Distance and tube zeta functions of fractals and arbitrary compact sets

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• 作者：M.L. Lapidus^a ; ¹ ; ^{lapidus@math.ucr.edu} ; G. Radunović^b ; ² ; ^{goran.radunovic@fer.hr} ; D. Žubrinić^b ; ² ; ^{darko.zubrinic@fer.hr}
• 关键词：Zeta function ; Distance zeta function ; Tube zeta function ; Fractal set ; Fractal string ; Box dimension ; Complex dimensions ; Principal complex dimensions ; Minkowski content ; Minkowski measurable set ; Residue ; Dirichlet-type integral ; Transcendentally quasiperiodic set ; Fractality and complex dimensions
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：307
• 期：Complete
• 页码：1215-1267
• 全文大小：928 K
• 卷排序：307

文摘

Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets A of the N  -dimensional Euclidean space RNRN, for any integer N≥1N≥1. It is defined by the Lebesgue integral ζA(s)=∫Aδd(x,A)s−NdxζA(s)=∫Aδd(x,A)s−Ndx, for all s∈Cs∈C with Res sufficiently large, and we call it the distance zeta function of A  . Here, d(x,A)d(x,A) denotes the Euclidean distance from x to A   and AδAδ is the δ-neighborhood of A, where δ   is a fixed positive real number. We prove that the abscissa of absolute convergence of ζAζA is equal to dim‾BA, the upper box (or Minkowski) dimension of A. Particular attention is payed to the principal complex dimensions of A  , defined as the set of poles of ζAζA located on the critical line {Res=dim‾BA}, provided ζAζA possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, ζ˜A(s)=∫0δts−N−1|At|dt, called the tube zeta function of A. Assuming that A   is Minkowski measurable, we show that, under some mild conditions, the residue of ζ˜A computed at D=dimB⁡AD=dimB⁡A (the box dimension of A) is equal to the Minkowski content of A. More generally, without assuming that A is Minkowski measurable, we show that the residue is squeezed between the lower and upper Minkowski contents of A. We also introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets, along with Baker's theorem from the theory of transcendental numbers.