Given a topological dynamical system (X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functiona...Given a topological dynamical system (X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of (X,T) is the system (S(X),FT), where FT is defined by FT(φ) = T o φ for any φ ∈ S(X). We show that (1) If (∑, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where ∑ is any closed subset of a Cantor set and T a selfmap of ∑ (2) If (S(∑),Fσ) is transitive then it is Devaney chaos, where (∑, σ) is a subshift of finite type; (3) If (∑, T) has shadowing property, then (Su (∑), FT) has shadowing property, where ∑ is any closed subset of a Cantor set and T a selfmap of ∑; (4) If (X,T) is sensitive, where X is an interval or any closed subset of a Cantor set and T: X → X is continuous, then (Su(X),FT) is sensitive; (5) If ∑, is a closed subset of a Cantor set with infinite points and T :∑ →∑ is positively expansive then the entropy entv(FT) of the functional envelope of (∑, T) is infinity.展开更多
It is well known that a totally disconnected compact metric space without isolated points is a Cantor set.In this note me give a simple proof of this theorem.
Using powerful concepts and tools borrowed from the seminal arsenal connecting physics fundamentals with esoteric set theoretical operations developed in recent years by Alexandria E-infinity theoretician M. S. El Nas...Using powerful concepts and tools borrowed from the seminal arsenal connecting physics fundamentals with esoteric set theoretical operations developed in recent years by Alexandria E-infinity theoretician M. S. El Naschie, this paper explores the deep implications of some of the dualities Dr El Naschie has identified and analyzed in his exposes, connecting them with our own Xonic Quantum Physics (XQP) which places dynamical action rather than spacetime and energy at the core of the System of the World.展开更多
基金Supported by National Nature Science Funds of China(Grant No.11471125)
文摘Given a topological dynamical system (X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of (X,T) is the system (S(X),FT), where FT is defined by FT(φ) = T o φ for any φ ∈ S(X). We show that (1) If (∑, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where ∑ is any closed subset of a Cantor set and T a selfmap of ∑ (2) If (S(∑),Fσ) is transitive then it is Devaney chaos, where (∑, σ) is a subshift of finite type; (3) If (∑, T) has shadowing property, then (Su (∑), FT) has shadowing property, where ∑ is any closed subset of a Cantor set and T a selfmap of ∑; (4) If (X,T) is sensitive, where X is an interval or any closed subset of a Cantor set and T: X → X is continuous, then (Su(X),FT) is sensitive; (5) If ∑, is a closed subset of a Cantor set with infinite points and T :∑ →∑ is positively expansive then the entropy entv(FT) of the functional envelope of (∑, T) is infinity.
文摘It is well known that a totally disconnected compact metric space without isolated points is a Cantor set.In this note me give a simple proof of this theorem.
文摘Using powerful concepts and tools borrowed from the seminal arsenal connecting physics fundamentals with esoteric set theoretical operations developed in recent years by Alexandria E-infinity theoretician M. S. El Naschie, this paper explores the deep implications of some of the dualities Dr El Naschie has identified and analyzed in his exposes, connecting them with our own Xonic Quantum Physics (XQP) which places dynamical action rather than spacetime and energy at the core of the System of the World.