Related papers: Quantum Unique Ergodicity for maps on the torus

We consider some analogs of the quantum unique ergodicity conjecture for geodesics, horocycles, or ``shrinking'' families of sets. In particular, we prove the analog of the QUE conjecture for Eisenstein series restricted to the infinite…

We examine the consequences of classical ergodicity for the localization properties of individual quantum eigenstates in the classical limit. We note that the well known Schnirelman result is a weaker form of quantum ergodicity than the one…

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take the L^2…

For manifolds with geodesic flow that is ergodic on the unit tangent bundle, the quantum ergodicity theorem implies that almost all Laplacian eigenfunctions become equidistributed as the eigenvalue goes to infinity. For a locally symmetric…

We consider the quantized hyperbolic automorphisms on the 2-dimensional torus (or generalized quantum cat maps), and study the localization properties of their eigenstates in phase space, in the semiclassical limit. We prove that if the…

In this paper, we prove a uniform version of quantum unique ergodicity for high-frequency eigensections of a certain series of unitary flat bundles over arithmetic surfaces.

We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random…

We prove the existence of topological quantum error correcting codes with encoding rates $k/n$ asymptotically approaching the maximum possible value. Explicit constructions of these topological codes are presented using surfaces of…

Consider a sequence of finite regular graphs (GN) converging, in the sense of Benjamini-Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non-zero…

We prove that any ergodic endomorphism on torus admits a sequence of periodic orbits uniformly distributed in the metric sense. As a corollary, an endomorphism on torus is ergodic if and only if the Haar measure can be approximated by…

We propose a generalization of the model of classical baker map on the torus, in which the images of two parts of the phase space do overlap. This transformation is irreversible and cannot be quantized by means of a unitary Floquet…

We study the ergodic optimization problem over a real analytic expanding circle map. We show that in both the topological and the measure-theoretical senses, a typical $C^r$ performance function has a unique maximizing measure and the…

We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in…

Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface $H$ so that restrictions $\phi_j |_H$ to $H$ of $\Delta$-eigenfunctions of Riemannian manifolds $(M, g)$ with ergodic geodesic flow are quantum ergodic…

A new proof is given of Quantum Ergodicity for Eisenstein Series for cusped hyperbolic surfaces. This result is also extended to higher dimensional examples, with variable curvature.

In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski.

This short note proves that a Laplacian cannot be quantum uniquely ergodic if it possesses a quasimode of order zero which (i) has a singular limit, and (ii) is a linear combination of a uniformly bounded number of eigenfunctions (modulo an…

We define and study the properties of the infinite dimensional quantized Kronecker flow. This $\bC^*$-dynamical system arises as a quantization of the corresponding flow on an infinite dimensional torus. We prove an ergodic theorem for a…

Consider a sequence of possibly random graphs $G_N=(V_N, E_N)$, $N\ge 1$, whose vertices's have i.i.d. weights $\{W^N_x : x\in V_N\}$ with a distribution belonging to the basin of attraction of an $\alpha$-stable law, $0<\alpha<1$. Let…

We show that the topological entropy is monotonic for unimodal interval maps which are obtained from the restriction of quadratic rational maps with real coefficients. This is done by ruling out the existence of certain post-critical curves…