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We study semi-classical limits of eigenfunctions of a quantized linear hyperbolic automorphism of the torus (" cat map "). For some values of Planck's constant, the spectrum of the quan-tized map has large degeneracies. Our first goal in this paper is to show that these degeneracies are coupled to the existence of quantum symmetries. There is a commutative(More)
We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (" cat maps "). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence, all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly distributed. A key step in the argument is to show that(More)
The number of points on a hyperelliptic curve over a field of q elements may be expressed as q + 1 + S where S is a certain character sum. We study fluctuations of S as the curve varies over a large family of hyperelliptic curves of genus g. For fixed genus and growing q, Katz and Sarnak showed that S/ √ q is distributed as the trace of a random 2g × 2g(More)
Let K be a number field, let ϕ ∈ K (t) be a rational map of degree at least 2, and let α, β ∈ K. We show that if α is not in the forward orbit of β, then there is a positive proportion of primes p of K such that α mod p is not in the forward orbit of β mod p. Moreover, we show that a similar result holds for several maps and several points. We also present(More)
Let f 1 ,. .. , f g ∈ C(z) be rational functions, let Φ = (f 1 ,. .. , f g) denote their coordinate-wise action on (P 1) g , let V ⊂ (P 1) g be a proper subvariety, and let P be a point in (P 1) g (C). We show that if S = {n 0 : Φ n (P) ∈ V (C)} does not contain any infinite arithmetic progressions, then S must be a very sparse set of integers. In(More)