JungChao Ban^{*}
Department of Mathematics
The National Center for Theoretical Sciences
National Tsing Hua University
Hsinchu, Taiwan 30043, Taiwan
jcban@math.cts.nthu.edu.tw
SongSun Lin
Department of Applied Mathematics,
National ChiaoTung University,
HsinChu, Taiwan
sslin@math.nctu.edu.tw
YinHeng Lin
Department of Applied Mathematics,
National ChiaoTung University,
HsinChu, Taiwan
yhlin.am89g@nctu.edu.tw
Abstract : In this talk, we introduce the transition matrices in multidimensional lattice model, in the previous result , the recursive formulas for transition matrices \Bbb A_{n} on \Bbb Z_{2×n} are defined and of size 2^{n1}×2^{n1}.
It's of nature and interest to know whether \Bbb A_{n} is strongly transitive if \Bbb A_{2} possesses this property. Our purpose is to give some necessary and sufficient conditions for \Bbb A_{2} to guarantee the ST property for \Bbb A_{n} and the relations between the ST property and positivity of entropy are also discussed.
YiChiuan Chen
Institute of Mathematics
Academia Sinica
Taipei 11529, Taiwan
YCChen@math.sinica.edu.tw
Abstract : The embedding of Bernoulli shift into general timedependent systems was first obtained by Stoffer, Meyer and Sell by using the generalised Melnikov function. Later on, Wiggins developed the nonstationary ConleyMoser conditions and showed for nonautonomous maps the embedding of a subshift of finite type. In this talk, we extend the antiintegrability to nonautonomous symplectic twist maps to show the shift dynamics can be embedded in a natural way. We use nonautonomous standardlike and Hénonlike maps as examples to illustrate that the embedded shift can be a full shift, a subshift of finite type or of infinite type.
XinRong Dai^{*}
Department of Applied Mathematics
Zhejiang University of Technology
Hangzhou, 310014,P.R.China
Dai_xinrong@hotmail.com
Dejun Feng
Department of Mathematical Sciences
Tsinghua University
Beijing, 100084, P. R. China
dfeng@math.tsinghua.edu.cn
Yang Wang
School of Mathematics
Georgia Institute of Technology
Atlanta, Georgia 30332, USA.
wang@math.gatech.edu
Abstract : A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well known Bsplines, 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 noninteger dilations and arbitrary translations are studied in this paper. We classify completely all refinable splines with integer translations with arbitrary dilations.
DeJun Feng
Department of Mathematical Sciences
Tsinghua University
Beijing, 100084, P.R. China
dfeng@math.tsinghua.edu.cn
Abstract : Let m_{l} be the Bernoulli convolution associated with l Î (1,2). It is a longtime unsolved problem to classify all the parameters l for which m_{l} 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 nonPisot numbers such that the corresponding Bernoullii convolutions have some ßingular" behavior (having no L^{2} densities). In the another direction, the author proved a modified complete multifractal formalism for the Bernoulli convolutions associated with all Pisot numbers.
Xinchu Fu
Department of Mathematics
Shanghai University
Shanghai, 200436, P. R. China
xcfu@staff.shu.edu.cn
Abstract : We discuss dynamical behaviour of a class of discontinuous maps including piecewise linear maps on the 2torus 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 roundoff 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 longstanding 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.
Shangjiang Guo
College of Mathematics and Econometrics
Hunan University
Changsha, Hunan 410082,P. R. China
shangjguo@etang.com
Abstract : The focus of this talk is to study issues related to stability and bifurcation of a ring of neurons with selffeedback and delays, which has an oncenter offsurround 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 infinitedimensional nature of the problem caused by the synaptic delay and the possible spatial structure of the system (equivariant with respect to a D_{n}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 oncenter offsurround 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 D_{n}; (iii) calculation of the socalled 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 KreinRutman 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 spatiotemporal patterns (i.e., mirrorreflecting 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.
XingGang He
Department of Mathematics
Central China Normal University
Wuhan, 430079, P. R. China
xingganghe@sina.com
Abstract : The aim of this note is to study the construction of the boundary of a selfsimilar tile, which generates by an iterated function system {f_{i}(x) = 1/N(x+d_{i})}_{i = 1}^{N}. We will show that the boundary has complicated structure (no simple points), however, it is a regular fractal set.
Jiaxin Hu
Department of Mathematical Science
Tsinghua University
Beijing 100084 China
JXHu@math.tsinghua.edu.cn
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 twosided 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.
TuSen Huang
Institute of Mathematics
Zhejiang University of Sciences
Hangzhou City
Zhejiang Province, 310018, P.R.China
huangtusen@hotmail.com
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
KingShun Leung^{*}
Department of Mathematics
The Hong Kong Institute of Education
Tai Po, Hong Kong
ksleung@ied.edu.hk
KaSing Lau
Department of Mathematics
The Chinese University of Hong Kong
Shatin, Hong Kong
kslau@math.cuhk.edu.hk
Abstract : We study the disklikeness of the planar selfaffine tile T generated by an integral expanding matrix A and a consecutive collinear digit set D = {0, v, 2v, ¼, (q1)v } Ì \BbbZ^{2}. Let f(x) = x^{2}+ p x+ q be the characteristic polynomial of A, we show that the tile T is disklike if and only if 2p £ 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.
YinHeng Lin
Department of Applied Mathematics
National Chiao Tung University
Hsinchu 300, Taiwan
yhlin.am91g@nctu.edu.tw
YinWei Lin^{*} and TzonTzer Lu
Department of Applied Mathematics
National Sun Yatsen University
Kaohsiung, Taiwan, 80424 R.O.C.
timothy@math.nsysu.edu.tw
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

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



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


Chunqing Lu
Department of Mathematics and Statistics
Southern Illinois University at Edwardsville
Edwardsville, IL 62026, U.S.A.
clu@siue.edu
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.
JieHua Mai
Institute of Mathematics
Shantou University
Shantou,
Guangdong, 515063, P.R.China
jhmai@stu.edu.cn
Abstract : In this paper we study the asymptotically almost periodic points and the uniformly segmentrecurrent points of a continuous map f from a metric space (X,d) to itself. We prove that f is sensitive if there exist nonempty subsets V and W of with d(V,W) > 0 such that the stable manifolds M_{s}(V,f) º {x Î X:w(x,f)ÇV ¹ Æ} and M_{s}(W,f) º {y Î X:w(y,f)ÇW ¹ Æ} are dense in X, and one of the following two conditions holds: (1) W is finvariant, and the set of asymptotically almost periodic points of f is dense in X; (2) Both W and V are finvariant, and the set of uniformly segmentrecurrent points of f is dense in X. In particular, we show that if the set of wtransitive points and the set of asymptotically almost periodic points of f are dense in X, but fw(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 segmentrecurrent point, stable manifold.
^{f} The work was supported by the Special Foundation of National Prior Basis Research of China (Grant No. G1999075108).
KaSing Lau
Department of Mathematics
The Chinese University of Hong Kong
Shatin, Hong Kong
kslau@math.cuhk.edu.hk
SzeMan Ngai^{*}
Department of Mathematical Sciences
Georgia Southern University
Statesboro, GA 30460, U.S.A.
ngai@gsu.mat.georgiasouthern.edu
Abstract : For iterated function systems (IFS) of contractive similitudes on R^{d}, 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.
Christoph Bandt and Nguyen Viet Hung
Institute for Mathematics and Informatics
Arndt University, 17487 Greifswald, Germany
bandt@unigreifswald.de
Hui Rao^{*}
Department of Mathematics
Tsinghua University
P.O. Box 100084, Beijing, China
HRao@math.tsinghua.edu.cn
Abstract : Let f_{1},...,f_{m} be contracting similarity maps on \Bbb R^{n}. Let A be the unique compact set such that

 (1) 
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 socalled central open set. We show that the f_{i} satisfy OSC if and only if the central open set is nonempty.
The other one is related to fixed points of neighbor maps and the ``forbidden points'', the points of \Bbb R^{n} which do not belong to any feasible open set. For maps f_{i} 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.
Yong Lin
Department of Mathematics
School of
Information
Renmin University of China
Beijing, 100872, China
linyong01@ruc.edu.cn
HuoJun Ruan^{*}
Department of Mathematics
Zhejiang University
Hangzhou, 310027, China
ruanhj@zju.edu.cn
Abstract : A selfsimilar measure on \mathbbR^{n} is defined to be a probability measure satisfying

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

When M > 0, m is a nonlinear selfsimilar measure, we get some results of L^{p} boundedness for maximal operators of m from the pointwise asymptotic estimate of the Fourier transform of m made by Strichartz.
Qiyan Shi^{*}
Department of Mathematical Sciences
Tsinghua University
Beijing 100084, China
qshi@mail.tsinghua.edu.cn
ZhiYing Wen
Department of Mathematical Sciences
Tsinghua University
Beijing 100084, China wenzy@mail.tsinghua.edu.cn
Abstract : Let W = {1, 1}^{N} and { w_{j} } 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 socalled Cantor group. Let a, b be two real number with a + b < 1.
Riesz type products on W

Jianzhong Su
Department of Mathematics
The University of Texas at Arlington
Arlington, Texas 76019, USA
su@uta.edu
Abstract : Elliptic bursting arises from fastlow 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 fastlow 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 S^{1} to S^{1} 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 Shishkovaeishtadt theory on delayed bifurcation or delay of stability loss. These dynamics phenomena come from HodgkinHuxley equation in neuronal electrophysiological dynamics.
Weiyi Su
Department of Mathematics
Nanjing University
Nanjing , 210093, P.R.China
suqiu@nju.edu.cn
Abstract : We introduce certain spaces on local fields, such as Holder spaces C^{s} (K),s Î R, Sobolev spaces W^{s}, s Î R, Triebel Btype spaces


SzeMan Ngai
Department of Mathematical Sciences
Georgia Southern University
Statesboro, GA 304608093, USA.
ngai@gsu.cs.gasou.edu
TaiMan Tang^{*}
Department of Mathematics
Xiangtan University
Hunan 411105, P.R.China
tmtang@xtu.edu.cn
Abstract : Let T be a planar connected selfsimilar 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 disklikedness 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 LuoRaoTan's work on a problem of ConwayGrünbaum on planar selfsimilar tiles.
Yun Tang^{*} and Fenghong Yang
Department of Mathematical Sciences
Tsinghua University
Beijing 100084, China
ytang@math.tsinghua.edu.cn
Abstract : Classification of homoclinic tangencies for nharmonic 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 multiharmonic perturbed system to show the homoclinic bifurcation and give the bifurcation diagrams.
Wanyi Wang^{*}
Department of Mathematics
Shanghai Jiao Tong University
Shanghai,200240, P.R. China
and
Department of Mathematics
Inner Mongolia Normal University
Huhhot, 010022, Inner Mongolia, P. R. China
wwy@imnu.edu.cn
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 firstorder Melnikov function with multiple parameters.
Shengyou Wen
Department of Mathematics
Hubei University
Hubei 430062,
P.R.China
sywen_65@163.com
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.
Min Wu
School of Mathematics
South China University of Technology
Guangzhou, Guangdong 510640, P. R. China
wumin_ scut@mail.edu.cn
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.
Jun Xian^{*} and Song
Li
Department of Mathematics
Zhejiang University
Hangzhou, China
mathxj@zju.edu.cn
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 nonbandlimited spaces and irregularly sampling points in practical applications. It is wellknown that in the sampling and reconstruction problem for nonbandlimited spaces, signal is often assumed to belong to a shiftinvariant spaces.
Clearly we hope signal spaces be sufficiently large to accommodate a large number of possible models. So Aldroubi and Feichtinger introduced latticeinvariant space. The latticeinvariant spaces are a sufficiently large and value family of signal spaces. We will show new reconstruction formula in the latticeinvariant 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 AP iterative algorithm. In this paper, we improve the AP iterative algorithm in common shiftinvariant 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.
YuanLing Ye
Department of Mathematics
South China Normal University
Guangzhou 510631, P.R. China
ylye@scnu.edu.cn
Abstract : For a summable variation potential function on a subshift of finite type, Pollicott (2000 Trans. Amer. Math. Soc. 352 843853) 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.
ZuGuo Yu^{*}
School of Mathematics and Computational Science,
Xiangtan University,
Hunan 411105, China.
yuzuguo@yahoo.com.cn
Vo Anh
Program in Statistics and Operations Research,
Queensland University of Technology,
GPO Box 2434, Brisbane, Queensland 4001, Australia.
v.anh@qut.edu.au
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 D_{q} spectra of all organisms studied are multifractallike and sufficiently smooth for the C_{q} curves to be meaningful. The C_{q} 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.