Session 7: Dynamical System and Fractals, Wavelets Strongly Transitivity of Transition Matrices in Multi-Dimensional Lattice Model

Jung-Chao Ban*
Department of Mathematics
The National Center for Theoretical Sciences
National Tsing Hua University
Hsinchu, Taiwan 30043, Taiwan

Song-Sun Lin
Department of Applied Mathematics,
National Chiao-Tung University,
Hsin-Chu, Taiwan

Yin-Heng Lin
Department of Applied Mathematics,
National Chiao-Tung University,
Hsin-Chu, Taiwan

Abstract : In this talk, we introduce the transition matrices in multi-dimensional lattice model, in the previous result , the recursive formulas for transition matrices \Bbb An on \Bbb Z2×n are defined and of size 2n-1×2n-1.

It's of nature and interest to know whether \Bbb An is strongly transitive if \Bbb A2 possesses this property. Our purpose is to give some necessary and sufficient conditions for \Bbb A2 to guarantee the ST property for \Bbb An and the relations between the ST property and positivity of entropy are also discussed.

Symbolic Dynamics for Non-autonomous Symplectic Twist Maps near the Anti-integrable Limit

Yi-Chiuan Chen
Institute of Mathematics
Academia Sinica
Taipei 11529, Taiwan

Abstract : The embedding of Bernoulli shift into general time-dependent systems was first obtained by Stoffer, Meyer and Sell by using the generalised Melnikov function. Later on, Wiggins developed the non-stationary Conley-Moser conditions and showed for non-autonomous maps the embedding of a subshift of finite type. In this talk, we extend the anti-integrability to non-autonomous symplectic twist maps to show the shift dynamics can be embedded in a natural way. We use non-autonomous standard-like and Hénon-like maps as examples to illustrate that the embedded shift can be a full shift, a subshift of finite type or of infinite type.

Classification of Refinable Splines

Xin-Rong Dai*
Department of Applied Mathematics
Zhejiang University of Technology
Hangzhou, 310014,P.R.China

De-jun Feng
Department of Mathematical Sciences
Tsinghua University
Beijing, 100084, P. R. China

Yang Wang
School of Mathematics
Georgia Institute of Technology
Atlanta, Georgia 30332, USA.

Abstract : A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well known B-splines, play a key role in computer aided geometric designs. So far all studies on refinable splines have focused on positive integer dilations and integer translations, and under this setting a rather complete classification was obtained by Lawton, Lee and Shen. However, refinable splines do not have to have integer dilations and integer translations. The classification of refinable splines with non-integer dilations and arbitrary translations are studied in this paper. We classify completely all refinable splines with integer translations with arbitrary dilations.

Bernoulli convolutions associated with certain algebraic numbers

De-Jun Feng
Department of Mathematical Sciences
Tsinghua University
Beijing, 100084, P.R. China

Abstract : Let ml be the Bernoulli convolution associated with l (1,2). It is a long-time unsolved problem to classify all the parameters l for which ml are totally singular. The only known class of such numbers, found by P. Erdos in 1930's, is the family of all Pisot numbers. Along this direction, in a joint paper with Y. Wang, we constructed some non-Pisot numbers such that the corresponding Bernoullii convolutions have some ßingular" behavior (having no L2 densities). In the another direction, the author proved a modified complete multifractal formalism for the Bernoulli convolutions associated with all Pisot numbers.

Discontinuous Dynamical Systems: Progress and Problems

Xinchu Fu
Department of Mathematics
Shanghai University
Shanghai, 200436, P. R. China

Abstract : We discuss dynamical behaviour of a class of discontinuous maps including piecewise linear maps on the 2-torus and planar piecewise isometries. We also discuss coding and symbolic description of these maps. For piecewise linear parabolic maps on the torus, when the entries in the matrix are rational we show that the maximal invariant set has positive Lebesgue measure. For linear elliptic maps with round-off and quantization discontinuities, two examples arising from digital signal processing are examined, and are shown to have the dynamics of piecewise isometries of a union of convex polygons on the plane. Properties of invariant disk packings for invertible piecewise isometries are also discussed. It is shown that tangencies between such disks in the packings are very rare. This support the long-standing conjecture that the exceptional sets possess positive Lebesgue measure. Some results about the topological entropy and dynamical complexity of piecewise isometries are given; and global attractors for planar PWIs are characterized via invariant measures and positive continuous functions on phase space. And finally, some typical open problems in this new research area are presented.

Stability and Bifurcation in a Ring Neural Network

Shangjiang Guo
College of Mathematics and Econometrics
Hunan University
Changsha, Hunan 410082,P. R. China

Abstract : The focus of this talk is to study issues related to stability and bifurcation of a ring of neurons with self-feedback and delays, which has an on-center off-surround characteristic and can be identified with a Lie group. Such a network has been found in a variety of neural structures, such as neocortex, cerebellum, hippocampus, and even in chemistry and electrical engineering, and can be studied to gain insight into the mechanisms underlying the behavior of recurrent network. Needless to say, this is a very difficult task due to the infinite-dimensional nature of the problem caused by the synaptic delay and the possible spatial structure of the system (equivariant with respect to a Dn-action). Some general theorems are available about the existence and global continuation of periodic solutions in symmetric delay differential equations. However, applications of these general results to concrete systems such as on-center off-surround networks involve several highly nontrivial tasks: (i) distribution of zeros in characteristic equations which are usually transcendental and depend on parameters; (ii) symmetry analysis on certain generalized eigenspaces of the generator of a linearized system and identifcation of these spaces with a direct sum of two identical absolute irreducible representations of Dn; (iii) calculation of the so-called crossing numbers which are related to the usual transversality condition in a standard Hopf bifurcation theory; (iv) a priori estimation of the period and of the norm of a periodic solution. This dissertation is organized as follows:

Firstly, linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. By means of space decomposition, we subtly discuss the distribution of zeros of the characteristic equation, and then we have derived some sufficient conditions to ensure that all the characteristic roots have negative real parts. Hence, the zero solution of the model is asymptotically stable.

Secondly, by means of the standard Hopf bifurcation theory, we obtain a branch of periodic solutions and its continuation. Based on the normal form approach and the center manifold theory, we derive the formula for determining the properties of Hopf bifurcating periodic orbit for a ring of neurons with delays, such as the direction of Hopf bifurcation, stability of the Hopf bifurcating periodic orbits and so on.

Thirdly, under some suitable conditions, such a network has a slowly oscilatory synchronous solution which is completely characterized by a scalar delay differential equation. Despite the fact that the slowly oscillatory periodic solution of the scalar equation is stable, by making use of Floquet theory and Krein-Rutman theorem, we show that the associated synchronous periodic solution is unstable if the size of the network is large.

Forthly, by using of the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups, we not only investigate the effect of synaptic delay of signal transmission on the pattern formation, but also obtain some important results about the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns (i.e., mirror-reflecting waves, standing waves and discrete waves). Moreover, we analyze the stability of the bifurcated periodic solutions. In addition, we show that spontaneous bifurcations of multiple branches of periodic solutions exist for all large delay (global continuation), and consider the coincidence of these periodic solutions.

Some Properties of Self-similar Tiles in One Dimension

Xing-Gang He
Department of Mathematics
Central China Normal University
Wuhan, 430079, P. R. China

Abstract : The aim of this note is to study the construction of the boundary of a self-similar tile, which generates by an iterated function system {fi(x) = 1/N(x+di)}i = 1N. We will show that the boundary has complicated structure (no simple points), however, it is a regular fractal set.

Potential Spaces on Fractals

Jiaxin Hu
Department of Mathematical Science
Tsinghua University
Beijing 100084 China

Abstract : We introduce potential spaces on fractal metric spaces, investigate their embedding theorems, and derive various Besov spaces. Our starting point is that there exists a local, stochastically complete heat kernel satisfying a two-sided estimate on the fractal considered. The results of this paper are among the marvelous consequences of the heat kernel on the fractal. This is a joint work with M. Zaehle.

Structural Stability of Isolated Invariant Set

Tu-Sen Huang
Institute of Mathematics
Zhejiang University of Sciences
Hangzhou City
Zhejiang Province, 310018, P.R.China

Abstract : In this article, we give some properties of attractors for a flow, introduce the concept of strong stability, and prove that any attractor in the Conley's sense is strongly stable. We discuss the relation between the existence of Morse decompositions for an isolated invariant set and that of Liapunov functions. We also introduce the notion of strong continuation about an isolated invariant set, and give some types of isolated invariant set which are related by strong continution. last, we discuss the Conley's conjecture about the semicontinuity of the nonwandering set function, and give its partial answer

Disklikeness of Planar Self-affine Tiles

King-Shun Leung*
Department of Mathematics
The Hong Kong Institute of Education
Tai Po, Hong Kong

Ka-Sing Lau
Department of Mathematics
The Chinese University of Hong Kong
Shatin, Hong Kong

Abstract : We study the disklikeness of the planar self-affine tile T generated by an integral expanding matrix A and a consecutive collinear digit set D = {0, v, 2v, , (|q|-1)v } \BbbZ2. Let f(x) = x2+ p x+ q be the characteristic polynomial of A, we show that the tile T is disklike if and only if  2|p| |q+2|. Moreover, T is a hexagonal tile for all the cases except when p = 0, T is a square tile. The proof depends on certain special devices to count the numbers of nodal points and neighbors of T and a criterion on disklikeness devised by Bandt and Wang. We also describe the boundary of such a disklike T by means of a labelled directed graph.

Patterns Generation and Transition Matrices in Higher Dimensional Lattice Models

Yin-Heng Lin
Department of Applied Mathematics
National Chiao Tung University
Hsinchu 300, Taiwan

Abstract : The aim of this paper is to study the pattern generation problems for more symbols on larger lattice with edge 2l in d-dimensional models, d 3. Defining orderings for pattern U on S2l×2l××2l on Z2l×2l××2l Zd+1 enable us to derive simple recursion formulas for generating ordering matrices and the corresponding transition matrices. Furthermore, the spatial entropy can be computed through the maximum eigenvalue of transition matrices.

Qualitative Behavior of an Iterative Functional Differential Equation

Yin-Wei Lin* and Tzon-Tzer Lu
Department of Applied Mathematics
National Sun Yat-sen University
Kaohsiung, Taiwan, 80424 R.O.C.

Abstract : Functional differential equations with delay have long been studied due to their practical applications. Among them many researches study the case when the deviating argument depending on the state variable. For example, early in 1967 Cooke proposed

x(t)+ax(t-r(x(t))) = 0
for some population and infection models. Such type of differential equations with state dependent delay unavoidable contain certain composition of unknown function. In literature they are called the iterative functional differential equations. There are many research directions on these equations. For instance, Si, Wang and Cheng found the analytic solutions of many such iterative functional differential equations.

The existence and uniqueness of solution is one of the major problem. For examples, Dunkel worked on

x(t) = f(x(h(x(t)))),
and Fiagbedzi and Gebeily on
x(t) = 1
by using Banach or Schauder fixed point theorem. In this talk, we use the Weierstrass M-test to establish our existence and uniqueness of solution of iterative functional differential equation
x(t) = f(t,x(t),x[m](t)),
where x[m](z) = x(x(x(z))) denotes the m-th iterate of the function x(z).

Also great attention has been paid to the qualitative properties of these iterative functional differential equations. For example, Stanek (1995) proved every solution of

x(t) = x(x(t))+x(t),
either vanishes identically or is strictly monotonic. Eder (1984) classified all the solutions of
x(t) = x(x(t))
as four types. We have more detailed analysis on Eder's work. In fact, two types of the increasing solutions are identical. In some situations, we have surprising non-uniqueness of the solutions. For the other two types of decreasing solutions, we also have some deeper results to clarify their behavior.

Chaotic Solutions of the Equation of an Undamped
Pendulum with Oscillatory Forcing

Chunqing Lu
Department of Mathematics and Statistics
Southern Illinois University at Edwardsville
Edwardsville, IL 62026, U.S.A.

Abstract : The paper studies the chaotic motion of an undamped pendulum with oscillatory forcing, which is governed by a nonlinear second order equation, u+ (1+a sinb t)sinu = 0. It rigorously proves the existence of infinitely many periodic and nonperiodic solutions. The shooting method is applied in the proof.

Limiting Behaviors of Maps and Sensitivity

Jie-Hua Mai
Institute of Mathematics
Shantou University
Shantou, Guangdong, 515063, P.R.China

Abstract : In this paper we study the asymptotically almost periodic points and the uniformly segment-recurrent points of a continuous map f from a metric space (X,d) to itself.  We prove that f is sensitive if there exist non-empty subsets V and W of with d(V,W) > 0 such that the stable manifolds Ms(V,f) {x X:w(x,f)V } and Ms(W,f) {y X:w(y,f)W } are dense in X, and one of the following two conditions holds: (1) W is f-invariant, and the set of asymptotically almost periodic points of f is dense in X; (2) Both W and V are f-invariant, and the set of uniformly segment-recurrent points of f is dense in X.  In particular, we show that if the set of w-transitive points and the set of asymptotically almost periodic points of f are dense in X, but f|w(f) is not minimal, then f is sensitive.

2000 Mathematics Subject Classification.  37D45, 37B25, 54H20.

Key words and phrases. Metric space,continuous map,sensitivity, asymptotically almost periodic point, uniformly segment-recurrent point, stable manifold.

f The work was supported by the Special Foundation of National Prior Basis Research of China (Grant No. G1999075108).

Iterated function systems of generalized finite type

Ka-Sing Lau
Department of Mathematics
The Chinese University of Hong Kong
Shatin, Hong Kong

Sze-Man Ngai*
Department of Mathematical Sciences
Georgia Southern University
Statesboro, GA 30460, U.S.A.

Abstract : For iterated function systems (IFS) of contractive similitudes on Rd, we introduce a generalized finite type condition which allows us to include some IFS's of contractive similitudes with overlaps whose contraction ratios are not exponentially commensurable. We show that this condition extends the finite type condition and the open set condition. It also implies the weak separation property. We also show that under this condition, the Hausdorff dimension of the attractor can be computed in terms of the spectral radius of a weighted incidence matrix.

Some New Results on the Open Set Condition

Christoph Bandt and Nguyen Viet Hung
Institute for Mathematics and Informatics
Arndt University, 17487 Greifswald, Germany

Hui Rao*
Department of Mathematics
Tsinghua University
P.O. Box 100084, Beijing, China

Abstract : Let f1,...,fm be contracting similarity maps on \Bbb Rn. Let A be the unique compact set such that

A = f1(A)
The fi are said to satisfy the OSC if there exists a nonempty open set V \Bbb Rn such that

i = 1 
fi(V) V    and     fi(V)fj(V) =   for i j.
We call V a feasible open set of the fi, or of A.

The main problem with OSC is that it is difficult to check. So far, there is no general algorithm to construct a feasible open set, cf. []. In this note, we give two interesting characterizations of OSC. One depends on the construction of a so-called central open set. We show that the fi satisfy OSC if and only if the central open set is non-empty.

The other one is related to fixed points of neighbor maps and the ``forbidden points'', the points of \Bbb Rn which do not belong to any feasible open set. For maps fi on \Bbb R, we characterize the set of forbidden points and show that OSC holds if and only if the fixed points of neighbor maps are not dense in \Bbb R.

We provide a few examples and show that a feasible open set V can be unbounded, or can admit infinite Lebesgue measure.

Maximal Functions and Fourier Transforms of Self-similar Measures

Yong Lin
Department of Mathematics
School of Information
Renmin University of China
Beijing, 100872, China

Huo-Jun Ruan*
Department of Mathematics
Zhejiang University
Hangzhou, 310027, China

Abstract : A self-similar measure on \mathbbRn is defined to be a probability measure satisfying

m = N

j = 1 
pj mSj-1 + M

j = 1 
qj (m* m) Tj-1,
where Sj x = rj Rj x + bj, Tj x = hj Qj x + cj are contractive similarities, 0 < rj < 1, 0 < hj < 1/2, 0 < pj < 1, 0 < qj < 1, j = 1N pj +j = 1M qj = 1 , Rj, Qj are orthogonal matrix and m* m is the convolution of two measures.

When M = 0, m is a linear self-similar measure, we establish the asymptotic behavior of averages of the derivative of the Fourier transform of m such as

|x| R 





dx = O(Rn-b)
for any order derivation of [^(m)](x) as R under certain additional hypotheses.

When M > 0, m is a nonlinear self-similar measure, we get some results of Lp boundedness for maximal operators of m from the pointwise asymptotic estimate of the Fourier transform of m made by Strichartz.

A Riesz Product Type Measure on the Cantor Group

Qi-yan Shi*
Department of Mathematical Sciences
Tsinghua University
Beijing 100084, China

Zhi-Ying Wen
Department of Mathematical Sciences
Tsinghua University
Beijing 100084, China

Abstract : Let W = {-1, 1}N and { wj } be independent random variables taking values in {-1, 1} with equal probability. Endowed with the product topology and under the operation of pointwise product, W is a compact abelian group, the so-called Cantor group. Let a, b be two real number with |a| + |b| < 1.

Riesz type products on W

Pn = n

j = 1 
(1 + a wj + b wj+1)
is studied. We show that the weak limit of { \dfracPndwinfw Pn dw } exists, thus defined a probability measure on W; the measure is continuous and singular with respect to the normalized Haar measure on W; it has multifractal structure, and the multifractal formalism holds for some cases.

Multiple-time-scale Dynamical Systems in Neuron Models

Jianzhong Su
Department of Mathematics
The University of Texas at Arlington
Arlington, Texas 76019, USA

Abstract : Elliptic bursting arises from fast-low systems and involves recurrent alternation between active phases of large amplitude oscillations and silent phases of small amplitude oscillations. We discuss in this talk a geometric analysis of elliptic bursting with and without noise. We first prove the existence of elliptic bursting solutions for a class of fast-low systems without noise by establishing an invariant region for the return map of the solutions. We further conclude all essential dynamics can be described by a S1 to S1 mapping. For noisy elliptic bursters, the bursting patterns depend on random variations associated with delayed bifurcations. We provide an exact formulation of the duration of delay and analyse its distribution. The duration of the delay, and consequently the durations of active and silent phases, is shown to be closely related to the logarithm of the amplitude of the noise. The treatment of noisy delayed bifurcation here is a general theory of delayed bifurcation valid for other systems involving delayed bifurcation as well and is a continuation of the rigorous Shishkova-eishtadt theory on delayed bifurcation or delay of stability loss. These dynamics phenomena come from Hodgkin-Huxley equation in neuronal electro-physiological dynamics.

Function Spaces and Fractals on Local Fields

Weiyi Su
Department of Mathematics
Nanjing University
Nanjing , 210093, P.R.China

Abstract : We introduce certain spaces on local fields, such as Holder spaces Cs (K),s R, Sobolev spaces Ws, s R, Triebel B-type spaces

Bsr,t (K), s R, 0 < r, t ,
F-type spaces
Fsr,t (K), s R, 0 < r, t ,
and study some properties of these spaces. Moreover, fractals defined on these spaces are considered, including Hausdorff dimensions, distribution dimensions, Fourier dimensions and capacity dimensions, and their relationship. Finally, differentiability of fractals are discussed in this note.

Topology of Planar Connected Self-similar Tiles

Sze-Man Ngai
Department of Mathematical Sciences
Georgia Southern University
Statesboro, GA 30460-8093, USA.

Tai-Man Tang*
Department of Mathematics
Xiangtan University
Hunan 411105, P.R.China

Abstract : Let T be a planar connected self-similar tile. We show that the closure of each component of T is locally connected. From considerations on the set of cut points of T, we show that the closure of the components of T are disks for some T. This work answers the questions on the disk-likedness of the `components' of some classical fractals including the Levy dragon, the Eisenstein set and the fundamental domain of a canonical number system. It extends Luo-Rao-Tan's work on a problem of Conway-Grünbaum on planar self-similar tiles.

Classification of Homoclinic Tangencies for Periodic Perturbed Systems

Yun Tang* and Fenghong Yang

Department of Mathematical Sciences
Tsinghua University
Beijing 100084, China

Abstract : Classification of homoclinic tangencies for n-harmonic perturbed systems is discussed. The relationship of the order of Melnikov function's zero with the harmonic component is given. By applying the singularity theory to Melnikov function, possible types of homoclinic tangencies is studied and the principle for their classification is established. In addition, we focus on a certain multi-harmonic perturbed system to show the homoclinic bifurcation and give the bifurcation diagrams.

On Hopf Cyclicity of Planar Systems with Multiple Parameters

Wanyi Wang*
Department of Mathematics
Shanghai Jiao Tong University
Shanghai,200240, P.R. China
Department of Mathematics
Inner Mongolia Normal University
Huhhot, 010022, Inner Mongolia, P. R. China

Maoan Han
Department of Mathematics,
Shanghai Jiao Tong University
Shanghai,200240, P.R. China

Jiong Sun
Department of Mathematics
Inner Mongolia University
Huhhot, 010021, Inner Mongolia, P.R. China

Abstract : In this paper, we discuss the maximal number of limit cycles which appear under perturbations in Hopf bifurcations by using degenerate first-order Melnikov function with multiple parameters.

On Regular Properties of Method I Construction and Packing Measures

Shengyou Wen
Department of Mathematics
Hubei University
Hubei 430062, P.R.China

Abstract : Let t be a premeasure on a complete separable metric space and let t* be the Method I measure constructed from t. We will show that under some reasonable conditions the Method I measure t* has some regular properties which can be described in terms of the premesure t. We also show that a packing premeasure fulfills these conditions if and only if it is locally finite.

The Multifractal Spectrum of some Moran Measures

Min Wu
School of Mathematics
South China University of Technology
Guangzhou, Guangdong 510640, P. R. China

Abstract : The Multifractal formalism is shown to hold for a class of Moran measures supported on the Moran fractals associated with the sequences of which the frequency of the litter exists.

This work is supported by Natural Science Foundation of China under grant number 10171028 and the Special Founds for Major State Basic Research Projects of China.

Sampling and Reconstruction in some Non-bandlimited Spaces

Jun Xian* and Song Li
Department of Mathematics
Zhejiang University
Hangzhou, China

Abstract : In digital signal and image processing, digital communication, etc., a continuous signal is usually represented and processed by using its discrete samples. For a bandlimited signal of finite energy, it is completely described by the famous classical Shannon sampling theorem. However, we often meet the reconstruction problem for non-bandlimited spaces and irregularly sampling points in practical applications. It is well-known that in the sampling and reconstruction problem for non-bandlimited spaces, signal is often assumed to belong to a shift-invariant spaces.

Clearly we hope signal spaces be sufficiently large to accommodate a large number of possible models. So Aldroubi and Feichtinger introduced lattice-invariant space. The lattice-invariant spaces are a sufficiently large and value family of signal spaces. We will show new reconstruction formula in the lattice-invariant spaces with new method. This result is generalized and improved form of Chen's result.

For practical application and computation of reconstruction, Aldroubi et al., present a A-P iterative algorithm. In this paper, we improve the A-P iterative algorithm in common shift-invariant spaces. The improved algorithm occupies better convergence than the old one. We study the improved algorithm with emphasis on its implementation in spline subspaces. Then we obtain explicit convergence rate of the algorithm in spline subspaces. Numerical results are furnished to illustrate the improved algorithm.

Decay of Correlations for Weakly Expansive Dynamical Systems

Yuan-Ling Ye
Department of Mathematics
South China Normal University
Guangzhou 510631, P.R. China

Abstract : For a summable variation potential function on a subshift of finite type, Pollicott (2000 Trans. Amer. Math. Soc. 352 843-853) gave an estimate of the decay of correlations. It was known that the systems he considered have the bounded distortion property (BDP), and that is a key condition on the systems. In this paper we study weakly expansive Dini dynamical systems that may not have the BDP. Under some assumptions, our theorem gives an estimate of the decay of correlations.

Application of Fractal to Genome Analysis

Zu-Guo Yu*
School of Mathematics and Computational Science,
Xiangtan University,
Hunan 411105, China.

Vo Anh
Program in Statistics and Operations Research,
Queensland University of Technology,
GPO Box 2434, Brisbane, Queensland 4001, Australia.

Abstract : The notion of measure representation of complete genomes is introduced. The iterated function systems (IFS) model has been studied in fractal geometry for a long time. The IFS model and recurrent IFS (RIFS) model are proposed to simulate the measure representation of complete genomes and protein sequences. We found that the RIFS model is better than the IFS model for the simulation of the measure representation of complete genomes.The estimated parameters in RIFS model can be used to discuss some biological problems related to classification and evolutionary tree of living organisms.

A new chaos game representation (CGR) of protein sequences based on the detailed HP model is proposed. Multifractal and correlation analyses of the measures based on the CGR of protein sequences from complete genomes are performed. The Dq spectra of all organisms studied are multifractal-like and sufficiently smooth for the Cq curves to be meaningful. The Cq curves of bacteria resemble a classical phase transition at a critical point. The correlation distance of the difference between the measure based on the CGR of protein sequences and its fractal background is also proposed to construct a more precise phylogenetic tree of bacteria.