Spectral zeta functions of fractals and the complex dynamics of polynomials

Teplyaev, Alexander

doi:10.1090/s0002-9947-07-04150-5

Cited by 48 publications

(107 citation statements)

References 44 publications

(105 reference statements)

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“…In fact, it is known that the heat kernel trace for a Laplacian on fractals displays log-oscillations in the scale [153,154]. Oscillatory behaviour has been found analytically and numerically for various fractals [155,156,157,158], and (4.28) illustrates a rather universal phenomenon. This is one of the most crucial points of the physical scenario that will emerge in [42], where it shall be given adequate space.…”

Section: Fractional Measures As Approximations Of Fractalsmentioning

confidence: 96%

Geometry of fractional spaces

Calcagni

2012

Advances in Theoretical and Mathematical Physics

105

261

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We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool is fractional calculus, which is cast in a way convenient for the definition of the differential structure, distances, volumes, and symmetries. By an extensive use of concepts and techniques of fractal geometry, we clarify the relation between fractional calculus and fractals, showing that fractional spaces can be regarded as fractals when the ratio of their Hausdorff and spectral dimension is greater than one. All the results are analytic and constitute the foundation for field theories living on multi-fractal spacetimes, which are presented in a companion paper.

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Section: Fractional Measures As Approximations Of Fractalsmentioning

confidence: 96%

Geometry of fractional spaces

Calcagni

2012

Advances in Theoretical and Mathematical Physics

105

261

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“…11 In the special case of ordinary fractal strings, Equation (3.1.10) was first observed in [La2] and used in [La3,La4,LaPo1,LaPo2,LaMa1,LaMa2,Tep,La5,LalLa1,LalLa2,HerLa1,HerLa2,HerLa3]. Furthermore, a formula analogous to the one given in Equation (3.1.10) exists for other generalizations of ordinary fractal strings, including the class of fractal sprays (also called higher-dimensional fractal strings); see [LaPo3,La2,La3,.…”

Section: Generalized Fractal Strings and The Spectral Operatormentioning

confidence: 99%

“…Many applications and extensions of fractal string theory and/or of the corresponding theory of complex fractal dimensions can be found throughout the books La5] and in [La1,La2,La3,La4,LaPo1,LaPo2,LaPo3,LaMa1,LaMa2,HeLa,HamLa,Tep,LaPe,LaPeWi,LaLeRo,ElLaMaRo,LaLu1,LaLu2,LalLa1,LalLa2,LaRaZu,HerLa1,HerLa2,HerLa3,HerLa4,La6]. These include, in particular, applications to various aspects of number theory and arithmetic geometry, dynamical systems, spectral geometry, geometric measure theory, noncommutative geometry, mathematical physics and nonarchimedean analysis.…”

Section: Generalized Fractal Strings and The Spectral Operatormentioning

confidence: 99%

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Herichi¹,

Lapidus²

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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A Christophe Soulé, avec une profonde amitié et admiration,à l'occasion de ses 60 ans Abstract. Nous rappelons quelques unes des principales propriétés d'universalité de la fonction zêta de Riemann ζ(s). De plus, nous expliquons comment obtenir une quantification naturelle du théorème d'universalité de Voronin (et de ses généralizations). Notre travail est basé sur la théorie des cordes fractales développée par le deuxiéme auteur et M. van Frankenhuijsen dans . Nous utilisonségalement la théorie développée dans [HerLa1-3] par les auteurs de cet article pourétudier de façon rigoreuse l'opérateur spectral (qui relie la géométrie et le spectre des cordes fractales généralizées). Cet opérateur spectral est representée comme le composé de la fonction zêta de Riemann du 'shift infinitesimal' ∂ : a = ζ(∂). Dans le processus du quantification du théorème d'universalité de la fonction zêta de Riemann, le rôle joué par la variable s (au sens classique du théoréme d'universalité) (dans le théorème classique d'universalité) est joué par la famille des shifts infinitésimaux tronqués afin d'étudier l'opérateur spectral en lien avec la reformulation spectrale de l'hypothése de Riemann vue comme un problème spectral inverse pour les cordes fractales. Ce dernier résultat fournit une version opératorielle de la reformulation spectrale obtenue par le second auteur et H. Maier dans [LaMa2]. Notre présent travail au long terme, ainsi que [La5, La6], a en partie pour but d'obtenir une quantification naturelle de divers aspects de la théorie analytiques des nombres et de la géométrie arithmétiques.2010 Mathematics Subject Classification. Primary 11M06, 11M26, 11M41, 28A80, 32B40, 47A10, 47B25, 65N21, 81Q12, 82B27. Secondary 11M55, 28A75, 34L05, 34L20, 35P20, 47B44, 47D03, 81R40.Key words and phrases. Riemann zeta function, Riemann zeros, Riemann hypothesis, spectral reformulations, fractal strings, complex dimensions, explicit formulas, geometric and spectral zeta functions, geometric and spectral counting functions, inverse spectral problems, infinitesimal shift, truncated infinitesimal shifts, spectral operator, truncated spectral operators, universality of the Riemann zeta function, universality of the spectral operator.

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“…Remark 1.3. Equation (1.4) has since been given an interesting dynamical interpretation by Alexander Teplyaev in [Tep1,Tep2] in terms of the complex dymanics of the renormalization map [Ram,RamTo,Sh,FukSh,Sab1,Sab2,Sab3] associated with the decimation method for the Dirichlet Laplacian on a 'fractal interval' (i.e., an interval viewed as a nontrivial self-similar set or graph); see also Derfer et al [DerGrVo].…”

Section: Generalized Fractal Strings and Their Complex Dimensionsmentioning

confidence: 99%

“…For results concerning the higher-dimensional analog of the above inverse and direct spectral problems for " fractal drums"in R d (d ≥ 1), as well as for their physical motivations and their relationship with the Weyl-Berry conjecture [Berr1,Berr2], the reader may wish to consult [La1,La2,La3,La4,LaPo3], along with [La-vF3, §12.5] and the references therein, including [Berr1,Berr2,BroCa,FlVa,Ger,GerScm1,GerScm2,HeLa,. (See also [FukSh,Ham1,Ham2,Ki,KiLa1,KiLa2,La3,Sab3,Str,Tep1,Tep2] and the relevant references therein for the case of a drum with a fractal membrane instead of a fractal boundary. )…”

Section: 1])mentioning

confidence: 99%

Riemann zeros and phase transitions via the spectral operator on fractal strings

Herichi¹,

Lapidus²

2012

J. Phys. A: Math. Theor.

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The spectral operator was introduced by M. L. Lapidus and M. van Frankenhuijsen in their reinterpretation of the earlier work of M. L. Lapidus and H. Maier [LaMa2] on inverse spectral problems and the Riemann hypothesis. In essence, it is a map that sends the geometry of a fractal string onto its spectrum. In this survey paper, we present the rigorous functional analytic framework given by the authors in [HerLa1] and within which to study the spectral operator. Furthermore, we give a necessary and sufficient condition for the invertibility of the spectral operator (in the critical strip) and therefore obtain a new spectral and operator-theoretic reformulation of the Riemann hypothesis. More specifically, we show that the spectral operator is invertible (or equivalently, that zero does not belong to its spectrum) if and only if the Riemann zeta function ζ(s) does not have any zeroes on the vertical line Re(s) = c. Hence, it is not invertible in the mid-fractal case when c = 1 2 , and it is invertible everywhere else (i.e., for all c ∈ (0, 1) with c = 1 2 ) if and only if the Riemann hypothesis is true. We also show the existence of four types of (mathematical) phase transitions occurring for the spectral operator at the critical fractal dimension c = 1 2 and c = 1 concerning the shape of the spectrum, its boundedness, its invertibility as well as its quasi-invertibility.

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Spectral zeta functions of fractals and the complex dynamics of polynomials

Cited by 48 publications

References 44 publications

Geometry of fractional spaces

Geometry of fractional spaces

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Riemann zeros and phase transitions via the spectral operator on fractal strings

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