Sierpinski gasket

In this work we investigate force-induced desorption of linear polymers in good solvents in non-homogeneous environment, by applying the model of self-avoiding walk on two- and three-dimensional fractal lattices, obtained as generalization of the Sierpinski gasket fractal. For each of these lattices one of its boundaries represents an adsorbing wall, whereas along one of the fractal edges, not lying in the adsorbing wall, an external force acts on the self-avoiding walk. The hierarchical nature of the lattices under study enables an exact real-space renormalization group treatment, which yields the phase diagram of polymer critical behavior. We show that for this model there is no low-temperature reentrance in the cases of two-dimensional lattices, whereas in all studied three-dimensional cases the force-temperature dependance is reentrant. We also find that in all cases the force-induced desorption transition is of first order.

A sectoral Sierpinski Gasket fractal (SSGF) antenna is proposed for dualband operation with wide-bandwidth covering GPS, DCS-1800, PCS-1800, UMTS, IMT-2000, Wireless broadband Internet Services (WiBro), Bluetooth, and WLAN bands. The SSGF antenna consists of volume 65.5 Â 27 Â 1.6 mm 3 . To analyze its performance, measurements are carried out. The proposed antenna model exhibits resonances in 1.51-3.39 GHz (2:1 VSWR BW 76.6%) and 5.31-6.32 GHz (2:1 VSWR BW 17.3%) bands with 2.5-5 dBi gain. Very good agreement is obtained between simulation and experimental results.

We study the problem of polymer adsorption in a good solvent when the container of the polymer-solvent system is taken to be a member of the Sierpinski gasket (SG) family of fractals. Members of the SG family are enumerated by an integer b (2<~b~< or), and it is assumed that one side of each SG fi'actal is an impenetrable adsorbing boundary. We calculate the critical exponents ),j, ?l J, and ),~, which, within the self-avoiding walk model (SAW) of tile polymer chain, are associated with the numbers of all possible SAWs with one, both, and no ends anchored to the adsorbing impenetrable boundary, respectively. By applying the exact renormalization group (RG) method for 2 ~< b ~< 8 and the Monte Carlo renormalization group (MCRG) method for a sequence of fractals with 2 ~<b ~<80, we obtain specific values for these exponents. The obtained results show that all three critical exponents y~, ~'ll, and y.~, in both the bulk phase and crossover region are monotonically increasing functions with b. We discuss their mutual relations, their relations with other critical exponents pertinent to SAWs on the SG fractals, and their possible asymptotic behavior in the limit b ~ or, when the fractal dimension of the SG fractals approaches the Euclidean value 2.

A brief historical perspective is first given concerning financial crashes, -from the 17th till the 20th century. In modern times, it seems that log periodic oscillations are found before crashes in several financial indices. The same is found in sand pile avalanches on Sierpinski gaskets. A discussion pertains to the after shock period with illustrations from the DAX index. The factual financial observations and the laboratory ones allow us some conjecture on symptoms and remedies for discussing financial crashes along econophysics lines.

We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V 0 = {p 1 , p 2 , p 3 } be the set of vertices of SG and u i (x) = 1 2 (x + p i ) the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations u w = u w 1 u w 2 · · · u w n for any sequence w = (w 1 , w 2 , . . . , w n ) ∈ {1, 2, 3} n . The union of the images of V 0 under these iterations is the set of nth stage vertices V n of SG. Let F : V n → R be any function. Given any numbers α w (w ∈ {1, 2, 3} n ) with 0 < |α w | < 1, there exists a unique continuous extension f : SG → R of F , such that

The Sierpinski fractal or Sierpinski gasket § is a familiar object stud- ied by specialists in dynamical systems and probability. In this paper, we consider a graph Sn derived from the first n iterations of the pro- cess that leads to §, and study some of its properties, including its cycle structure, domination number and pebbling number. Various open questions

We prove there exist exponentially decaying generalized eigenfunctions on a blow-up of the Sierpinski gasket with boundary. These are used to show a Borel-type theorem, specifically that for a prescribed jet at the boundary point there is a smooth function having that jet.

This paper introduces a new approach to represent logic functions in the form of Sierpinski Gaskets. The structure of the gasket allows to manipulate with the corresponding logic expression using recursive essence of fractals. Thus, the Sierpinski gasket's pattern has myriad useful properties which can enhance practical features of other graphic representations like decision diagrams. We have covered possible applications of Sierpinski gaskets in logic design and justified our assumptions in logic function minimization (both Boolean and multiple-valued cases). The experimental results on benchmarks with advances in the novel structure are considered as well.

This paper continues the study of fundamental properties of elementary functions on the Sierpinski gasket (SG) related to the Laplacian defined by Kigami: harmonic functions, multiharmonic functions, and eigenfunctions of the Laplacian. We describe the possible point singularities of such functions, and we use the description at certain periodic points to motivate the definition of local derivatives at these points. We study the global behavior of eigenfunctions on all generic infinite blow-ups of SG, and construct eigenfunctions that decay at infinity in certain directions. We study the asymptotic behavior of normal derivatives of Dirichlet eigenfunctions at boundary points, and give experimental evidence for the behavior of the normal derivatives of the heat kernel at boundary points.

A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite set almost surely if the initial value is compact: the key to both of these facts is the observation that, because of the topology of the real line and the continuity of Brownian sample paths, at the time when two particles collide one or the other of them must have already collided with each particle that was initially between them. We investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous. We give a quite general criterion for a coalescing system of particles on a compact state space to coalesce to a finite set at all positive times almost surely and show that there is almost sure instantaneous coalescence to a locally finite set for systems of Brownian motions on the Sierpinski gasket and stable processes on the real line with stable index greater than one.

Irregular objects are often modeled by fractals sets. In order to formulate partial differential equations on these nowhere differentiable sets the development of a "new analysis" is necessary. With the help of the model case of the Sierpinski gasket the definition of energy forms and Laplacians on self-similar finitely ramified fractals is explained. Moreover, some results for certain classes of non-self-similar fractals are presented.

We refute the claims made by Riera and Chalub [Phys. Rev. E 58, 4001 (1998)] by demonstrating that they have not provided enough data (requisite in their series expansion method) to draw reliable conclusions about criticality of self-avoiding walks on the Sierpinski gasket family of fractals.

2014 Nous donnons les relations de récurrence exactes pour les fonctions génératrices décrivant les marches aléatoires sans recoupement et avec volume exclu sur la famille des tamis de Sierpinski. Leur étude révèle que la moyenne des distances parcourues pour des marches de N pas varie comme N03BD avec 03BD = In (2)/ln { (7 -5)/2}, indépendamment de la force (limitée) d'interaction.

A stacked configuration of Sierpinski microstrip antenna, with the patches on each layer complementing each other, shows prominent reflection coefficient (⌫) in lower and upper frequency bands; and better performance with respect to impedance bandwidth and gain. This article reports and compares the simulation and experiment results to validate the above claims.

Graph transformation is concerned with the manipulation of graphs by means of rules. Graph grammars have been traditionally studied using techniques from category theory. In previous works, we introduced Matrix Graph Grammars (MGGs) as a purely algebraic approach for the study of graph grammars and graph dynamics, based on the representation of graphs by means of their adjacency matrices. MGGs have been succesfully applied to problems such as applicability of rule sequences, sequentialization and reachability, providing new analysis techniques and generalizing and improving previous results. Our next objective is to generalize MGGs in order to approach computational complexity theory and static properties of graphs out of the dynamics of certain grammars. In the present work, we start building bridges between MGGs and complexity by introducing what we call Monotone Complex Logic, which allows establishing a (bijective) link between MGGs and complex analysis. We use this logic to recast the formulation and basic building blocks of MGGs as more proper geometric and analytic concepts (scalar products, norms, distances). MGG rules can also be interpreted -via operators -as complex numbers. Interestingly, the subset they define can be characterized as the Sierpinski gasket. and vectors and the rewriting can be expressed using Boolean operators only. One important point of MGGs is that, as a difference from other approaches , it explicitly represents the rule dynamics (addition and deletion of elements), instead of only the static parts (pre-and post-conditions). Apart from the practical implications, this fact facilitates new theoretical analysis techniques such as for example checking independence of a sequence of arbitrary length and a permutation of it, or obtaining the smallest graph able to fire a sequence. See for a detailed account.

We review our scaling results for the diffusion-limited reactions A + A ~ 0 and A + B ~ 0 on Euclidean and fractal geometries. These scaling results embody the anomalies that are observed in these reactions in low dimensions; we collect these observations under a single phenomenological umbrella. Although we are not able to fix all the exponents in our scaling expressions from first principles, we establish bounds that bracket the observed numerical results.

We investigate the spectral zeta function of a self-similar Sturm-Liouville operator associated with a fractal self-similar measure on the half-line and C. Sabot's work connecting the spectrum of this operator with the iteration of a rational map of several complex variables. We obtain a factorization of the spectral zeta function expressed in terms of the zeta function associated with the dynamics of the corresponding renormalization map, viewed as a rational function on the complex projective plane, P 2 (C). The result generalizes to several complex variables and to the case of fractal Sturm-Liouville operators a factorization formula obtained by the second author for the spectral zeta function of a fractal string and later extended to the Sierpinski gasket and some other decimable fractals by A. Teplyaev. As a corollary, in the very special case when the underlying self-similar measure is Lebesgue measure on [0, 1], we obtain a representation of the Riemann zeta function in terms of the dynamics of a certain polynomial in P 2 (C), thereby extending to several variables an analogous result by A. Teplyaev. The above fractal Hamiltonians and their spectra are relevant to the study of diffusions on fractals and to aspects of condensed matters physics, including to the key notion of density of states.

We establish an analogue of WeyΓs classical theorem for the asymptotics of eigenvalues of Laplacians on a finitely ramified (i.e., p.c.f.) self-similar fractal K, such as, for example, the Sierpinski gasket. We consider both Dirichlet and Neumann boundary conditions, as well as Laplacians associated with Bernoulli-type ("multifractal") measures on K. From a physical point of view, we study the density of states for diffusions or for wave propagation in fractal media. More precisely, let Q(X) be the number of eigenvalues less than x. Then we show that ρ(x) is of the order of x ds / 2 as x-» +00, where the "spectral exponent" d s is computed in terms of the geometric as well as analytic structures of K. Further, we give an effective condition that guarantees the existence of the limit of x~d s / 2 ρ(x) as x->-hoc; this condition is, in some sense, "generic". In addition, we define in terms of the above "spectral exponents" and calculate explicitly the "spectral dimension" of K.

We propose families of infinitely ramified fractals, which we call the m-sheet Sierpinski gasket with side b [(mSG)b], on which the q-state Potts model can be exactly solvable through a real-space renormalization-group (RSRG) technique for which there are phase transitions at finite temperatures for m>1. We also propose, within a cell-to-cell RSRG scheme, a criterion for a suitable choice of cells in the study of antiferromagnetic (AF) classical spin models defined on (or approximated by) multirooted hierarchical lattices, and apply it for the AF Potts model on some (mSG)b fractals. Concerning the Ising model on the (mSG)2 family, we obtain the exact para (P)-ferromagnetic (F) critical temperature as a function of m and verify that, for m=1 and 2, there is no AF order (not even at zero temperature). We calculated the exact P-F and possibly exact P-AF critical frontiers and the corresponding correlation-length critical exponents for q=2-, 3-, and 4-state Potts model on the (mSG)4. The AF q=2 and 4 cases have highly degenerate ground states and each one presents, above a certain critical fractal dimension Dcf(q,m), an unusual low-temperature phase whose attractor occurs at a non-null temperature. For q=2 and Df(2,m)>=5.1, we prove that the correlations have a power-law decay with distance along this entire phase.

We prove that the free energy per spin of the nearest-neighbour classical n-vector spin model situated on the Sierpinski gasket (SG) family of fractals tends to the free energy per spin of the same model situated on the standard triangular lattice, when the linear size b of the SG fractal generator tends to infinity. This finding can be regarded as the first step in the analytical investigation of the spin system thermodynamics at the fractal to Euclidean crossover.

We investigate asymptotical behavior of numbers of long Hamiltonian walks (HWs), i.e., self-avoiding random walks that visit every site of a lattice, on various fractal lattices. By applying an exact recursive technique we obtain scaling forms for open HWs on three-simplex lattice, Sierpinski gasket, and their generalizations: Given-Mandelbrot (GM), modified Sierpinski gasket (MSG), and n -simplex fractal families. For GM, MSG and n -simplex lattices with odd values of n , the number of open HWs ZN , for the lattice with N≫1 sites, varies as ωNNγ . We explicitly calculate the exponent γ for several members of GM and MSG families, as well as for n -simplices with n=3 , 5, and 7. For n -simplex fractals with even n we find different scaling form: ZN˜ωNμN1/df , where df is the fractal dimension of the lattice, which also differs from the formula expected for homogeneous lattices. We discuss possible implications of our results on studies of real compact polymers.

We study Hamiltonian walks (HWs) on the family of three-dimensional modified Sierpinski gasket fractals, as a model for compact polymers in nonhomogeneous media in three dimensions. Each member of this fractal family is labeled with an integer b >= 2. We apply an exact recursive method which allows for explicit enumeration of extremely long Hamiltonian walks of different types: closed and open, with end-points anywhere in the lattice, or with one or both ends fixed at the corner sites, as well as some Hamiltonian conformations consisting of two or three strands. Analyzing large sets of data obtained for b = 2, 3 and 4, we find that numbers ZN of Hamiltonian walks, on fractal lattice with N sites, for N\gg 1 behave as ZN ~ ωNμNσ. The leading term ωN is characterized by the value of the connectivity constant ω > 1, which depends on b, but not on the type of HW. In contrast to that, the stretched exponential term μNσ depends on the type of HW through the constant μ < 1, whereas the exponent σ is determined by b alone. For larger b values, using some general features of the applied recursive relations, without explicit enumeration of HWs, we argue that the asymptotical behavior of ZN should be the same, with σ = ln3/ln[b(b + 1)(b + 2)/6], valid for all b > 2. This differs from the formulae obtained recently for Hamiltonian walks on other fractal lattices, as well as from the formula expected for homogeneous lattices. We discuss the possible origins and implications of such a result.

A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite

The problem of the decimation of a network of impedances on the threedimensional Sierpinski gasket is solved: the exact map M is given and its asymptotic behaviours are studied. The most significant invariant subspaces of M and the associated submaps are considered. This also allows us to address the problem of small-size phenomena, such as oscillating asymptotic behaviour, on this kind of fractal. The set of the resonances of the system and the frequency dependence of the total impedance are studied both in the thermodynamic limit and in mesoscopic systems.

We investigate the sandpile model on the two-dimensional Sierpinski gasket fractal. We find that the model displays interesting critical behavior, and we analyze the distribution functions of avalanche sizes, lifetimes, and topplings and calculate the associated critical exponents ϭ1.51Ϯ0.04, ␣ϭ1.63Ϯ0.04, and ϭ1.36Ϯ0.04. The avalanche size distribution shows power-law behavior modulated by logarithmic oscillations which can be related to the discrete scale invariance of the underlying lattice. Such a distribution can be formally described by introducing a complex scaling exponent *ϵϩi␦, where the real part corresponds to the power law and the imaginary part ␦ is related to the period of the logarithmic oscillations.

Fractals are very often illustrated by digital images, this is a consequence of the widespread availability of electronic and computational equipment. These images are always limited by their pixel resolution. Subsequently the determination of dimensions for such images is restricted practically because for their calculation the range of scales is not infinite. Downscaling the pixel resolution of images to get images with reduced resolution worsens the achievable absolute computational results but surprisingly the changes in these values and especially the changes of the courses of the calculated generalised dimensions give very promising results. This is due to the qualitative nature of these changes with respect to pixel downscaling. This opens the possibility of determining the quality of the estimation of the fractal dimensions of any image with any particular pixel resolution. Furthermore the pattern of changes itself caused by downscaling is different between the individual fractals. Several geometrical fractals (Sierpinski gasket, a fern, a Menger gasket and a modified Menger gasket with circles) as well as an example of a measured fractal, the digital images of cancer invasion studies in vitro were investigated to study the influence of downscaling the pixel resolution of digital images on the estimation of fractal dimensions. (H. Ahammer).

Abstract— A fractal monopole antenna based on the Sierpinski gasket is studied in this paper. The monopole antenna based on the Sierpinski gasket constructed through three iterations displays a multiband behaviour with three bands that are log-periodically spaced by a ...

We perform extensive simulations of the sandpile model on a Sierpinski gasket. Critical exponents for waves and avalanches are determined. We extend the existing theory of waves to the present case. This leads to an exact value for the exponent w which describes the distribution of wave sizes: w = ln(9=5)=ln 3. Numerically, it is found that the number of waves in an avalanche is proportional to the number of distinct sites toppled in the avalanche. This leads to a conjecture for the exponent that determines the distribution of avalanche sizes: = 1+ w = ln (27=5)=ln 3. Our predictions are in good agreement with the numerical results.

A multiband diamond-shaped Sierpinski gasket monopole antenna is described. The radiation pattern of wideband monopole antennas like the circular-disk and diamond monopole antennas degrades as frequency increases. In this paper, several cutouts based on the Sierpinski gasket geometry are made in the diamond monopole antenna in order to improve the radiation pattern of the antenna. It is experimentally shown that a multiband diamond-shaped Sierpinski gasket monopole antenna with a significant improved radiation pattern can be achieved by making cutouts in the diamond monopole antenna. Figure 4 Diamond-shaped Sierpinski gasket monopole antenna with flare angle of 90°F igure 5 Measured return loss of the diamond monopole antennas: (a) flare angle of 60°; (b) flare angle of 90°F igure 6 Measured return loss of the diamond shaped Sierpinski gasket monopole antennas (a) flare angle of 60°; (b) flare angle of 90°1 070 Figure 7 Measured radiation patterns of the diamond-shaped Sierpinski gasket and diamond monopole antennas with flare angle of 60°in the E-plane (solid line: co-polarization, dashed line: cross-polarization) ABSTRACT: Dyadic Green functions for time-harmonic fields in a homogeneous, isotropic, dielectric-magnetic medium, moving with constant velocity, are derived by first implementing a simple transformation and then using the dyadic Green functions available for uniaxial dielectricmagnetic mediums.

From this relation it can be shown, using exact results and a scaling assumption, that the dissipative sandpiles' correlation length exponent \nu always equals 1/d_w, where d_w is the fractal dimension of the random walker. This leads to a new understanding of the known results that \nu=1/2 on any Euclidean lattice. Our result is however more general and as an example we also present exact data for finite Sierpinski gaskets which fully confirm our predictions.

The scaling properties of waves of topplings in the sandpile model on the Sierpinski gasket are investigated. The exponent describing the asymptotics of the distribution of last waves in an avalanche is found. Predictions for scaling exponents in the forward and backward conditional probabilites for two consecutive waves are given. All predictions were examined by numerical simulations and were found to be in reasonable agreement with the obtained data.

We study the problem of adsorption of self-interacting linear polymers situated in fractal containers that belong to the three-dimensional (3d) Sierpinski gasket (SG) family of fractals. Each member of the 3d SG fractal family has a fractal impenetrable 2d adsorbing surface (which is, in fact, 2d SG fractal) and can be labelled by an integer $b$ ($2\le b\le\infty$). By applying the exact and Monte Carlo renormalization group (MCRG) method, we calculate the critical exponents $\nu$ (associated with the mean squared end-to-end distance of polymers) and $\phi$ (associated with the number of adsorbed monomers), for a sequence of fractals with $2\le b\le4$ (exactly) and $2\le b\le40$ (Monte Carlo). We find that both $\nu$ and $\phi$ monotonically decrease with increasing $b$ (that is, with increasing of the container fractal dimension $d_f$), and the interesting fact that both functions, $\nu(b)$ and $\phi(b)$, cross the estimated Euclidean values. Besides, we establish the phase diagrams, for fractals with $b=3$ and $b=4$, which reveal existence of six different phases that merge together at a multi-critical point, whose nature depends on the value of the monomer energy in the layer adjacent to the adsorbing surface.

A fractal monopole antenna based on the Sierpinski gasket is studied in this paper. The monopole antenna based on the Sierpinski gasket constructed through three iterations displays a multiband behaviour with three bands that are log-periodically spaced by a factor of 2, the same scale factor that defines the geometrical self-similarity of the Sierpinski fractal. The simulated as well as measured input return loss and radiation patterns all display multi-band behaviour. The geometrical scale factor of the Sierpinski fractal is then changed to see whether the bands are shifted according to the new scale factor or not. The simulated results showed that the band positions could be controlled by changing the scale factor but on account of poor input matching. This poor input matching of Sierpinski fractal monopole antenna with modified scale factor is rectified using microstrip feed technique. The simulated as well as measured results of the antenna are in good agreement.

In this paper we prove the existence of a new type of Sierpinski curve Julia set for certain families of rational maps of the complex plane. In these families, the complementary domains consist of open sets that are preimages of the basin at ∞ as well as preimages of other basins of attracting cycles.

We address the problem of evaluating the number S N (t) of distinct sites visited up to time t by N noninteracting random walkers all starting from the same origin in fractal media. For a wide class of fractals ͑of which the percolation cluster at criticality and the Sierpinski gasket are typical examples͒ we propose, for large N and after the short-time compact regime, an asymptotic series for S N (t) analogous to that found for Euclidean media: S N (t)ϷŜ N (t)(1Ϫ⌬). Here Ŝ N (t) is the number of sites ͑volume͒ inside a hypersphere of radius L͓ln(N)/c͔ 1/v where L is the root-mean-square chemical displacement of a single random walker, and v and c determine how fast 1Ϫ⌫ t (l ) ͑the probability that a given site at chemical distance l from the origin is visited by a single random walker by time t) decays for large values of l /L: 1Ϫ⌫ t (l )ϳexp͓Ϫc(l /L) v ͔. For the fractals considered in this paper, vϭd w l /(d w l Ϫ1), d w l being the chemical-diffusion exponent. The corrective term ⌬ is expressed as a series in ln Ϫn (N)ln m ln(N) ͑with nу1 and 0рmрn), which is given explicitly up to nϭ2. This corrective term contributes substantially to the final value of S N (t) even for relatively large values of N.

We consider self-similar sets in the plane for which a cyclic group acts transitively on the pieces. Examples like n-gon Sierpiński gaskets, Gosper snowflake and terdragon are well-known, but we study the whole family. For each n our family is parametrized by the points in the unit disk. Due to a connectedness criterion, there are corresponding Mandelbrot sets which are used to find various new examples with interesting properties. The Mandelbrot sets for n > 2 are regular-closed, and the open set condition holds for all parameters on their boundary, which is not known for the case n = 2.

We describe a geometric approach for studying phase transitions, based upon the analysis of the "density of states" (DOS) functions (exact partition functions) for finite Ising systems. This approach presents a complementary method to the standard Monte Carlo method, since with a single calculation of the density of states (which is independent of parameters and depends only on the topology of the system), the entire range of parameter values can be studied with minimal additional effort. We calculate the DOS functions for the nearest-neighbor (nn) lsing model in nonzero field for square lattices up to 12 × 12 spins, and for triangular lattices up to * Corresponding author. ~" This work is based in part on the Ph.D. thesis of S. Sastry.

Abstract— A fractal monopole antenna based on the Sierpinski gasket is studied in this paper. The monopole antenna based on the Sierpinski gasket constructed through three iterations displays a multiband behaviour with three bands that are log-periodically spaced by a ...

Abstract. In this paper we prove the existence of a new type of Sierpinski curve Julia set for certain families of rational maps of the complex plane. In these families, the complementary domains consist of open sets that are preimages of the basin at ∞ as well as preimages of other basins of attracting cycles. In recent years the families of rational maps of the complex plane given by zn + λ/zd have been shown to exhibit a rich array of both dynamical and topological phenomena. The principal focus of these studies has most often been the Julia sets for such maps. As is well known, the Julia set is the set on which all of the “interesting ” dynamics occurs. For many λ-values, the Julia sets of these maps are also quite interesting from a topological point of view. For example, for each n ≥ 2, it is known that there are infinitely many λ-values for which the Julia set is a generalized Sierpinski gasket (see [5]), and none of these Julia sets are homeomorphic to each other. As another...

We prove that the generator of the renormalization group of Potts models on hierarchical lattices can be represented by a rational map acting on a finite-dimensional product of complex projective spaces. In this framework we can also consider models with an applied external magnetic field and multiple-spin interactions. We use recent results regarding iteration of rational maps in several complex variables to show that, for some class of hierarchical lattices, Lee-Yang and Fisher zeros belong to the unstable set of the renormalization map.