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Perron-Frobenius op...
Perron-Frobenius operators and the Klein-Gordon equation
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Canto-Martín, Francisco (författare)
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- Hedenmalm, Håkan (författare)
- KTH,Matematik (Avd.)
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Montes-Rodríguez, Alfonso (författare)
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KTH Matematik (Avd) (creator_code:org_t)
- 2014
- 2014
- Engelska.
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Ingår i: Journal of the European Mathematical Society (Print). - 1435-9855 .- 1435-9863. ; 16:1, s. 31-66
- Relaterad länk:
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https://urn.kb.se/re...
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visa fler...
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https://doi.org/10.4...
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visa färre...
Abstract
Ämnesord
Stäng
- For a smooth curve Gamma and a set Lambda in the plane R-2, let AC(Gamma; Lambda) be the space of finite Borel measures in the plane supported on Gamma, absolutely continuous with respect to arc length and whose Fourier transform vanishes on Lambda. Following [12], we say that (Gamma, Lambda) is a Heisenberg uniqueness pair if AC(Gamma; Lambda) = {0}. In the context of a hyperbola Gamma, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Gamma of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the issue of finding the dimension of AC(Gamma; Lambda) when it is nonzero. We will fix the curve Gamma to be the hyperbola x(1)x(2) = 1, and the set Lambda = Lambda(alpha,beta) to be the lattice-cross Lambda(alpha,beta) = (alpha Zeta x {0}) boolean OR ({0} x beta Z), where alpha, beta are positive reals. We will also consider Gamma(+), the branch of x(1)x(2) = 1 where x(1) > 0. In [12], it is shown that AC(Gamma; Lambda(alpha,beta)) = {0} if and only if alpha beta <= 1. Here, we show that for alpha beta > 1, we get a rather drastic "phase transition": AC(Gamma; Lambda(alpha,beta)) is infinite-dimensional whenever alpha beta > 1. It is shown in [13] that AC(Gamma(+); Lambda(alpha,beta)) = {0} if and only if alpha beta < 4. Moreover, at the edge alpha beta = 4, the behavior is more exotic: the space AC(Gamma(+); Lambda(alpha,beta)) is one-dimensional. Here, we show that the dimension of AC(Gamma(+); Lambda(alpha,beta)) is infinite whenever alpha beta > 4. Dynamical systems, and more specifically Perron-Frobenius operators, play a prominent role in the presentation.
Ämnesord
- NATURVETENSKAP -- Matematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics (hsv//eng)
Nyckelord
- Trigonometric system
- inversion
- Perron-Frobenius operator
- Koopman operator
- invariant measure
- Klein-Gordon equation
- ergodic theory
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- art (ämneskategori)
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