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
For the shift a of the symbolic space ∑N there exists a subset (called a chaotic set for σ) C of ∑N whose Hausdorff dimension is 1 everywhere (i.e. the Hausdorff dimension of the intersection of C and every non-emp...For the shift a of the symbolic space ∑N there exists a subset (called a chaotic set for σ) C of ∑N whose Hausdorff dimension is 1 everywhere (i.e. the Hausdorff dimension of the intersection of C and every non-empty open set of the symbolic space ∑N is 1), satisfying the condition for any non-empty subset A of the set C, and for any continuous map F: A→∑N there exists a strictly increasing sequence {rn} of positive integers such that the sequence {σ (x)} converges to F(x) for any x∈A. On the other hand, it is shown that in ∑N every chaotic set for σ has 1-dimensional Hausdorff measure 0.展开更多
Let I be the interval [0,1].We denote by (I) the space of all continuous mapsfrom I into itself with the C^0 topology,i.e.the topology induced by the metric ρ(f,g) =sup{|f(x)-g(x)||x∈I}. Let f∈(I).A subset C of I i...Let I be the interval [0,1].We denote by (I) the space of all continuous mapsfrom I into itself with the C^0 topology,i.e.the topology induced by the metric ρ(f,g) =sup{|f(x)-g(x)||x∈I}. Let f∈(I).A subset C of I is said to be Li-Yorke chaotic in respect to f if forany two points x,y∈C with x≠y,展开更多
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.
A novel La Shalle＇s invariant set theory （LSIST） based adaptive asymptotic synchronization （LSISAAS） method is proposed to asymptotically synchronize Duffing system with unknown parameters which also are consider...A novel La Shalle＇s invariant set theory （LSIST） based adaptive asymptotic synchronization （LSISAAS） method is proposed to asymptotically synchronize Duffing system with unknown parameters which also are considered as system states. The LSISASS strategy depends on the only information, i.e. one state of the master system. According to the LSIST, the LSISASS method can asymptotically synchronize fully the states of the master system and the unknown system parameters as well. Simulation results also validate that the LSISAAS approach can obtain asymptotic synchronization.展开更多
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Project supported by the National Natural Science Foundation of China.
文摘For the shift a of the symbolic space ∑N there exists a subset (called a chaotic set for σ) C of ∑N whose Hausdorff dimension is 1 everywhere (i.e. the Hausdorff dimension of the intersection of C and every non-empty open set of the symbolic space ∑N is 1), satisfying the condition for any non-empty subset A of the set C, and for any continuous map F: A→∑N there exists a strictly increasing sequence {rn} of positive integers such that the sequence {σ (x)} converges to F(x) for any x∈A. On the other hand, it is shown that in ∑N every chaotic set for σ has 1-dimensional Hausdorff measure 0.
基金Project supported by the Youth Science Foundation of Anhui University.
文摘Let I be the interval [0,1].We denote by (I) the space of all continuous mapsfrom I into itself with the C^0 topology,i.e.the topology induced by the metric ρ(f,g) =sup{|f(x)-g(x)||x∈I}. Let f∈(I).A subset C of I is said to be Li-Yorke chaotic in respect to f if forany two points x,y∈C with x≠y,
基金The Foundation (A0424619) of National Science Mathematics TanYuan
文摘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.
文摘A novel La Shalle＇s invariant set theory （LSIST） based adaptive asymptotic synchronization （LSISAAS） method is proposed to asymptotically synchronize Duffing system with unknown parameters which also are considered as system states. The LSISASS strategy depends on the only information, i.e. one state of the master system. According to the LSIST, the LSISASS method can asymptotically synchronize fully the states of the master system and the unknown system parameters as well. Simulation results also validate that the LSISAAS approach can obtain asymptotic synchronization.
