• Corpus ID: 232068740

Entropy of logarithmic modes

  title={Entropy of logarithmic modes},
  author={Suresh Eswarathasan},
Consider (M, g) a compact, boundaryless Riemannian manifold admitting an Anosov geodesic flow. Let ε < min{1, λmax 2 } where λmax is the maximal expansion rate for (M, g). We study the semiclassical measures μsc of ε-logarithmic modes, which are quasimodes spectrally supported in intervals of width ε ~ | log ~| , of the Laplace-Beltrami operator on M . Under a technical assumption on the support of a signed measure related to μsc, we show that the lower bound for the Kolmogorov-Sinai entropy of… 


We study the large eigenvalue limit for the eigenfunctions of the Laplacian, on a compact manifold of negative curvature - in fact, we only assume that the geodesic flow has the Anosov property. In
Strong Scarring of Logarithmic Quasimodes
We consider a semiclassical (pseudo)differential operator on a compact surface $(M,g)$, such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit $\gamma$ at
Hecke theory and equidistribution for the quantization of linear maps of the torus
1.1. Background. One of the key issues of “Quantum Chaos” is the nature of the semiclassical limit of eigenstates of classically chaotic systems. When the classical system is given by the geodesic
Joint quasimodes, positive entropy, and quantum unique ergodicity
We study joint quasimodes of the Laplacian and one Hecke operator on compact congruence surfaces, and give conditions on the orders of the quasimodes that guarantee positive entropy on almost every
Logarithmic-Scale Quasimodes that do not Equidistribute
Given any compact hyperbolic surface $M$, and a closed geodesic on $M$, we construct of a sequence of quasimodes on $M$ whose microlocal lifts concentrate positive mass on the geodesic. Thus, the
Spectral gaps without the pressure condition
For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many
On the Entropy of Quantum Limits for 2-Dimensional Cat Maps
We study semiclassical measures, or quantum limits, for quantized hyperbolic automorphisms of $${\mathbb{T}^2}$$ . We show that any quantum limit has the following property: if a weight α is carried
Scarred Eigenstates for Quantum Cat Maps of Minimal Periods
In this paper we construct a sequence of eigenfunctions of the ``quantum Arnold's cat map'' that, in the semiclassical limit, shows a strong scarring phenomenon on the periodic orbits of the
A Haar component for quantum limits on locally symmetric spaces
We prove lower bounds for the entropy of limit measures associated to non-degenerate sequences of eigenfunctions on locally symmetric spaces of non-positive curvature. In the case of certain compact
Invariant measures and arithmetic quantum unique ergodicity
We classify measures on the locally homogeneous space ?i\ SL(2,R) ?~ L which are invariant and have positive entropy under the diagonal subgroup of SL(2,R) and recurrent under L. This classification