Fractality and Lapidus zeta functions at infinity

We study fractality of unbounded sets of finite Lebesgue measure at infinity by introducing the notions of Minkowski dimension and content at infinity. We also introduce the Lapidus zeta function at infinity, study its properties and demonstrate its use in analysis of fractal properties of unbounded sets at infinity.Comment: 19 pages, 1 figur

Quasiperiodic sets at infinity and meromorphic extensions of their fractal zeta functions

In this paper we introduce an interesting family of relative fractal drums (RFDs in short) at infinity and study their complex dimensions which are defined as the poles of their associated Lapidus (distance) fractal zeta functions introduced in a previous work by the author. We define the tube zeta function at infinity and obtain a functional equation connecting it to the distance zeta function at infinity much as in the classical setting. Furthermore, under suitable assumptions, we provide general results about existence of meromorphic extensions of fractal zeta functions at infinity in the Minkowski measurable and nonmeasurable case. We also provide a sufficiency condition for Minkowski measurability as well as an upper bound for the upper Minkowski content, both in terms of the complex dimensions of the associated RFD. We show that complex dimensions of quasiperiodic sets at infinity posses a quasiperiodic structure which can be either algebraic or transcedental. Furthermore, we provide an example of a maximally hyperfractal set at infinity with prescribed Minkowski dimension, i.e., a set such that the abscissa of convergence of the corresponding fractal zeta function is in fact its natural boundary.Comment: 31 pages, 2 figure

Essential singularities of fractal zeta functions

We study the essential singularities of geometric zeta functions $\zeta_{\mathcal L}$, associated with bounded fractal strings $\mathcal L$. For any three prescribed real numbers $D_{\infty}$, $D_1$ and $D$ in $[0,1]$, such that $D_{\infty}<D_1\le D$, we construct a bounded fractal string $\mathcal L$ such that $D_{\rm par}(\zeta_{\mathcal L})=D_{\infty}$, $D_{\rm mer}(\zeta_{\mathcal L})=D_1$ and $D(\zeta_{\mathcal L})=D$. Here, $D(\zeta_{\mathcal L})$ is the abscissa of absolute convergence of $\zeta_{\mathcal L}$, $D_{\rm mer}(\zeta_{\mathcal L})$ is the abscissa of meromorphic continuation of $\zeta_{\mathcal L}$, while $D_{\rm par}(\zeta_{\mathcal L})$ is the infimum of all positive real numbers $\alpha$ such that $\zeta_{\mathcal L}$ is holomorphic in the open right half-plane $\{{\rm Re}\, s>\alpha\}$, except for possible isolated singularities in this half-plane. Defining $\mathcal L$ as the disjoint union of a sequence of suitable generalized Cantor strings, we show that the set of accumulation points of the set $S_{\infty}$ of essential singularities of $\zeta_{\mathcal L}$, contained in the open right half-plane $\{{\rm Re}\, s>D_{\infty}\}$, coincides with the vertical line $\{{\rm Re}\, s=D_{\infty}\}$. We extend this construction to the case of distance zeta functions $\zeta_A$ of compact sets $A$ in $\mathbb{R}^N$, for any positive integer $N$.Comment: Theorem 3.2 (b) was wrong in the previous version, so we have decided to omit it and pursue this issue at some future time. Part (b) of Theorem 3.2. was not used anywhere else in the paper. Theorem 3.2. is now called Proposition 3.2. on page 12. Corrected minor typos and added new references To appear in: Pure and Applied Functional Analysis; issue 5 of volume 5 (2020

Fractal Tube Formulas for Compact Sets and Relative Fractal Drums: Oscillations, Complex Dimensions and Fractality

We establish pointwise and distributional fractal tube formulas for a large class of relative fractal drums in Euclidean spaces of arbitrary dimensions. A relative fractal drum (or RFD, in short) is an ordered pair $(A,\Omega)$ of subsets of the Euclidean space (under some mild assumptions) which generalizes the notion of a (compact) subset and that of a fractal string. By a fractal tube formula for an RFD $(A,\Omega)$, we mean an explicit expression for the volume of the $t$-neighborhood of $A$ intersected by $\Omega$ as a sum of residues of a suitable meromorphic function (here, a fractal zeta function) over the complex dimensions of the RFD $(A,\Omega)$. The complex dimensions of an RFD are defined as the poles of its meromorphically continued fractal zeta function (namely, the distance or the tube zeta function), which generalizes the well-known geometric zeta function for fractal strings. These fractal tube formulas generalize in a significant way to higher dimensions the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen and later on, by the first author, Pearse and Winter in the case of fractal sprays. They are illustrated by several interesting examples. These examples include fractal strings, the Sierpi\'nski gasket and the 3-dimensional carpet, fractal nests and geometric chirps, as well as self-similar fractal sprays. We also propose a new definition of fractality according to which a bounded set (or RFD) is considered to be fractal if it possesses at least one nonreal complex dimension or if its fractal zeta function possesses a natural boundary. This definition, which extends to RFDs and arbitrary bounded subsets of $\mathbb{R}^N$ the previous one introduced in the context of fractal strings, is illustrated by the Cantor graph (or devil's staircase) RFD, which is shown to be `subcritically fractal'.Comment: 90 pages (because of different style file), 5 figures, corrected typos, updated reference

Complex dimensions of fractals and meromorphic extensions of fractal zeta functions

We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function $\zeta_A(s):=\int_{A_\delta} d(x,A)^{s-N}\mathrm{d}x$, where $\delta>0$ is fixed and $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ extends the definition of the zeta function associated with bounded fractal strings to arbitrary bounded subsets $A$ of $\mathbb{R}^N$. The abscissa of Lebesgue convergence $D(\zeta_A)$ coincides with $D:=\overline\dim_BA$, the upper box dimension of $A$. The complex dimensions of $A$ are the poles of the meromorphic continuation of the fractal zeta function of $A$ to a suitable connected neighborhood of the "critical line" $\{\Re(s)=D\}$. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function $|A_t|$ as $t\to0^+$, where $A_t$ is the Euclidean $t$-neighborhood of $A$. We pay particular attention to a class of Minkowski measurable sets, such that $|A_t|=t^{N-D}(\mathcal M+O(t^\gamma))$ as $t\to0^+$, with $\gamma>0$, and to a class of Minkowski nonmeasurable sets, such that $|A_t|=t^{N-D}(G(\log t^{-1})+O(t^\gamma))$ as $t\to0^+$, where $G$ is a nonconstant periodic function and $\gamma>0$. In both cases, we show that $\zeta_A$ can be meromorphically extended (at least) to the open right half-plane $\{\Re(s)>D-\gamma\}$. Furthermore, up to a multiplicative constant, the residue of $\zeta_A$ evaluated at $s=D$ is shown to be equal to $\mathcal M$ (the Minkowski content of $A$) and to the mean value of $G$ (the average Minkowski content of $A$), respectively. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line $\{\Re(s)=D\}$.Comment: 30 pages, 2 figures, improved parts of the paper and shortened the paper by reducing background material, to appear in Journal of mathematical analysis and applications in 201

Fraktalna analiza neomeđenih skupova u euklidskim prostorima i Lapidusove zeta funkcije

In this thesis , we consider relative fractal drums and their corresponding Lapidus fractal zeta functions, as well as a generalization of this notions to the case of unbounded sets at infinity. Relative fractal drums themselves are a generalization of the notion of a bounded subset in an Euclidean space. Here, we continue the ongoing research into their properties and the higher-dimensional theory of their fractal zeta functions and complex dimensions which started as a collaboration between M. L. Lapidus and D. Žubrinić in 2009 with the later addition of the author of this thesis. The theory of complex dimensions is already well developed for fractal strings; that is, for fractal subsets of the real line. The complex dimensions of a relative fractal drum are defined as poles of a meromorphic continuation of its corresponding distance or tube zeta function. Complex dimensions of a relative fractal drum generalize, in a way, the notion its box (or Minkowski) dimension. More precisely, under some mild conditions, the value of the box dimension of a relative fractal drum is a pole of its corresponding fractal zeta function with maximal real part. Moreover, the residue computed at this pole is closely related to its Minkowski content. Here we derive important results which further justify the notion of ‘complex dimensions’ and connect it to fractal properties of a given relative fractal drum. More precisely, we establish fractal tube formulas for a class of relative fractal drums which express their relative tube function; that is, the Lebesgue measure of their relative δ-neighborhood for small values of δ, as a sum over the residues of their fractal zeta function. These formulas are given with or without an error term and hold pointwise or distributionally depending on the growth properties of the corresponding fractal zeta function. The importance of these formulas is that they show how the complex dimensions are related to the asymptotic development of the relative tube function of a given relative fractal drum. As an application we derive a Minkowski measurability criterion for a large class of relative fractal drums. Furthermore, we also show that the complex dimensions of a relative fractal drum are, as expected, invariant to the dimension of the ambient space. We introduce a further generalization of the theory of complex dimensions to the context of unbounded sets at infinity which can be used as a new approach of applying fractal analysis to unbounded subsets in Euclidean spaces. This is done for unbounded sets of finite Lebesgue measure by introducing the notions of Minkowski content at infinity and Minkowski (or box) dimension at infinity which describe their fractal properties. Furthermore, we proceed by introducing an appropriate Lapidus (or distance) zeta function at infinity and show that it is well connected with the fractal properties of unbounded sets. We proceed by constructing interesting examples of quasiperiodic sets at infinity with arbitrary number (even infinite) of quasiperiods that exhibit complex fractal behavior. We also address the natural question which arises when dealing with unbounded sets and their fractal properties; that is, establish results about the fractal properties of their images under the one-point compactification and under the geometric inversion. Furthermore, we also investigate fractal properties of unbounded sets of infinite Lebesgue measure by introducing notions of the parametric φ-shell Minkowski content at infinity and the corresponding parametric φ-shell Minkowski (or box) dimension at infinity and we establish results connecting these notions with the distance zeta function at infinity. Finally we demonstrate how fractal analysis of unbounded sets via the geometric inversion may be applied to investigate bifurcations of dynamical systems occurring at infinity.U ovoj disertaciji bavimo se relativnim fraktalnim bubnjevima i njihovim fraktalnim zeta funkcijama Lapidusovog tipa, kao i generalizacijama ovih pojmova za slučaj neomeđenih skupova u beskonačnosti. Relativni fraktalni bubnjevi su sami po sebi generalizacija pojma omeđenog skupa u Euklidskom prostoru. Ovdje nastavljamo istraživanje njihovih svojstava i višedimenzionalne teorije njihovih fraktalnih zeta funkcija te pripadajućih kompleksnih dimenzija koje je započeto suradnjom M. L. Lapidusa i D. Žubrinića 2009. godine a kojoj se autor disertacije pridružio nešto kasnije. Teorija kompleksnih dimenzija već je vrlo dobro razvijena za slučaj fraktalnih struna, odnosno, fraktalnih podskupova realnog pravca. Kompleksne dimenzije relativnog fraktalnog bubnja definirane su kao polovi meromorfnog proširenja pripadajuće razdaljinske ili cijevne zeta funkcije. Na određeni način kompleksne dimenzije relativnog fraktalnog bubnja generaliziraju pojam njegove box dimenzije (ili dimenzije Minkowskog). Preciznije, uz neke blage uvjete, vrijednost box dimenzije relativnog fraktalnog bubnja jest pol njegove pripadajuće fraktalne zeta funkcije s maksimalnom vrijednošću realnog dijela. Štoviše, reziduum u tom polu usko je povezan sa sadržajem Minkowskog danog relativnog fraktalnog bubnja. U ovoj radnji izvodimo važne rezultate koji donose daljnje opravdanje pojma ‘kompleksnih dimenzija’ i povezuju ga s fraktalnim svojstvima danog relativnog fraktalnog bubnja. Preciznije, kao rezultat dobivamo fraktalne cijevne formule za klasu relativnih fraktalnih bubnjeva koje izražavaju njihovu relativnu cijevnu funkciju, odnosno, Lebesgueovu mjeru njihove relativne δ-okoline za male vrijednosti δ, kao sumu po reziduumima njihove fraktalne zeta funkcije. Te formule su dane s greškom ili bez greške i vrijede po točkama ili distribucijski ovisno svojstvima rasta pripadajuće fraktalne zeta funkcije. Važnost ovih formula je u tome što pokazuju kako su kompleksne dimenzije povezane s asimptotikom relativne cijevne funkcije danog relativnog fraktalnog bubnja. Kao primjenu izvodimo kriterij za Minkowskivljevu izmjerivost velike klase relativnih fraktalnih bubnjeva. Nadalje, očekivano, pokazujemo da su kompleksne dimenzije danog relativnog fraktalnog bubnja invarijantne u odnosu na dimenziju ambijentnog prostora. U nastavku radnje uvodimo generalizaciju teorije kompleksnih dimenzija u kontekstu neomeđenih skupova u beskonačnosti koja može poslužiti kao novi pristup primjeni fraktalne analize na neomeđene skupove u Euklidskim prostorima. U slučaju neomeđenih skup ova konačne Lebesgueove mjere, generalizaciju provodimo uvođenjem pojmova sadržaja Minkowskog u beskonačnosti i box dimenzije u beskonačnosti (ili dimenzije Minkowskog u beskonačnosti) koji opisuju njihova fraktalna svojstva. Nadalje, uvodimo i pripadajuću Lapidusovu (ili razdaljinsku) zeta funkciju u beskonačnosti te pokazujemo da je dobro povezana s fraktalnim svojstvima neomeđenih skupova. Nastavljamo s konstrukcijom zanimljivih primjera kvaziperiodičkih skupova u beskonačnosti s proizvoljnim brojem (moguće i beskonačnim) kvaziperioda koji posjeduju složena fraktalna svojstva. Također se bavimo i prirodnim pitanjem koje se postavlja prilikom istraživanja neomeđenih skupova i njihovih fraktalnih svojstava, u vidu pronalaženja rezultata koji ih povezuju s fraktalnim svojstvima njihovih slika po jednotočkovnoj kompaktifikaciji i po geometrijskoj inverziji. Nadalje, također istražujemo i fraktalna svojstva neomeđenih skupova beskonačne Lebesgueove mjere uvođenjem pojmova parametarskog φ-omotačkog sadržaja Minkowskog u beskonačnosti i pripadajuće parametarske φ-omotačke dimenzije Minkowskog u beskonačnosti (ili φ-omotačke box dimenzije u beskonačnosti) te izvodimo rezultate koji povezuju ove pojmove s razdaljinskom zeta funkcijom u beskonačnosti. Naposljetku, demonstriramo kako se fraktalna analiza neomeđenih skupova preko geometrijske inverzije može primijeniti u istraživanju bifurkacija dinamičkih sustava koje se događaju u beskonačnosti