We prove two types of nodal results for density one subsequences of an orthonormal basis {ϕj } of eigenfunctions of the Laplacian on a negatively curved compact surface. The first type of result involves the intersections Zϕ j ∩H of the nodal set Zϕ j of ϕj with a smooth curve H. Using recent results on quantum ergodic restriction theorems and prior results on periods of eigenfunctions over curves, we prove that the number of intersection points tends to infinity for a density one subsequence of the ϕj, and furthermore that the number of such points where ϕj |H changes sign tends to infinity. We also prove that the number of zeros of the normal derivative ∂νϕj on H tends to infinity, also with sign changes. From these results we obtain a lower bound on the number of nodal domains of even and odd eigenfunctions on surfaces with an orientation-reversing isometric involution with non-empty and separating fixed point set. Using (and generalizing) a geometric argument of Ghosh-Reznikov-Sarnak, we show that the number of nodal domains of even or odd eigenfunctions tends to infinity for a density one subsequence of eigenfunctions.

Individuals are constantly processing external information and translating it into actions. This draws on limited resources of attention and requires economizing on attention devoted to signals related to economic behaviour. A natural measure of such costs is based on Shannon’s “channel capacity”. Modelling economic agents as constrained by Shannon capacity as they process freely available information turns out to imply that discretely distributed actions, and thus actions that persist across repetitions of the same decision problem, are very likely to emerge in settings that without information costs would imply continuously distributed behaviour. We show how these results apply to the behaviour of an investor choosing portfolio allocations, as well as to some mathematically simpler “tracking” problems that illustrate the mechanism. Trying to use costs of adjustment to explain “stickiness” of actions when interpreting the behaviour in our economic examples would lead to mistaken conclusions.

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Abstract.It is an open problem in general to prove that there exists a sequence of ∆g-eigenfunctions ϕj k on a Riemannian manifold (M, g) for which the number N (ϕj k ) of nodal domains tends to infinity with the eigenvalue. Our main result is that N (ϕj k ) → ∞ along a subsequence of eigenvalues of density 1 if the (M, g) is a non-positively curved surface with concave boundary, i.e. a generalized Sinai or Lorentz billiard. Unlike the recent closely related work of Ghosh-Reznikov-Sarnak and of the authors on the nodal domain counting problem, the surfaces need not have any symmetries.