in the case of purely deterministic perturbations, the long-time behavior of such flows can be stochastic. The stochasticity is caused by instabilities near the saddle points as well as by the ergodic component of the locally Hamiltonian system on the torus.

SELF-SIMILAR CORRECTIONS TO THE ERGODIC THEOREM FOR THE PASCAL-ADIC TRANSFORMATION

Let T be the Pascal-adic transformation. For any measurable function g, we consider the corrections to the ergodic theorem [Formula: see text] When seen as graphs of functions defined on {0,…,ℓ - 1}, we show for a suitable class of functions g that these quantities, once properly renormalized, converge to (part of) the graph of a self-affine function. The latter only depends on the ergodic component of x, and is a deformation of the so-called Blancmange function. We also briefly describe the links with a series of works on Conway recursive $10,000 sequence.

Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dyanmical systems

We establish general criteria for ergodicity and Bernoulliness for volume preserving diffeormorphisms and flows on compact manifolds. We prove that every ergodic component with non-zero Lyapunov exponents of a contact flow is Bernoulli. As an application of our general results, we construct on every compact 3-dimensional manifold a C∞ Riemannian metric whose geodesic flow is Bernoulli.

Loose Bernoullicity is preserved under exponentiation by integrable functions

It is known that if Ω is a Lebesgue space, T:Ω→Ω is a loosely Bernoulli transformation, and L is a fixed non-zero integer, then the transformation S = TL will again be loosely Bernoulli on each ergodic component. In this note, the above stated result is extended to include the case where L is an arbitrary integrable integer-valued function on Ω.