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- PÄR KURLBERG
- 1999

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)

- PÄR KURLBERG
- 1999

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)

- Pär Kurlberg
- 2004

- PÄR KURLBERG
- 2008

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)

- PÄR KURLBERG
- 2001

We study the value distribution and extreme values of eigenfunctions for the " quantized cat map ". This is the quantiza-tion of a hyperbolic linear map of the torus. In a previous paper it was observed that there are quantum symmetries of the quantum map-a commutative group of unitary operators which commute with the map, which we called " Hecke operators… (More)

We consider two standard pseudorandom number generators from number theory: the linear congruential generator and the power generator. For the former, we are given integers e, b, n (with e, n > 1) and a seed u = u 0 , and we compute the sequence u i+1 = eu i + b (mod n). This sequence was first considered as a pseudorandom number generator by D. H. Lehmer.… (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)

We consider the distribution of spacings between consecutive elements in subsets of Z/qZ where q is highly composite and the subsets are defined via the Chinese remainder theorem. We give a sufficient criterion for the spacing distribution to be Pois-sonian as the number of prime factors of q tends to infinity, and as an application we show that the value… (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)