A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions

In this paper we establish the complete multifractal formalism for equilibrium measures for Hölder continuous conformal expanding maps andexpanding Markov Moran-like geometric constructions. Examples include Markov maps of an interval, beta transformations of an interval, rational maps with hyperbolic Julia sets, and conformal toral endomorphisms. We also construct a Hölder continuous homeomorphism of a compact metric space with an ergodic invariant measure of positive entropy for which the dimension spectrum is not convex, and hence the multifractal formalism fails.     

Multifractal Analysis of Asymptotically Additive Sequences

For saturated maps, we effect a complete multifractal analysis of the dimension spectra obtained from asymptotically additive sequences of continuous functions. This includes, for example, the class of maps with the specification property. We consider also the more general cases of ratios of sequences and of multidimensional spectra in which a single sequence is replaced by a vector of sequences. In addition, we establish a conditional variational principle for the topological pressure of a continuous function on the level sets of an asymptotically additive sequence (again in the former general setting). Finally, we apply our results to the dimension spectra of an average conformal repeller. In particular, we obtain almost automatically a conditional variational principle for the Hausdorff dimension of the level sets obtained from an asymptotically additive sequence.     

SIMULATION OF MULTIPLE FRACTAL AND COMPACT GROWTH OF ULTRA-THIN FILMS ON HEXAGONAL SUBSTRATE

The multiple cluster growth of ultra-thin films with different deposition rate and different substrate temperature has been studied by kinetic Monte-Carlo simulation. With increasing diffusion rate along cluster edges (corresponding to an increasing substrate temperature), pattern structures change smoothly from fractal islands, compact islands with random shapes, to regular islands, and the average branch width of clusters increases continuously up to some constant value in the compact island limit. The formation of the multiple fractal and compact clusters can be described quantitatively by multifractal. The results of multifractal analysis show that with pattern change from fractal to compact islands, the Hausdorff dimension D₀, the information dimension D₁, and the correlation dimension D₂ decrease, while the width and height of the multifractal spectra increase.     

Diffusion on two-dimensional percolation clusters with multifractal jump probabilities

By means of Monte Carlo simulations we studied the properties of diffusion limited recombination reactions (DLRR's) and random walks on two dimensional incipient percolation clusters with multifractal jump probabilities. We claim that, for these kind of geometric and energetic heterogeneous substrata, the long time behavior of the particle density in a DLRR is determined by a random walk exponent. It is also suggested that the exploration of a random walk is compact. It is considered a general case of intersection ind euclidean dimension of a random fractal of dimension D_F and a multifractal distribution of probabilities of dimensionsD _q (q real), where the two dimensional incipient percolation clusters with multifractal jump probabilities are particular examples. We argue that the object formed by this intersection is a multifractal of dimensionsD' _q =D _q +D _F -d, for a finite interval ofq.     

Boundary critical behaviour of two-dimensional random Potts models

Using extensive Monte Carlo simulations, transfer matrix techniques and conformal invariance, ferromagnetic random q-state Potts models for are studied in the vicinity of the critical temperature. In particular the surface and bulk magnetization exponents and are found monotonically increasing with q. At the critical temperature, different moments (n) of the magnetization profiles are calculated which are all found to accurately follow predictions of conformal invariance. The critical correlation functions show multifractal behaviour, the decay exponents of the different moments both in the volume and at the surface, are n-dependent. Received 4 June 1999     

Geometry and scaling laws of excursion and iso-sets of enstrophy and dissipation in isotropic turbulence

Motivated by interest in the geometry of high intensity events of turbulent flows, we examine the spatial correlation functions of sets where turbulent events are particularly intense. These sets are defined using indicator functions on excursion and iso-value sets. Their geometric scaling properties are analysed by examining possible power-law decay of their radial correlation function. We apply the analysis to enstrophy, dissipation and velocity gradient invariants Q and R and their joint spatial distributions, using data from a direct numerical simulation of isotropic turbulence at Re_λ ≈ 430. While no fractal scaling is found in the inertial range using box-counting in the finite Reynolds number flow considered here, power-law scaling in the inertial range is found in the radial correlation functions. Thus, a geometric characterisation in terms of these sets’ correlation dimension is possible. Strong dependence on the enstrophy and dissipation threshold is found, consistent with multifractal behaviour. Nevertheless, the lack of scaling of the box-counting analysis precludes direct quantitative comparisons with earlier work based on multifractal formalism. Surprising trends, such as a lower correlation dimension for strong dissipation events compared to strong enstrophy events, are observed and interpreted in terms of spatial coherence of vortices in the flow.     

Finitized Conformal Spectra of the Ising Model on the Klein Bottle and Möbius Strip

We study the conformal spectra of the critical square lattice Ising model on the Klein bottle and Möbius strip using Yang–Baxter techniques and the solution of functional equations. In particular, we obtain expressions for the finitized conformal partition functions in terms of finitized Virasoro characters. This demonstrates that Yang–Baxter techniques and functional equations can be used to study the conformal spectra of more general exactly solvable lattice models in these topologies. The results rely on certain properties of the eigenvalues which are confirmed numerically.     

Generic Metrics and Connections on Spin- and Spin c -Manifolds

We study the dependence of the dimension h ⁰(g,A) of the kernel of the Atyiah-Singer Dirac operator on a spin ^c -manifold M on the metric g and the connection A. The main result is that in the case of spin-structures the value of h ⁰(g) for the generic metric is given by the absolute value of the index provided . In dimension 2 the mod-2 index theorems have to be taken into a account and we obtain an extension of a classical result in the theory of Riemann surfaces. In the spin ^c -case we also discuss upper bounds on h ⁰(g,A) for generic metrics, and we obtain a complete result in dimension 2. The much simpler dependence on the connection A and applications to Seiberg–Witten theory are also discussed. Received: 3 July 1996 / Accepted: 27 February 1997     

Inheriting geodesic flows

We investigate the propagation equations for the expansion, vorticity and shear for perfect fluid space-times which are geodesic. It is assumed that space-time admits a conformal Killing vector which is inheriting so that fluid flow lines are mapped conformally. Simple constraints on the electric and magnetic parts of the Weyl tensor are found for conformal symmetry. For homothetic vectors the vorticity and shear are free; they vanish for nonhomothetic vectors. We prove a conjecture for conformal symmetries in the special case of inheriting geodesic flows: there exist no proper conformal Killing vectors (ψ _;ab ≠ 0) for perfect fluids except for Robertson-Walker space-times. For a nonhomothetic vector field the propagation of the quantity ln (R _ab u ^a u ^b ) along the integral curves of the symmetry vector is homogeneous.     

Multifractality of nonlinear iterative processes

The Havlin-Bunde multifractal hypothesis [Physica D 38:184 (1989)] is expanded (in the form of the dimension-invariance approach) to nonlinear iterative (recursion) processes such as dielectric breakdown, phase transitions from periodic attractors to chaos, and cascades in turbulence. Comparison with model and laboratory data of different authors shows that for strong nonlinearity the dimension invariance is broken     

Solution of functional equations of restricted An−1(1) fused lattice models

Functional equations, in the form of fusion hierarchies, are studied for the transfer matrices of the fused restricted A_n−1⁽¹⁾ lattice models of Jimbo, Miwa and Okado. Specifically, these equations are solved analytically for the finite-size scaling spectra, central charges and some conformal weights. The results are obtained in terms of Rogers dilogarithm and correspond to coset conformal field theories based on the affine Lie algebra A_n−1⁽¹⁾ with GKO pair A_n−1⁽¹⁾ ⊕ A_n−1⁽¹⁾ ⊃ A_n−1⁽¹⁾.     

Suspension Flows Over Countable Markov Shifts

We introduce the notion of topological pressure for suspension flows over countable Markov shifts, and we develop the associated thermodynamic formalism. In particular, we establish a variational principle for the topological pressure, and an approximation property in terms of the pressure on compact invariant sets. As an application we present a multifractal analysis for the entropy spectrum of Birkhoff averages for suspension flows over countable Markov shifts. The domain of the spectrum may be unbounded and the spectrum may not be analytic. We provide explicit examples where this happens. We also discuss the existence of full measures on the level sets of the multifractal decomposition.     

Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis

Multifractal detrended fluctuation analysis (MF-DFA) is a relatively new method of multifractal analysis. It is extended from detrended fluctuation analysis (DFA), which was developed for detecting the long-range correlation and the fractal properties in stationary and non-stationary time series. Although MF-DFA has become a widely used method, some relationships among the exponents established in the original paper seem to be incorrect under the general situation. In this paper, we theoretically and experimentally demonstrate the invalidity of the expression τ(q)=qh(q)-1 stipulating the relationship between the multifractal exponent τ(q) and the generalized Hurst exponent h(q). As a replacement, a general relationship is established on the basis of the universal multifractal formalism for the stationary series as τ(q)=qh(q)-qH'-1, where H' is the nonconservation parameter in the universal multifractal formalism. The singular spectra, α and f(α), are also derived according to this new relationship.     

A Comment on Differential Identities for the Weyl Spinor Perturbations

It is shown that in a type-D vacuum space-time with cosmological constant, the components of the Weyl spinor perturbations along the principal spinors of the background conformal curvature satisfy differential identities, which are valid in all the normalized spin frames {o ^A , ^A } such that o ^A and ^A are double principal spinors of the background conformal curvature.     

Effects of Monomer Size Distribution on the fractal dimensionality of diffusion-limited aggregates

Computer simulations of diffusion-limited aggregation (DLA) for monomers to investigate the effects of size and of lognormal distribution on the fractal dimensionality of the aggregates were conducted on a two-dimensional lattice. The results show the DLA clusters posses multifractal characteristics. For clusters consisting of monodisperse monomers, the bifurcation point on the graph of the pair correlation function (PCF) for each cluster is located right at the monomers size under investigation The textural dimension (D_f1) has a stable value of about 1.65, whereas the structural dimension (D_f2) decreased with increase in monomer size. For the cases with monomers in log-normal distributions, the textural dimension is around 1.67; however, the structural dimension decreases with increasing polydispersity of monomer size.     

Thermal States in Conformal QFT. I

We analyze the set of locally normal KMS states w.r.t. the translation group for a local conformal net A{{\mathcal A}} of von Neumann algebras on \mathbb R{\mathbb R} . In this first part, we focus on the completely rational net A{{\mathcal A}} . Our main result here states that, if A{{\mathcal{A}}} is completely rational, there exists exactly one locally normal KMS state j{\varphi} . Moreover, j{\varphi} is canonically constructed by a geometric procedure. A crucial r?le is played by the analysis of the “thermal completion net” associated with a locally normal KMS state. A similar uniqueness result holds for KMS states of two-dimensional local conformal nets w.r.t. the time-translation one-parameter group.     

Conformal transformations of <Emphasis Type="Italic">S</Emphasis>-matrix in scalar field theory

In this paper, three methods for describing the conformal transformations of the S-matrix in quantum field theory are proposed. They are illustrated by applying the algebraic renormalization procedure to the quantum scalar field theory, defined by the LSZ reduction mechanism in the BPHZ renormalization scheme. Central results are shown to be independent of scheme choices and derived to all orders in loop expansions. Firstly, the local Callan-Symanzik equation is constructed, in which the insertion of the trace of the energy-momentum tensor is related to the beta function and the anomalous dimension. With this result, the Ward identities for the conformal transformations of the Green functions are derived. Then the conformal transformations of the S-matrix defined by the LSZ reduction procedure are calculated. Secondly, the conformal transformations of the S-matrix in the functional formalism are related to charge constructions. The commutators between the charges and the S-matrix operator are written in a compact way to represent the conformal transformations of the S-matrix. Lastly, the massive scalar field theory with local coupling is introduced in order to control breaking of the conformal invariance further. The conformal transformations of the S-matrix with local coupling are calculatedReceived: 3 June 2003, Revised: 24 July 2003, Published online: 2 October 2003Yong Zhang: Supported by Graduiertenkolleg Quantenfeldtheorie: Mathematische Struktur und physikalische Anwendungen, University Leipzig.