ENTROPY OF SEMICLASSICAL MEASURES

Description

ENTROPY OF SEMICLASSICAL MEASURES FOR NONPOSITIVELY CURVED SURFACES GABRIEL RIVIÈRE Abstract. We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. To do this, we look at se- quences of distributions associated to them and we study the entropic properties of their accumu- lation points, the so-called semiclassical measures. Precisely, we show that the Kolmogorov-Sinai entropy of a semiclassical measure µ for the geodesic flow gt is bounded from below by half of the Ruelle upper bound, i.e. hKS(µ, g) ≥ 1 2 ∫ S?M ?+(?)dµ(?), where ?+(?) is the upper Lyapunov exponent at point ?. The main strategy is the same as in [17] except that we have to deal with weakly chaotic behavior. 1. Introduction Let M be a compact, connected, C∞ riemannian manifold. For all x ? M , T ?xM is endowed with a norm ?.?x given by the metric over M . The geodesic flow gt over T ?M is defined as the Hamiltonian flow corresponding to H(x, ?) := ??? 2 x 2 . This quantity corresponds to the classical kinetic energy in the case of the absence of potential.

• also ask


• quantum unique


• geodesic flow


• points need


• flow gt over


• semiclassical measure


• measure cannot


• liouville measure

Sujets

• Geodesic
• Symplectic manifold

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 25

Extrait

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M
: S M ! M
(x;) := x V = (x;) d
1 S Mx
Z 2 T S M
0c(t) = (a(t);b(t)) t2 ( ; ) S M c(0) = c (0) =Z
H K (Z) =r 0 b(0) =r b(0) a (0) d (Z)
r X ()H
2
T S M =H V
S M 18
= (x;)2S M X;Y 2T S M
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M S M

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0R(X;Y )Z X Y Z J (t) =
r 0 J(t) (t)
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@c =c Y (t) = (c (t))0 s js=0@s
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0 0c 8s2 ( ; ) c M (t) t (t)s

(V;W ) T S M H V