A diffeomorphism is non-uniformly partially hyperbolic if it preserves an ergodic measure with at least one zero Lyapunov exponent.We prove that a C^(1)-smooth Z^(d)-action has the quasishadowing property if one of th...A diffeomorphism is non-uniformly partially hyperbolic if it preserves an ergodic measure with at least one zero Lyapunov exponent.We prove that a C^(1)-smooth Z^(d)-action has the quasishadowing property if one of the generators is C^(1+α)(α>0)non-uniformly partially hyperbolic.展开更多
In this paper,forward expansiveness and entropies of"subsystems"2)of Z^(k)_(+)-actions are investigated.Letαbe a Z^(k)_(+)-action on a compact metric space.For each 1≤j≤k-1,denote G^(j)_(+)={V+:=V∩R^(k)_...In this paper,forward expansiveness and entropies of"subsystems"2)of Z^(k)_(+)-actions are investigated.Letαbe a Z^(k)_(+)-action on a compact metric space.For each 1≤j≤k-1,denote G^(j)_(+)={V+:=V∩R^(k)_(+):V is a j-dimensional subspace of R^(k)}.We consider the forward expansiveness and entropies forαalong V+∈G^(j)_(+).Adapting the technique of"coding",which was introduced by M.Boyle and D.Lind to investigate expansive subdynamics of Z^(k)-actions,to the Z^(k)_(+)cases,we show that the set E^(j)_(+)(α)of forward expansive j-dimensional V_(+)is open in G^(j)_(+).The topological entropy and measure-theoretic entropy of j-dimensional subsystems ofαare both continuous in E^(j)_(+)(α),and moreover,a variational principle relating them is obtained.For a 1-dimensional ray L∈G^(+)_(1),we relate the 1-dimensional subsystem ofαalong L to an i.i.d.random transformation.Applying the techniques of random dynamical systems we investigate the entropy theory of 1-dimensional subsystems.In particular,we propose the notion of preimage entropy(including topological and measure-theoretical versions)via the preimage structure ofαalong L.We show that the preimage entropy coincides with the classical entropy along any L∈E1+(α)for topological and measure-theoretical versions respectively.Meanwhile,a formula relating the measure-theoretical directional preimage entropy and the folding entropy of the generators is obtained.展开更多
Special generators of the unoriented cobordism ring MO* are constructed to determine the groups J<sub>n,k</sub><sup>τ</sup> of n-dimensional cobordism classes in MO<sub>n</sub> con...Special generators of the unoriented cobordism ring MO* are constructed to determine the groups J<sub>n,k</sub><sup>τ</sup> of n-dimensional cobordism classes in MO<sub>n</sub> containing a representative M<sup>n</sup> admitting a （Z<sub>2</sub>）<sup>k</sup> -action with fixed point set of constant codimension.展开更多
Special generators of the unoriented cobordism ring MO<sub>*</sub> are constructed to determine some groups J<sub>n,k</sub><sup>l<sub>1</sub>,l<sub>2</sub>,…,l<...Special generators of the unoriented cobordism ring MO<sub>*</sub> are constructed to determine some groups J<sub>n,k</sub><sup>l<sub>1</sub>,l<sub>2</sub>,…,l<sub>m</sub></sup> of cobordism classes in MO<sub>n</sub> containing a representative M<sup>n</sup> admitting a （Z<sub>2</sub>）<sup>k</sup>-action with the fixed point set of（n-l<sub>i</sub>）-dimensional submanifolds of M<sup>n</sup>.展开更多
基金supported by the Science and Technology Research Program of Chongqing Municipal Education Commission(Grant No.KJQN202300802)the second author is supported by NSFC(Grant Nos.11801261,12071285)+1 种基金the third author is supported by NSFC(Grant Nos.11871120,12071082)Natural Science Foundation of Chongqing(Grant No.cstc2021jcyj-msxmX0299)。
文摘A diffeomorphism is non-uniformly partially hyperbolic if it preserves an ergodic measure with at least one zero Lyapunov exponent.We prove that a C^(1)-smooth Z^(d)-action has the quasishadowing property if one of the generators is C^(1+α)(α>0)non-uniformly partially hyperbolic.
基金Wang and Zhu are supported by NSFC (Grant Nos.11771118,11801336,12171400)Wang is also supported by China Postdoctoral Science Foundation (No.2021M691889)。
文摘In this paper,forward expansiveness and entropies of"subsystems"2)of Z^(k)_(+)-actions are investigated.Letαbe a Z^(k)_(+)-action on a compact metric space.For each 1≤j≤k-1,denote G^(j)_(+)={V+:=V∩R^(k)_(+):V is a j-dimensional subspace of R^(k)}.We consider the forward expansiveness and entropies forαalong V+∈G^(j)_(+).Adapting the technique of"coding",which was introduced by M.Boyle and D.Lind to investigate expansive subdynamics of Z^(k)-actions,to the Z^(k)_(+)cases,we show that the set E^(j)_(+)(α)of forward expansive j-dimensional V_(+)is open in G^(j)_(+).The topological entropy and measure-theoretic entropy of j-dimensional subsystems ofαare both continuous in E^(j)_(+)(α),and moreover,a variational principle relating them is obtained.For a 1-dimensional ray L∈G^(+)_(1),we relate the 1-dimensional subsystem ofαalong L to an i.i.d.random transformation.Applying the techniques of random dynamical systems we investigate the entropy theory of 1-dimensional subsystems.In particular,we propose the notion of preimage entropy(including topological and measure-theoretical versions)via the preimage structure ofαalong L.We show that the preimage entropy coincides with the classical entropy along any L∈E1+(α)for topological and measure-theoretical versions respectively.Meanwhile,a formula relating the measure-theoretical directional preimage entropy and the folding entropy of the generators is obtained.
文摘Special generators of the unoriented cobordism ring MO* are constructed to determine the groups J<sub>n,k</sub><sup>τ</sup> of n-dimensional cobordism classes in MO<sub>n</sub> containing a representative M<sup>n</sup> admitting a （Z<sub>2</sub>）<sup>k</sup> -action with fixed point set of constant codimension.
文摘Special generators of the unoriented cobordism ring MO<sub>*</sub> are constructed to determine some groups J<sub>n,k</sub><sup>l<sub>1</sub>,l<sub>2</sub>,…,l<sub>m</sub></sup> of cobordism classes in MO<sub>n</sub> containing a representative M<sup>n</sup> admitting a （Z<sub>2</sub>）<sup>k</sup>-action with the fixed point set of（n-l<sub>i</sub>）-dimensional submanifolds of M<sup>n</sup>.
