Fractality, self-similarity and complex dimensions

Lapidus, Michel L.; Frankenhuijsen, Machiel van

doi:10.1090/pspum/072.1/2112111

Cited by 7 publications

(9 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Further, multifractal zeta functions provide the motivation for the zeta functions which appear in [31,32,35,47] and are discussed in the epilogue of this paper, Section 8. Other approaches to multifractal analysis can be found in [1,3,4,[6][7][8][12][13][14][15][16][30][31][32][33][34][35][36][37][39][40][41][42][43][44][45]47].…”

Section: Introductionmentioning

confidence: 99%

Fractal Strings and Multifractal Zeta Functions

2009

Self Cite

View full text Add to dashboard Cite

Abstract. For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.

show abstract

Section: Introductionmentioning

confidence: 99%

Fractal Strings and Multifractal Zeta Functions

2009

Self Cite

View full text Add to dashboard Cite

show abstract

“…Both of these features, the existence of a meromorphic continuation and a product formula, are reminiscent of the theory of one-dimensional fractal strings first defined and studied by M. Lapidus in [26]. We present basic definitions and results in Section 2, and the reader can find more details and references in [27,28,29,30,31,32]. In particular, equations (5.2)-(5.3) in [26] (see also section 1.3 of [27]) contain the product formula We give examples where the λ j are actually eigenvalues of a Laplacian, but it is possible to define λ j in terms of R and z 0 only, without reference to any Laplacian.…”

Section: Introductionmentioning

confidence: 99%

“…This means that the spectrum has a product structure which does not manifest itself in the structure of the underlying space. The poles of a spectral zeta function are called the complex spectral dimensions (see [26,27,28]). They are 1 and log 2+2inπ log 3…”

Section: Introductionmentioning

confidence: 99%

Spectral zeta functions of fractals and the complex dynamics of polynomials

Teplyaev

2007

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals, such as the Sierpiński gasket, and a fractal Laplacian on the interval. These formulas contain a new type of zeta function associated with a polynomial (rational functions also can appear in this context). It is proved that this zeta function has a meromorphic continuation to a half-plane with poles contained in an arithmetic progression. It is shown as an example that the Riemann zeta function is the zeta function of a quadratic polynomial, which is associated with the Laplacian on an interval. The spectral zeta function of the Sierpiński gasket is a product of the zeta function of a polynomial and a geometric part; the poles of the former are canceled by the zeros of the latter. A similar product structure was discovered by M.L. Lapidus for self-similar fractal strings.

show abstract

“…Lapidus and Van Frankenhuysen [1][2][3] consider the functions known as nonlattice Dirichlet polynomials, which are exponential polynomials of the form…”

Section: Introductionmentioning

confidence: 99%

“…Also, the set of dimensions of fractality of a fractal string is defined as the closure of the set of real parts of its complex dimensions. In fact, in [1][2][3], the authors give a conjecture about the density of the real parts of the complex dimensions for the case of nonlattice strings (associated to the nonlattice Dirichlet polynomials (1.1)) which they formulate, respectively, in the following form.…”

Section: Introductionmentioning

confidence: 99%

On the existence of exponential polynomials with prefixed gaps

Mora¹,

Sepulcre²,

Vidal³

2013

Bulletin of the London Mathematical Society

View full text Add to dashboard Cite

This paper shows that the conjecture of Lapidus and Van Frankenhuysen on the set of dimensions of fractality associated with a nonlattice fractal string is true in the important special case of a generic nonlattice self-similar string, but in general is false. The proof and the counterexample of this have been given by virtue of a result on exponential polynomials P (z), with real frequencies linearly independent over the rationals, that establishes a bound for the number of gaps of RP , the closure of the set of the real projections of its zeros, and the reason for which these gaps are produced.

show abstract

Fractality, self-similarity and complex dimensions

Cited by 7 publications

References 18 publications

Fractal Strings and Multifractal Zeta Functions

Fractal Strings and Multifractal Zeta Functions

Spectral zeta functions of fractals and the complex dynamics of polynomials

On the existence of exponential polynomials with prefixed gaps

Contact Info

Product

Resources

About