In this paper we introduce two metrics:the max metric d_(n,q)and the mean metric d_(n,q).We give an equivalent characterization of rigid measure preserving systems by the two metrics.It turns out that an invariant mea...In this paper we introduce two metrics:the max metric d_(n,q)and the mean metric d_(n,q).We give an equivalent characterization of rigid measure preserving systems by the two metrics.It turns out that an invariant measureμon a topological dynamical system(X,T)has bounded complexity with respect to d_(n,q)if and only ifμhas bounded complexity with respect to d_(n,q)if and only if(X,B_X,μ,T)is rigid.We also obtain computation formulas of the measure-theoretic entropy of an ergodic measure preserving system(resp.the topological entropy of a topological dynamical system)by the two metrics dn,q and dn,q.展开更多
This paper deals with representations of groups by "affine" automorphisms of compact, convex spaces, with special focus on "irreducible" representations: equivalently"minimal" actions. Wh...This paper deals with representations of groups by "affine" automorphisms of compact, convex spaces, with special focus on "irreducible" representations: equivalently"minimal" actions. When the group in question is P SL(2, R), the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations. This analysis shows that, surprisingly, all these representations are equivalent. In fact, it is found that all irreducible affine representations of this group are equivalent. The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces. If it holds for the "universal strongly proximal space"of the group(to be defined), then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.展开更多
基金Supported by NNSF of China(Grant Nos.11971455,11801538,11801193,11871188,11731003 and 12090012)supported by STU Scientific Research Foundation for Talents(Grant No.NTF19047)。
文摘In this paper we introduce two metrics:the max metric d_(n,q)and the mean metric d_(n,q).We give an equivalent characterization of rigid measure preserving systems by the two metrics.It turns out that an invariant measureμon a topological dynamical system(X,T)has bounded complexity with respect to d_(n,q)if and only ifμhas bounded complexity with respect to d_(n,q)if and only if(X,B_X,μ,T)is rigid.We also obtain computation formulas of the measure-theoretic entropy of an ergodic measure preserving system(resp.the topological entropy of a topological dynamical system)by the two metrics dn,q and dn,q.
基金supported by National Science CenterPoland(Grant No.2018/30/M/ST1/00061)+1 种基金the Wroc law University of Science and Technology(Grant No.049U/0052/19)supported by National Natural Science Foundation of China(Grants Nos.11671094,11722103 and 11731003)。
文摘In this survey we will present the symbolic extension theory in topological dynamics,which was built over the past twenty years.
文摘This paper deals with representations of groups by "affine" automorphisms of compact, convex spaces, with special focus on "irreducible" representations: equivalently"minimal" actions. When the group in question is P SL(2, R), the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations. This analysis shows that, surprisingly, all these representations are equivalent. In fact, it is found that all irreducible affine representations of this group are equivalent. The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces. If it holds for the "universal strongly proximal space"of the group(to be defined), then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.
