# Centre for Nonlinear Dynamics Zagreb

## Fractal Zeta Functions

Focus of research:

The emergence of fractal zeta functions is a result of the natural development of the theory of geometric zeta functions of fractal strings and the related theory of complex dimensions, both established by Michel L. Lapidus (University of California, Riverside) in the early 1990s, and studied with numerous collaborators (and in three research monographs with Machiel van Frankenhuijsen). Of particular interest are Lapidus zeta functions (the name has been suggested by Darko Žubrinić) which covers distance zeta functions, introduced in 2009, associated with arbitrary bounded fractal sets in Euclidean spaces, as well as tube zeta functions and already existing geometric zeta functions (of fractal strings). Fractal zeta functions represent a bridge connecting the geometry of fractal sets with complex analysis. Their poles, called complex dimensions of fractal sets, are important for understanding the oscillatory nature of the inner geometry of fractal sets.

In 2009 Professor Michel L. Lapidus offered a joint research project to Darko Žubrinić (University of Zagreb), dedicated to the study of fractal zeta functions and related topics, in which also Goran Radunović (PhD student of D.Ž.) participates. It is expected that the project will result, among others, in a joint research monograph, Fractal Zeta Functions and Fractal Drums / Higher Dimensional Theory of Complex Dimensions (approx. 620 pp.).

The distance zeta function associated with a given bounded subset $A$ of $\mathbb{R}^N$, introduced in 2009 by M. L. Lapidus, or the Lapidus zeta function (as we suggest to call it in the future), is defined by

$$\zeta_A(s)=\int_{A_\delta}d(x,A)^{s-N} \mathrm{d} x$$

where $\delta$ is an arbitrary fixed positive real number, $A_{\delta}$ is the $\delta$-neighbourhood of $A$ in $\mathbb{R}^N$, $d(x,A)$ is the Euclidean distance from $x$ to $A$ and $s$ is a complex number. The integral is understood in the sense of Lebesgue. The zeta function (absolutely) converges when

$$\operatorname{Re}\,s>\overline{\dim}_BA,$$

and the lower bound (that is, the value of $\overline{\dim}_BA$, known as the upper box dimension or the upper Minkowski dimension of $A$) is optimal. This result can be found in reference [2] below, which contains many additional details, as well as in the survey paper [4]. The Lapidus zeta function establishes a bridge between the geometry of fractal sets and complex analysis. It enables a far-reaching generalization of the theory of (geometric) zeta functions of fractal strings (introduced by M. L. Lapidus in the early 1990s) and in particular, of the classical Riemann zeta function.

Plenary lectures

Michel L. Lapidus (University of California, United States): Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory (joint work with G. Radunović and D. Žubrinić, University of Zagreb, Croatia), Fractal Geometry and Stochastics V, 2014. Abstract of the talk: [PDF]

Michel L. Lapidus (University of California, United States): Fractal Zeta Functions and Fractal Drums [PDF], (joint work with G. Radunović and D. Žubrinić, University of Zagreb, Croatia), 33nd Annual Western States Mathematical Physics Meeting, California Institute of Technology (Caltech), USA, 16-17 February 2015; 60 min. lecture; slides of the lecture; more information at Croatian scientific bibliography

Doctoral dissertation

On 25th March 2015, Goran Radunović (University of Zagreb), sucessfully defended his PhD thesis in front of the following exam comittee: Professor Siniša Slijepčević (University of Zagreb), Professor Michel L. Lapidus, second mentor (University of California, Riverside), Professor Darko Žubrinić, first mentor (University of Zagreb), Professor Vesna Županović (University of Zagreb), Domagoj Kovačević, Assistant Professor (University of Zagreb). The title of his extensive and high quality thesis was

Goran Radunović: Fractal analysis of unbounded sets in Euclidean spaces and Lapidus zeta functions, [PDF], University of Zagreb, Croatia, 249 pp.

Dr. Goran Radunović with Professors Michel L. Lapidus and Drako Žubrinić, at the University of Zagreb, 25 March 2015, immediately after Goran's successful defense of his PhD thesis.

Darko Žubrinić: Lapidus zeta functions and applications, a short survey of some of the basic results, Zagreb, 2015

#### Key publications:

1. M. L. Lapidus, J. A. Rock and D. Žubrinić, Box-counting fractal strings, zeta functions and equivalent forms of Minkowski dimension, in: Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I : Fractals in Pure Mathematics (D. Carfi, M. L. Lapidus, E. P. J. Pearse and M. van Frankenhuijsen, eds.), Contemporary Mathematics vol 600, Amer. Math. Soc., Providence, 2013, 239-271. (Also: arXiv:1207.6681).
2. M. L. Lapidus, G. Radunović and D. Žubrinić, Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions, research monograph, Springer, New York, 2016, to appear, approx. 620 pages.
3. Goran Radunović: Fractal analysis of unbounded sets in Euclidean spaces and Lapidus zeta functions[PDF] PhD thesis, University of Zagreb, under supervision of Professors Darko Žubrinić (University of Zagreb) and Michel L. Lapidus (University of California, Riverside), 249 pp.
4. M. L. Lapidus, G. Radunović and D. Žubrinić: Fractal zeta functions and complex dimensions of relative fractal drums, J. of Fixed Point Theory and Appl. 15 (2014), 321-378. Festschrift issue in honor of Haim Brezis' 70th birthday. (DOI: 10.1007/s11784-014-0207-y.) (Also: e-print, arXiv:1407.8094v3 [math-ph], 2014.)
5. M. L. Lapidus, G. Radunović and D. Žubrinić, Fractal zeta functions and complex dimensions: A general higher-dimensional theory, survey article,
in:  Fractal Geometry and Stochastics V (C. Bandt, K. Falconer and M. Zähle, eds.),  Proc. Fifth Internat. Conf. (Tabarz, Germany, March 2014), Progress in Probability, Birkhäuser, Basel, Boston and Berlin, 2015, in press. (Based on a plenary lecture given by the first author at that conference.)
6. M. L. Lapidus, G. Radunović and D. Žubrinić: Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces, arXiv:1411.5733 [math-ph], 2014.