Universal multifractal properties of circle maps from the point of view of critical phenomena I. Phenomenology

The strange attractor for maps of the circle at criticality has been shown to be characterized by a remarkable infinite set of exponents. This characterization by an infinite set of exponents has become known as the multifractal approach. The present paper reformulates the multifractal properties of the strange attractor in a way more akin to critical phenomena. This new approach allows one to study the universal properties of both the critical point and of its vicinity within the same framework, and it allows universal properties to be extracted from experimental data in a straightforward manner. Obtaining Feigenbaum's scaling function from the experimental data is, by contrast, much more difficult. In addition to the infinite set of exponents, universal amplitude ratios here appear naturally. To study the crossover region near criticality, a correlation time, which plays a role analogous to the correlation length in critical phenomena, is introduced. This new approach is based on the introduction of a joint probability distribution for the positive integer moments of the closest-return distances. This joint probability distribution is physically motivated by the large fluctuations of the multifractal moments with respect to the choice of origin. The joint probability distribution has scaling properties analogous to those of the free energy close to a critical point.     

Universal properties of the transition from quasi-periodicity to chaos in dissipative systems

An exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi-periodic regime is held at a fixed irrational value. Our approach is a natural extension of K.A.M. theory to strong coupling. Most of our analysis is for analytic circle maps. We have found a strong coupling fixed point where invertibility is lost, which describes the universal features of the transition to chaos. We find numerically that any two such critical maps with the same winding number are C¹ conjugate. It follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency.When the winding number has a periodic continued fraction, our renormalization transform has a fixed point and spectra are self similar in addition. For a set of non-periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub-space of all critical maps. Physically one finds singular and universal spectra that do not scale.     

Combination Laws for Scaling Exponents and Relation to the Geometry of Renormalization Operators

Renormalization group has become a standard tool for describing universal properties of different routes to chaos—period-doubling in unimodal maps, quasiperiodic transitions in circle maps, dynamics on the boundaries of Siegel disks, destruction of invariant circles of area-preserving twist maps, and others. The universal scaling exponents for each route are related to the properties of the corresponding renormalization operators.     

The world according to Rényi: thermodynamics of multifractal systems

We discuss basic statistical properties of systems with multifractal structure. This is possible by extending the notion of the usual Gibbs-Shannon entropy into more general framework—Rényi’s information entropy. We address the renormalization issue for Rényi’s entropy on (multi)fractal sets and consequently show how Rényi’s parameter is connected with multifractal singularity spectrum. The maximal entropy approach then provides a passage between Rényi’s information entropy and thermodynamics of multifractals. Important issues such as Rényi’s entropy versus Tsallis-Havrda-Charvat entropy and PDF reconstruction theorem are also studied. Finally, some further speculations on a possible relevance of our approach to cosmology are discussed.     

Renormalization Group for Strongly Coupled Maps

Systems of strongly coupled chaotic maps generically exhibit collective behavior emerging out of extensive chaos. We show how the well-known renormalization group (RG) of unimodal maps can be extended to the coupled systems, and in particular to coupled map lattices (CMLs) with local diffusive coupling. The RG relation derived for CMLs is nonperturbative, i.e., not restricted to a particular class of configurations nor to some vanishingly small region of parameter space. After defining the strong-coupling limit in which the RG applies to almost all asymptotic solutions, we first present the simple case of coupled tent maps. We then turn to the general case of unimodal maps coupled by diffusive coupling operators satisfying basic properties, extending the formal approach developed by Collet and Eckmann for single maps. We finally discuss and illustrate the general consequences of the RG: CMLs are shown to share universal properties in the space-continuous limit which emerges naturally as the group is iterated. We prove that the scaling properly ties of the local map carry to the coupled systems, with an additional scaling factor of length scales implied by the synchronous updating of these dynamical systems. This explains various scaling laws and self-similar features previously observed numerically.     

Reducible expansions and related sharp crossovers in Feigenbaum's renormalization field

We discuss reducible aspects of Mao and Hu's multiple scaling expansion [J. Stat. Phys. 46, 111 (1987); Int. J. Mod. Phys. B 2, 65 (1988)] in the framework of renormalization theory. After establishing a suitable form of reduced expansion, we present numerical evidence showing sharp crossovers from Feigenbaum's constant (delta) to Mao and Hu's constant (delta (')) in the first-order reduced expansion. We find that the crossover is caused by the universal scaling relation existing in constant coefficients of Mao and Hu's expansion. Special attention is paid to constant coefficients corresponding to scaling terms including delta ('). We show numerically that they converge to zero in universal ways with convergence ratios larger than delta. Here, the convergence direction is transversal to the unstable eigendirection of the linearized renormalization operator. From this observation, we propose a concise form of expansion for Feigenbaum's universal function g(r)(x).     

Multifractality of Brownian motion near absorbing polymers

We characterize the multifractal behavior of Brownian motion in the vicinity of an absorbing star polymer. We map the problem to an O(M)-symmetric phi(4)-field theory relating higher moments of the Laplacian field of Brownian motion to corresponding composite operators. The resulting spectra of scaling dimensions of these operators display the convexity properties that are necessarily found for multifractal scaling but unusual for power of field operators in field theory. Using a field-theoretic renormalization group approach we obtain the multifractal spectrum for absorption at the core of a polymer star as an asymptotic series. We evaluate these series using resummation techniques.     

Universal Scaling Behavior of Directed Percolation Around the Upper Critical Dimension

In this work we consider the steady state scaling behavior of directed percolation around the upper critical dimension. In particular we determine numerically the order parameter, its fluctuations as well as the susceptibility as a function of the control parameter and the conjugated field. Additionally to the universal scaling functions, several universal amplitude combinations are considered. We compare our results with those of a renormalization group approach.     

On the spectrum of multifractal diffusion process

A quasi-multifractal model of stochastic processes is considered. In contrast to the more widely known multifractal random walk model, it is free of such substantial drawbacks as infinite variance of the modeled processes and time-dependent increments. An analysis of a multifractal diffusion-type process is presented, including the moments of increments and local scaling exponents of the process. A quasi-multifractal spectrum of the process is computed. A focus is on two new concepts in the theory of multifractal processes: effective scaling exponent and quasi-multifractal spectrum of a process.     

Parameter renormalization of maps based on potential function

A systematic way for deriving the parameter renormalization group equation for one-dimensional maps is presented and the critical behavior of periodic doubling is investigated. Introducing a formal potential function in one-parameter cases, it is shown that accumulation points correspond to local potential maxima and universal constants are easily determined. The estimates of accumulation points and universal constants match the known values asymptotically when the order of potential grows large. The potential function shows scaling in the parameter space with the universal convergent rate at the accumulation point similar to the Feigenbaum universal function. For two-parameter cases, a parameter reduction transformation is found to be useful to determine some important fixed points. A locally defined potential function is introduced and its scaling property is discussed. (c) 1997 American Institute of Physics.     

Hierarchy of critical exponents of currents on -simplex fractals

Using the symmetry of ( d +1)-simplex fractals with decimation number b =2, the current distribution has been determined. Then using the renormalization group technique, based on the independent Schur's invariant polynomials of current distributions, the multifractal spectrum of even moments of current distributions has been evaluated analytically up to order six for an arbitrary value of d. Also the scaling exponents of order 8 and order 10 have been calculated numerically up to d =30. Received: 19 November 1997 / Revised: 21 January 1998 / Accepted: 9 February 1998     

Fractional Fourier Transform of Cantor Sets

A new kind of multifractal is constructed by fractional Fourier transform of Cantor sets. The wavelet transform modulus maxima method is applied to calculate the singularity spectrum under an operational definition of multifractal. In particular, an analysing procedure to determine the spectrum is suggested for practice. Nonanalyticities of singularity spectra or phase transitions are discovered, which are interpreted as some indications on the range of Boltzmann temperature q, on which the scaling relation of partition function holds.     

Bicritical and tetracritical phenomena and scaling properties of the SO(5) theory

By Monte Carlo simulations on a 3D SO(5) rotator model it is shown that, in contrast to the epsilon expansions of the renormalization group, the bicritical point is stable to biquadratic perturbations of AF-SC (antiferromagnetism-superconductivity) repulsions, which are produced by quantum fluctuations originated from the Gutzwiller projection. Therefore, the present work completes the link from a physical projected SO(5) starting point to an asymptotic SO(5) symmetry point. The tetracritical point is stable for attractive AF-SC interactions. Critical exponents and ratios are evaluated by scaling analysis. Bicritical and tetracritical scaling functions are derived for the first time. Suggestions on experiments are given.     

Disorder averaging and finite-size scaling

We propose a new picture of the renormalization group (RG) approach in the presence of disorder, which considers the RG trajectories of each random sample (realization) separately instead of the usual renormalization of the averaged free energy. The main consequence of the theory is that the average over randomness has to be taken after finding the critical point of each realization. To demonstrate these concepts, we study the finite-size scaling properties of the two-dimensional random-bond Ising model. We find that most of the previously observed finite-size corrections are due to the sample-to-sample fluctuation of the critical temperature and scaling predictions are fulfilled only by the new average.     

三能级钾原子气体三维傅里叶变换频谱的解析解

利用投影切片定理、傅里叶位移定理和误差函数给出三能级钾原子气体三维傅里叶变换频谱在T=0界面的解析解.固定均匀线宽,非均匀展宽和对角线相关系数可以定量地识别,通过在适当方向上拟合三维傅里叶变换频谱谱峰的切片来确定.结果表明,非均匀展宽增大,频谱图沿着对角线方向延伸,对角线相关系数增大,频谱图逐渐变圆,振幅也逐渐变小.     

Scaling properties of ℤ k- 1 actions on the circle

We derive universal scaling properties for ^k–1 actions on the circle whose generators have rotation numbers algebraic of degreek. As fork=2 these properties can be explained for arbitraryk in terms of a renormalization group transformation. It has at least one trivial fixed point corresponding to an action whose generators are pure rotations. The spectrum of the linearized transformation in this fixed point is analyzed completely. The fixed point is hyperbolic with a (k–1)-dimensional unstable manifold. In the casek=2 the known results are therefore recovered.     

Multifractality at the quantum Hall transition: beyond the parabolic paradigm

We present an ultrahigh-precision numerical study of the spectrum of multifractal exponents Deltaq characterizing anomalous scaling of wave function moments |psi|2q at the quantum Hall transition. The result reads Deltaq=2q(1-q)[b0+b1(q-1/2)2+cdots, three dots, centered], with b0=0.1291+/-0.0002 and b1=0.0029+/-0.0003. The central finding is that the spectrum is not exactly parabolic: b1 not equal0. This rules out a class of theories of the Wess-Zumino-Witten type proposed recently as possible conformal field theories of the quantum Hall critical point.     

Cantori spectra for the sawtooth map

Summary We calculate analytically the Fourier spectrum for the cantori of the sawtooth maps. These maps are a one-parameter family of chaotic area-preserving maps. We show that the Fourier spectrum grows exponentially for parameters close to criticality, and that it exhibits self-similarity structure at all length scales. The self-similarity scales as the quotients of successive denominators of the convergents of irrational numbers. We compute exactly the scaling for quadratic irrationals. The behaviour of the spectrum for large values of the perturbation parameter is also investigated. The author of this paper has agreed to not receive the proofs for correction.