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Large Supremum Norms and Small Shannon Entropy for Hecke Eigenfunctions of Quantized Cat Maps
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
2009 (Engelska)Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 286, nr 3, s. 1051-1072Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

This paper concerns the behavior of eigenfunctions of quantized cat maps and in particular their supremum norm. We observe that for composite integer values of N, the inverse of Planck's constant, some of the desymmetrized eigenfunctions have very small support and hence very large supremum norm. We also prove an entropy estimate and show that our functions satisfy equality in this estimate. In the case when N is a prime power with even exponent we calculate the supremum norm for a large proportion of all desymmetrized eigenfunctions and we find that for a given N there is essentially at most four different values these assume.

Ort, förlag, år, upplaga, sidor
2009. Vol. 286, nr 3, s. 1051-1072
Nationell ämneskategori
Matematik
Identifikatorer
URN: urn:nbn:se:kth:diva-24329DOI: 10.1007/s00220-008-0627-xISI: 000263059600008Scopus ID: 2-s2.0-59449109020OAI: oai:DiVA.org:kth-24329DiVA, id: diva2:346576
Anmärkning
QC 20100901Tillgänglig från: 2010-09-01 Skapad: 2010-09-01 Senast uppdaterad: 2022-06-25Bibliografiskt granskad
Ingår i avhandling
1. Problems in Number Theory related to Mathematical Physics
Öppna denna publikation i ny flik eller fönster >>Problems in Number Theory related to Mathematical Physics
2008 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

This thesis consists of an introduction and four papers. All four papers are devoted to problems in Number Theory. In Paper I, a special class of local ζ-functions is studied. The main theorem states that the functions have all zeros on the line Re(s)=1/2.This is a natural generalization of the result of Bump and Ng stating that the zeros of the Mellin transform of Hermite functions have Re(s)=1/2.In Paper II and Paper III we study eigenfunctions of desymmetrized quantized cat maps.If N denotes the inverse of Planck's constant, we show that the behavior of the eigenfunctions is very dependent on the arithmetic properties of N. If N is a square, then there are normalized eigenfunctions with supremum norm equal to , but if N is a prime, the supremum norm of all eigenfunctions is uniformly bounded. We prove the sharp estimate for all normalized eigenfunctions and all $N$ outside of a small exceptional set. For normalized eigenfunctions of the cat map (not necessarily desymmetrized), we also prove an entropy estimate and show that our functions satisfy equality in this estimate.We call a special class of eigenfunctions newforms and for most of these we are able to calculate their supremum norm explicitly.For a given , with k>1, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about and the supremum norm of the newforms in the second half have at most three different values, all of the order . The only dependence of A is that the normalization factor is different if A has eigenvectors modulo p or not. We also calculate the joint value distribution of the absolute value of n different newforms.In Paper IV we prove a generalization of Mertens' theorem to Beurling primes, namely that

\lim_{n \to \infty}\frac{1}{\ln n}\prod_{p \leq n} \left(1-p^{-1}\right)^{-1}=Ae^{\gamma}where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit exists. We also show that this limit coincides with ; for ordinary primes this claim is called Meissel's theorem. Finally we will discuss a problem posed by Beurling, namely how small |N(x)-[x] | can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c>0 there exists a Q such that |N(x)-[x] |

Ort, förlag, år, upplaga, sidor
Stockholm: KTH, 2008. s. viii, 28
Serie
Trita-MAT. MA, ISSN 1401-2278 ; 08:12
Nyckelord
number theory, mathetical physics, local zeta functions, cat maps, Beurling primes
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:kth:diva-9514 (URN)978-91-7415-177-0 (ISBN)
Disputation
2008-12-08, F3, KTH, Lindstedtsvägen 26, Stockholm, 13:00 (Engelska)
Opponent
Handledare
Anmärkning
QC 20100902Tillgänglig från: 2008-11-21 Skapad: 2008-11-11 Senast uppdaterad: 2022-06-26Bibliografiskt granskad

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Olofsson, Rikard
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Matematik (Inst.)
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