满Parry测度的有限型子集的Hausdorff维数与测度

All the full Parry measure subsets of a given subshift of finite type determined by an irreducible 0-1 matrix have the same Hausdorrf dimension and Hausdorff measure which coincide with those of the set of finite type.     

Coupled-expanding maps and one-sided symbolic dynamical systems

This paper studies relationships between coupled-expanding maps and one-sided symbolic dynamical systems. The concept of coupled-expanding map is extended to a more general one: coupled-expansion for a transitive matrix. It is found that the subshift for a transitive matrix is strictly coupled-expanding for the matrix in certain disjoint compact subsets; the topological conjugacy of a continuous map in its compact invariant set of a metric space to a subshift for a transitive matrix has a close relationship with that the map is strictly coupled-expanding for the matrix in some disjoint compact subsets. A certain relationship between strictly coupled-expanding maps for a transitive matrix in disjoint bounded and closed subsets of a complete metric space and their topological conjugacy to the subshift for the matrix is also obtained. Dynamical behaviors of subshifts for irreducible matrices are then studied and several equivalent statements to chaos are obtained; especially, chaos in the sense of Li–Yorke is equivalent to chaos in the sense of Devaney for the subshift, and is also equivalent to that the domain of the subshift is infinite. Based on these results, several new criteria of chaos for maps are finally established via strict coupled-expansions for irreducible transitive matrices in compact subsets of metric spaces and in bounded and closed subsets of complete metric spaces, respectively, where their conditions are weaker than those existing in the literature.     

符号空间有限型子转移混沌集的Hausdorff测度与Parry测度

设A是一个每列至少有二个元素为1的不可约0，1方阵，(∑4，σA)为由A所决定的符号空间有限型子转移．在∑A上定义一个与其拓扑相容的度量d使得(∑A，d)的Hausdorff维数为1．若G是H^-可测的σA的Li—Yorke混沌集，则H^1(C)＝0；若A是本原的，则存在一个σA的有限型混沌集S使得H^1(S)＝1，其中H^1为1-维的Hausdorff测度．     

Simulation of Effective Subshifts by Two-dimensional Subshifts of Finite Type

In this article we study how a subshift can simulate another one, where the notion of simulation is given by operations on subshifts inspired by the dynamical systems theory (factor, projective subaction …). There exists a correspondence between the notion of simulation and the set of forbidden patterns. The main result of this paper states that any effective subshift of dimension d—that is a subshift whose set of forbidden patterns can be generated by a Turing machine—can be obtained by applying dynamical operations on a subshift of finite type of dimension d+1—a subshift that can be defined by a finite set of forbidden patterns. This result improves Hochman’s (Invent. Math. 176(1):131–167, 2009).     

The topological Markov chain

The topological Markov chain or the subshift of finite type is a restriction of the shift on an invariant subset determined by a 0, 1-matrix, which has some important applications in the theory of dynamical systems. In this paper, the topological Markov chain has been discussed. First, we introduce a structure of the directed gragh on a 0, 1-matrix, and then by using it as a tool, we give some equivalent conditions with respect to the relationship among topological entropy, chaos, the nonwandering set, the set of periodic points and the 0, 1-matrix involved. This work is supported in part by the Foundation of Advanced Research Centre, Zhongshan University.     

Strong mixing subshift of finite type and Hausdorff measure of its chaotic set

We present a sufficient and necessary condition for the subshift of finite type to be a measure-preserving transformation or to be a strong mixing measure-preserving transformation with respect to the Hausdorff measure. It is proved that a strong mixing subshift of finite type has a chaotic set with full Hausdorff measure.     

On phase transitions for subshifts of finite type

It has recently been shown that a strongly irreducible subshift of finite type in two or more dimensions may have more than one measure of maximal entropy. In this paper we obtain some results on when (i.e. for what kinds of subshifts of finite type) this happens, and when it does not. In particular, we show that the parameter of a certain subshift of finite type introduced by Burton and Steif has a critical value, below which we have a unique measure of maximal entropy, and above which we have non-uniqueness.     

Topological Entropy of a Class of Subshifts of Finite Type

In this paper, we construct a special class of subshifts of finite type. By studying the spectral radius of the transfer matrix associated with the subshift of finite type, we obtain an estimation of its topological entropy. Interestingly, we find that the topological entropy of this class of subshifts of finite type converges monotonically to log(n + 1) (a constant only depends on the structure of the transfer matrices) as the increasing of the order of the transfer matrices.     

Sofic systems

A symbolic flow is called a sofic system if it is a homomorphic image (factor) of a subshift of finite type. We show that every sofic system can be realized as a finite-to-one factor of a subshift of finite type with the same entropy. From this it follows that sofic systems share many properties with subshifts of finite type. We concentrate especially on the properties of TPPD (transitive with periodic points dense) sofic systems.     

Cantor集合上的连续自映射

本文证明了Ｃａｎｔｏｒ集Ｃ上所有拓扑共轭于（单边的）有限型子移位的连续自映射的集合在H中稠密，此外Ｃ上每个拓扑传递（混合）映射均可被拓扑共轭于有限型子移位的拓仆传递（混合）映射一致逼近．