Let X be a metric space. We say that a continuous surjection f:X→X is a topological Anosov map (abbrev. TA map) if f is expansive and has pseudo orbit tracing property with respect to some compatible me...Let X be a metric space. We say that a continuous surjection f:X→X is a topological Anosov map (abbrev. TA map) if f is expansive and has pseudo orbit tracing property with respect to some compatible metric for X . This paper studies the properties of TA maps of non compact metric spaces and gives some conditions for the map to be topologically mixing.展开更多
Consider C^(2)Anosov systems on a compact manifold driven by a quasi-periodic force.We study their dynamical complexity on various levels from the perspectives of both path-wise dynamics and stochastic processes.Assum...Consider C^(2)Anosov systems on a compact manifold driven by a quasi-periodic force.We study their dynamical complexity on various levels from the perspectives of both path-wise dynamics and stochastic processes.Assuming that these systems are non-wandering(i.e.,every point in the phase space is nonwandering),we prove a set of results:(1)the existence of abundance of random periodic points;(2)a random Livsic theorem;(3)a random Mañé-Bousch-Conze-Guivarc'h lemma;(4)the existence of strong random horseshoes.Additionally,a concrete example constructed on a 2-dimensional torus is also given to uncover some interesting phenomena of the systems.展开更多
Let X be a compact metric space and let f:X→X be an Anosov map,i.e.,an expansive selfmap with the pseudoorbit tracing property(abbr.POTP)(see Lemma 1).If Nn(f) denotes the number of fixed points of f^n which we name ...Let X be a compact metric space and let f:X→X be an Anosov map,i.e.,an expansive selfmap with the pseudoorbit tracing property(abbr.POTP)(see Lemma 1).If Nn(f) denotes the number of fixed points of f^n which we name here the n-periodic number then we prove in the case as n tends to infinity that n^M ≤N_n(f)≤H^n,where M and H are two positive integers.展开更多
In this paper we give a classification of special endomorphisms of nil-manifolds:Let f:N/Γ→N/Γbe a covering map of a nil-manifold and denote by A:N/Γ→N/Γthe nil-endomorphism which is homotopic to f.If f is a spe...In this paper we give a classification of special endomorphisms of nil-manifolds:Let f:N/Γ→N/Γbe a covering map of a nil-manifold and denote by A:N/Γ→N/Γthe nil-endomorphism which is homotopic to f.If f is a special TA-map,then A is a hyperbolic nil-endomorphism and f is topologically conjugate to A.展开更多
Sakai defined the Anosov maps on compact metric spaces. Sun proved that Anosovmaps have the orbit-topological stability, Markov partitions and ζ-funtions. Aformulation for the topological entropy of an Anosov map wes...Sakai defined the Anosov maps on compact metric spaces. Sun proved that Anosovmaps have the orbit-topological stability, Markov partitions and ζ-funtions. Aformulation for the topological entropy of an Anosov map wes given by Sun in reference. In this note we will also study the topological entropy of an Anosov map, but we willpay attention to the relation between the entropy and the number of periodic points. Thefollowing consequenee will be proved.展开更多
We prove that the restrictions of the conjugacy between two Anosov diffeomorphisms of the twotorus to the stable and unstable manifolds are quasisymmetric homeomorphisms.
In this paper, we prove that some volume-preserving almost Anosov systems are ergodic if they are essentially accessible. The key idea is that there are stable and unstable manifolds with uniform size on the orbits Of...In this paper, we prove that some volume-preserving almost Anosov systems are ergodic if they are essentially accessible. The key idea is that there are stable and unstable manifolds with uniform size on the orbits Of the hyperbolic points for these systems.展开更多
Let f : M → M be an Anosov diffeomorphism on a nilmanifold. We consider Birkhoff sums for a Holder continuous observation along periodic orbits. We show that if there are two Birkhoff sums distributed at both sides o...Let f : M → M be an Anosov diffeomorphism on a nilmanifold. We consider Birkhoff sums for a Holder continuous observation along periodic orbits. We show that if there are two Birkhoff sums distributed at both sides of zero, then the set of Birkhoff sums of all the periodic points is dense in R.展开更多
An earlier conjecture is settled with an immersion of a 2-dimensional branched manifold.Possible obstructions in linear algebra and tiling theory are studied rst.
文摘Let X be a metric space. We say that a continuous surjection f:X→X is a topological Anosov map (abbrev. TA map) if f is expansive and has pseudo orbit tracing property with respect to some compatible metric for X . This paper studies the properties of TA maps of non compact metric spaces and gives some conditions for the map to be topologically mixing.
基金supported by National Natural Science Foundation of China(Grant Nos.12090010,12090013,11971330,12090012,12031019,11725105,and 12226102).
文摘Consider C^(2)Anosov systems on a compact manifold driven by a quasi-periodic force.We study their dynamical complexity on various levels from the perspectives of both path-wise dynamics and stochastic processes.Assuming that these systems are non-wandering(i.e.,every point in the phase space is nonwandering),we prove a set of results:(1)the existence of abundance of random periodic points;(2)a random Livsic theorem;(3)a random Mañé-Bousch-Conze-Guivarc'h lemma;(4)the existence of strong random horseshoes.Additionally,a concrete example constructed on a 2-dimensional torus is also given to uncover some interesting phenomena of the systems.
文摘Let X be a compact metric space and let f:X→X be an Anosov map,i.e.,an expansive selfmap with the pseudoorbit tracing property(abbr.POTP)(see Lemma 1).If Nn(f) denotes the number of fixed points of f^n which we name here the n-periodic number then we prove in the case as n tends to infinity that n^M ≤N_n(f)≤H^n,where M and H are two positive integers.
文摘In this paper we give a classification of special endomorphisms of nil-manifolds:Let f:N/Γ→N/Γbe a covering map of a nil-manifold and denote by A:N/Γ→N/Γthe nil-endomorphism which is homotopic to f.If f is a special TA-map,then A is a hyperbolic nil-endomorphism and f is topologically conjugate to A.
基金Project supported in part by the Foundation of Zhongshan University.
文摘Sakai defined the Anosov maps on compact metric spaces. Sun proved that Anosovmaps have the orbit-topological stability, Markov partitions and ζ-funtions. Aformulation for the topological entropy of an Anosov map wes given by Sun in reference. In this note we will also study the topological entropy of an Anosov map, but we willpay attention to the relation between the entropy and the number of periodic points. Thefollowing consequenee will be proved.
文摘We prove that the restrictions of the conjugacy between two Anosov diffeomorphisms of the twotorus to the stable and unstable manifolds are quasisymmetric homeomorphisms.
基金Supported by National Natural Science Foundation of China(Grant No.11001284)Natural Science FoundationProject of CQ CSTC(Grant No.cstcjjA00003)Fundamental Research Funds for Central Universities(CDJZR10100006)
文摘In this paper, we prove that some volume-preserving almost Anosov systems are ergodic if they are essentially accessible. The key idea is that there are stable and unstable manifolds with uniform size on the orbits Of the hyperbolic points for these systems.
基金supported by National Natural Science Foundation of China(Grant Nos.11771025 and 11831001)supported by National Natural Science Foundation of China(Grant Nos.12071007 and 11831001)。
文摘Let f : M → M be an Anosov diffeomorphism on a nilmanifold. We consider Birkhoff sums for a Holder continuous observation along periodic orbits. We show that if there are two Birkhoff sums distributed at both sides of zero, then the set of Birkhoff sums of all the periodic points is dense in R.
文摘An earlier conjecture is settled with an immersion of a 2-dimensional branched manifold.Possible obstructions in linear algebra and tiling theory are studied rst.
