Rigidity in Subclasses of Transitive and Mixing Systems

ABSTRACT

We will present some restrictions for a rigidity sequence of a nontrivial topological dynamical system. For instance, any finite linear combination of a rigidity sequence by integers has upper Banach density zero. However, there are rigidity sequences for some uniformly rigid systems whose reciprocal sums are infinite. We also show that if F is a family of subsets of natural numbers whose dual F* is filter, then a minimal F*-mixing system does not have F_{+}-rigid factor for F∈F.

We will present some restrictions for a rigidity sequence of a nontrivial topological dynamical system. For instance, any finite linear combination of a rigidity sequence by integers has upper Banach density zero. However, there are rigidity sequences for some uniformly rigid systems whose reciprocal sums are infinite. We also show that if F is a family of subsets of natural numbers whose dual F* is filter, then a minimal F*-mixing system does not have F

Cite this paper

D. Dastjerdi and M. Amiri, "Rigidity in Subclasses of Transitive and Mixing Systems,"*Advances in Pure Mathematics*, Vol. 2 No. 6, 2012, pp. 441-445. doi: 10.4236/apm.2012.26066.

D. Dastjerdi and M. Amiri, "Rigidity in Subclasses of Transitive and Mixing Systems,"

References

[1] S. Glasner and D. Maon, “Rigidity in Topological Dynamics,” Ergodic Theory and Dynamical Systems, Vol. 9, No. 2, 1989, pp. 309-320.

[2] H. Furstenberg and B. Wiess, “The Finite Multipliers of Finite Transformation,” Lecture Note in Math, Vol. 688 1978, pp. 127-132.

[3] A. Katok and A. Stepin, “Approximation in Ergodic Theory,” Russian Mathematical Surveys, Vol. 22, No. 5, 1967, pp. 77-102. doi:10.1070/RM1967v022n05ABEH001227

[4] E. Glasner and D. Maon, “On the Interplay between Measure and Topolgical Dynamics,” In: Hasselblatt and Katok, Eds., Hand Book of Dynamical System, Elsevier, Amsterdam, 2006, pp. 597-648.

[5] V. Bergelson, A. Del Junco, M. Lemanczyk and J. Rosenblatt, “Rigidity and Non-Recurrence along Sequences,” arXive: 1103.0905v1 [math. DS].

[6] T. C. Brown and A. R. Freedman, “Arithmetic Progression in Lacunary Sets,” Rocky Mountain, Journal of Mathematics, Vol. 17, No. 17, 1987, pp. 578-596.

[7] D. A. Dastjerdi and M. Hosseini, “Difference Sets of Null Density Subsets of ,” Advances in Pure Mathematics, Vol. 2, No. 3, 2012, pp. 195-199. doi:10.4236/apm.2012.23027

[8] V. Bergelson and T. Downarowicz, “Large Sets of Integers and Hierarchy of Mixing Properties of Measure-Preserving Systems,” Colloquium Mathematicum, Vol. 110, No. 1, 2008, pp. 117-150. doi:10.4064/cm110-1-4

[9] W. Huang, S. Shao and X. Ye, “Mixing via Sequence Entropy,” Contemporary Mathematics, Vol. 385, 2005, pp. 101-122. doi:10.1090/conm/385/07193

[10] W. Huang and X. Ye, “Topological Complexity, Return Times and Weak Disjointness,” Ergodic Theory and Dynamical Systems, Vol. 24, No. 3, 2004, pp. 825-846. doi:10.1017/S0143385703000543

[1] S. Glasner and D. Maon, “Rigidity in Topological Dynamics,” Ergodic Theory and Dynamical Systems, Vol. 9, No. 2, 1989, pp. 309-320.

[2] H. Furstenberg and B. Wiess, “The Finite Multipliers of Finite Transformation,” Lecture Note in Math, Vol. 688 1978, pp. 127-132.

[3] A. Katok and A. Stepin, “Approximation in Ergodic Theory,” Russian Mathematical Surveys, Vol. 22, No. 5, 1967, pp. 77-102. doi:10.1070/RM1967v022n05ABEH001227

[4] E. Glasner and D. Maon, “On the Interplay between Measure and Topolgical Dynamics,” In: Hasselblatt and Katok, Eds., Hand Book of Dynamical System, Elsevier, Amsterdam, 2006, pp. 597-648.

[5] V. Bergelson, A. Del Junco, M. Lemanczyk and J. Rosenblatt, “Rigidity and Non-Recurrence along Sequences,” arXive: 1103.0905v1 [math. DS].

[6] T. C. Brown and A. R. Freedman, “Arithmetic Progression in Lacunary Sets,” Rocky Mountain, Journal of Mathematics, Vol. 17, No. 17, 1987, pp. 578-596.

[7] D. A. Dastjerdi and M. Hosseini, “Difference Sets of Null Density Subsets of ,” Advances in Pure Mathematics, Vol. 2, No. 3, 2012, pp. 195-199. doi:10.4236/apm.2012.23027

[8] V. Bergelson and T. Downarowicz, “Large Sets of Integers and Hierarchy of Mixing Properties of Measure-Preserving Systems,” Colloquium Mathematicum, Vol. 110, No. 1, 2008, pp. 117-150. doi:10.4064/cm110-1-4

[9] W. Huang, S. Shao and X. Ye, “Mixing via Sequence Entropy,” Contemporary Mathematics, Vol. 385, 2005, pp. 101-122. doi:10.1090/conm/385/07193

[10] W. Huang and X. Ye, “Topological Complexity, Return Times and Weak Disjointness,” Ergodic Theory and Dynamical Systems, Vol. 24, No. 3, 2004, pp. 825-846. doi:10.1017/S0143385703000543