A class of minimal subshifts which display Schweizer-Smital chaos and have zero topotogical entropy is constructed, and it is proved that for a compact system, the positive topological entropy is not generally equival...A class of minimal subshifts which display Schweizer-Smital chaos and have zero topotogical entropy is constructed, and it is proved that for a compact system, the positive topological entropy is not generally equivalent to Schweizer-Smital chaos occurring on the measure centre.展开更多
Consider the subshifts induced by constant-length primitive substitutions on two symbols. By investigating the equivalent version for the existence of Li-Yorke scrambled sets and by proving the non-existence of Schwei...Consider the subshifts induced by constant-length primitive substitutions on two symbols. By investigating the equivalent version for the existence of Li-Yorke scrambled sets and by proving the non-existence of Schweizer-Smítal scrambled sets, we completely reveal for this class of subshifts the chaotic behaviors possibly occurring in the sense of Li-Yorke and Schweizer-Smítal.展开更多
1 Definitions and resultsWITH the development of the theory of dynamical systems and fractal geometry, one provedthat symbolic dynamics is a powerful tool to study chaos and fractals. It is useful to study thefractal ...1 Definitions and resultsWITH the development of the theory of dynamical systems and fractal geometry, one provedthat symbolic dynamics is a powerful tool to study chaos and fractals. It is useful to study thefractal characteristics of symbolic dynamics. In the present note we shall point out the relationbetween the dimension and measure theoretic entropy of any subshift in symbolic space. Weprove here that the Bowen’s formula for Hausdorff dimension holds without Markov struc-ture. A variational principle for dimension is obtained.展开更多
This paper focus on the chaotic properties of minimal subshift of shift operators. It is proved that the minimal subshift of shift operators is uniformly distributional chaotic, distributional chaotic in a sequence, d...This paper focus on the chaotic properties of minimal subshift of shift operators. It is proved that the minimal subshift of shift operators is uniformly distributional chaotic, distributional chaotic in a sequence, distributional chaotic of type k ( k∈{ 1,2,2 1 2 ,3 } ), and ( 0,1 ) -distribution.展开更多
In this paper,we have formed a class of the minimal chaotic subshifts having zero topological entropy and proved that the positive topological entropy is not equivalent to chaos occurring on the measure centre.
Recent progress in symbolic dynamics of cellular automata (CA) shows that many CA exhibit rich and complicated Bernoulli-shift properties, such as positive topological entropy, topological transitivity and even mixing...Recent progress in symbolic dynamics of cellular automata (CA) shows that many CA exhibit rich and complicated Bernoulli-shift properties, such as positive topological entropy, topological transitivity and even mixing. Noticeably, some CA are only transitive, but not mixing on their subsystems. Yet, for one-dimensional CA, this paper proves that not only the shift transitivity guarantees the CA transitivity but also the CA with transitive non-trivial Bernoulli subshift of finite type have dense periodic points. It is concluded that, for one-dimensional CA, the transitivity implies chaos in the sense of Devaney on the non-trivial Bernoulli subshift of finite types.展开更多
Seeking conjugate invariant is one of significant and general topics on dynamical systems. Few invariants have been found so far. Topological entropy is one of such invariants. Studies about the topological entropy ar...Seeking conjugate invariant is one of significant and general topics on dynamical systems. Few invariants have been found so far. Topological entropy is one of such invariants. Studies about the topological entropy are concentrated on areas of homeomorphisms and one-dimensional continuous self-maps. In this note we consider Anosov maps, a sort of continuous maps on general compact metric spaces. By using a finite subshift and the largest positive eigenvalue we will calculate the topological entropy for Anosov maps.展开更多
This paper deals with chaos for subshifts of finite type. We show that for any subshift of finite type determined by an irreducible and aperiodic matrix, there is a finitely chaotic set with full Hausdorff dimension. ...This paper deals with chaos for subshifts of finite type. We show that for any subshift of finite type determined by an irreducible and aperiodic matrix, there is a finitely chaotic set with full Hausdorff dimension. Moreover, for any subshift of finite type determined by a matrix, we point out that the cases including positive topological entropy, distributional chaos, chaos and Devaney chaos are mutually equivalent.展开更多
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 measur...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.展开更多
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 e...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.展开更多
目的针对Robert S Mackay等人提出的若非周期回复点生成的子转移中存在周期点,则此子转移是否包含一个不可数混乱集的问题,通过动力系统中为描述系统复杂性提供反例的工具-Sturm ian系统构造反例.方法利用符号动力系统的相关概念和Sturm...目的针对Robert S Mackay等人提出的若非周期回复点生成的子转移中存在周期点,则此子转移是否包含一个不可数混乱集的问题,通过动力系统中为描述系统复杂性提供反例的工具-Sturm ian系统构造反例.方法利用符号动力系统的相关概念和Sturm ian系统的极小非Li-Yorke混沌属性,构造了一类由非周期回复点生成含有周期点的子转移系统,并研究了其性质.结果a是符号空间中的非周期回复点,且orb(a)包含一个子转移σ的周期点,则orb(a)不包含一个不可数混乱(scrambled)集.结论若非周期回复点生成的子转移中存在周期点,则此子转移不一定包含一个不可数混乱集.展开更多
文摘A class of minimal subshifts which display Schweizer-Smital chaos and have zero topotogical entropy is constructed, and it is proved that for a compact system, the positive topological entropy is not generally equivalent to Schweizer-Smital chaos occurring on the measure centre.
基金the National Natural Science Foundation of China (Grant No. 10771084)the Education Department Foundation of Jilin Province (Grant No. 200568)the Foundations of Dalian Nationalities University and Jilin Normal University
文摘Consider the subshifts induced by constant-length primitive substitutions on two symbols. By investigating the equivalent version for the existence of Li-Yorke scrambled sets and by proving the non-existence of Schweizer-Smítal scrambled sets, we completely reveal for this class of subshifts the chaotic behaviors possibly occurring in the sense of Li-Yorke and Schweizer-Smítal.
文摘1 Definitions and resultsWITH the development of the theory of dynamical systems and fractal geometry, one provedthat symbolic dynamics is a powerful tool to study chaos and fractals. It is useful to study thefractal characteristics of symbolic dynamics. In the present note we shall point out the relationbetween the dimension and measure theoretic entropy of any subshift in symbolic space. Weprove here that the Bowen’s formula for Hausdorff dimension holds without Markov struc-ture. A variational principle for dimension is obtained.
文摘This paper focus on the chaotic properties of minimal subshift of shift operators. It is proved that the minimal subshift of shift operators is uniformly distributional chaotic, distributional chaotic in a sequence, distributional chaotic of type k ( k∈{ 1,2,2 1 2 ,3 } ), and ( 0,1 ) -distribution.
基金supported by National Natural Science Foundation of China (Grant Nos. 11401236 and 11471132)Self-Determined Research Funds of Central China Normal University (Grant No. CCNU17QN0009)
文摘The dynamical structure of the rational map ax+1/x on the projective line P^1(Q_2) over the field Q_2 of 2-adic numbers, is fully described.
基金Project partially supported by the National Educational Foundation of Chinaby the National Basic Research Project "Nonlinear Science
文摘In this paper,we have formed a class of the minimal chaotic subshifts having zero topological entropy and proved that the positive topological entropy is not equivalent to chaos occurring on the measure centre.
文摘Recent progress in symbolic dynamics of cellular automata (CA) shows that many CA exhibit rich and complicated Bernoulli-shift properties, such as positive topological entropy, topological transitivity and even mixing. Noticeably, some CA are only transitive, but not mixing on their subsystems. Yet, for one-dimensional CA, this paper proves that not only the shift transitivity guarantees the CA transitivity but also the CA with transitive non-trivial Bernoulli subshift of finite type have dense periodic points. It is concluded that, for one-dimensional CA, the transitivity implies chaos in the sense of Devaney on the non-trivial Bernoulli subshift of finite types.
文摘Seeking conjugate invariant is one of significant and general topics on dynamical systems. Few invariants have been found so far. Topological entropy is one of such invariants. Studies about the topological entropy are concentrated on areas of homeomorphisms and one-dimensional continuous self-maps. In this note we consider Anosov maps, a sort of continuous maps on general compact metric spaces. By using a finite subshift and the largest positive eigenvalue we will calculate the topological entropy for Anosov maps.
基金National Natural Science Funds of China (10171034)
文摘This paper deals with chaos for subshifts of finite type. We show that for any subshift of finite type determined by an irreducible and aperiodic matrix, there is a finitely chaotic set with full Hausdorff dimension. Moreover, for any subshift of finite type determined by a matrix, we point out that the cases including positive topological entropy, distributional chaos, chaos and Devaney chaos are mutually equivalent.
基金Supported by National Natural Science Foundation of China （Grant No. 60763009）
文摘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.
基金Supported by National Science Foundation of China(Grant No.11371346)
文摘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.
文摘目的针对Robert S Mackay等人提出的若非周期回复点生成的子转移中存在周期点,则此子转移是否包含一个不可数混乱集的问题,通过动力系统中为描述系统复杂性提供反例的工具-Sturm ian系统构造反例.方法利用符号动力系统的相关概念和Sturm ian系统的极小非Li-Yorke混沌属性,构造了一类由非周期回复点生成含有周期点的子转移系统,并研究了其性质.结果a是符号空间中的非周期回复点,且orb(a)包含一个子转移σ的周期点,则orb(a)不包含一个不可数混乱(scrambled)集.结论若非周期回复点生成的子转移中存在周期点,则此子转移不一定包含一个不可数混乱集.
