Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Spectral asymptotics for $V$-variable Sierpinski gaskets

U. Freiberg, B. M. Hambly, and John E. Hutchinson

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Abstract

The family of $V$-variable fractals provides a means of interpolating between two families of random fractals previously considered in the literature; scale irregular fractals ($V=1$) and random recursive fractals ($V=\infty$). We consider a class of $V$-variable affine nested fractals based on the Sierpinski gasket with a general class of measures. We calculate the spectral exponent for a general measure and find the spectral dimension for these fractals. We show that the spectral properties and on-diagonal heat kernel estimates for $V$-variable fractals are closer to those of scale irregular fractals, in that it is the fluctuations in scale that determine their behaviour but that there are also effects of the spatial variability.

Résumé

La famille des fractales $V$-variables donne un moyen d’interpolation entre deux familles de fractales aléatoires étudiées dans la littérature : les fractales à échelle irrégulière ($V=1$) et les fractales récursives aléatoires ($V=\infty$). Nous considérons une classe de fractales $V$-variables affines emboîtées, construites à partir du tamis de Sierpinski muni d’une classe générale de mesures. Nous calculons l’exposant spectral d’une mesure générale, et déterminons la dimension spectrale de ces fractales. Nous montrons que les propriétés spectrales, de même que les estimées de noyau de la chaleur sur la diagonale, sont plus proches de celles des fractales à échelle irrégulière, du fait que ce sont les fluctuations d’échelle qui déterminent leurs comportements. Néanmoins, la variabilité spatiale a aussi une influence.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 2162-2213.

Dates
Received: 4 March 2013
Revised: 10 August 2016
Accepted: 12 August 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773742

Digital Object Identifier
doi:10.1214/16-AIHP787

Mathematical Reviews number (MathSciNet)
MR3729651

Zentralblatt MATH identifier
06847078

Subjects
Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Secondary: 28A80: Fractals [See also 37Fxx] 31C25: Dirichlet spaces 35K08: Heat kernel 60J60: Diffusion processes [See also 58J65]

Keywords
Random fractals Laplace operator Eigenvalue counting function Spectral dimension Heat kernel estimates Spectral asymptotics V-variable Sierpinski gasket

Citation

Freiberg, U.; Hambly, B. M.; Hutchinson, John E. Spectral asymptotics for $V$-variable Sierpinski gaskets. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 2162--2213. doi:10.1214/16-AIHP787. https://projecteuclid.org/euclid.aihp/1511773742


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