Locally equicontinuous dynamical systems

Eli Glasner; Benjamin Weiss

Colloquium Mathematicae (2000)

  • Volume: 84/85, Issue: 2, page 345-361
  • ISSN: 0010-1354

Abstract

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A new class of dynamical systems is defined, the class of “locally equicontinuous systems” (LE). We show that the property LE is inherited by factors as well as subsystems, and is closed under the operations of pointed products and inverse limits. In other words, the locally equicontinuous functions in l ∞ ( ℤ ) form a uniformly closed translation invariant subalgebra. We show that WAP ⊂ LE ⊂ AE, where WAP is the class of weakly almost periodic systems and AE the class of almost equicontinuous systems. Both of these inclusions are proper. The main result of the paper is to produce a family of examples of LE dynamical systems which are not WAP.

How to cite

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Glasner, Eli, and Weiss, Benjamin. "Locally equicontinuous dynamical systems." Colloquium Mathematicae 84/85.2 (2000): 345-361. <http://eudml.org/doc/210818>.

@article{Glasner2000,
abstract = {A new class of dynamical systems is defined, the class of “locally equicontinuous systems” (LE). We show that the property LE is inherited by factors as well as subsystems, and is closed under the operations of pointed products and inverse limits. In other words, the locally equicontinuous functions in $l_\{∞\}(ℤ)$ form a uniformly closed translation invariant subalgebra. We show that WAP ⊂ LE ⊂ AE, where WAP is the class of weakly almost periodic systems and AE the class of almost equicontinuous systems. Both of these inclusions are proper. The main result of the paper is to produce a family of examples of LE dynamical systems which are not WAP.},
author = {Glasner, Eli, Weiss, Benjamin},
journal = {Colloquium Mathematicae},
keywords = {almost equicontinuous dynamical systems; weakly almost periodic dynamical systems; equicontinuity point},
language = {eng},
number = {2},
pages = {345-361},
title = {Locally equicontinuous dynamical systems},
url = {http://eudml.org/doc/210818},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - Glasner, Eli
AU - Weiss, Benjamin
TI - Locally equicontinuous dynamical systems
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 2
SP - 345
EP - 361
AB - A new class of dynamical systems is defined, the class of “locally equicontinuous systems” (LE). We show that the property LE is inherited by factors as well as subsystems, and is closed under the operations of pointed products and inverse limits. In other words, the locally equicontinuous functions in $l_{∞}(ℤ)$ form a uniformly closed translation invariant subalgebra. We show that WAP ⊂ LE ⊂ AE, where WAP is the class of weakly almost periodic systems and AE the class of almost equicontinuous systems. Both of these inclusions are proper. The main result of the paper is to produce a family of examples of LE dynamical systems which are not WAP.
LA - eng
KW - almost equicontinuous dynamical systems; weakly almost periodic dynamical systems; equicontinuity point
UR - http://eudml.org/doc/210818
ER -

References

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  1. [AAB1] E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic, in: Convergence in Ergodic Theory and Probability Walter de Gruyter, 1996, 25-40. Zbl0861.54034
  2. [AAB2] E. Akin, J. Auslander and K. Berg, Almost equicontinuity and the enveloping semigroup, in: Topological Dynamics and Applications (Minneapolis, MN, 1995), Contemp. Math. 215, Amer. Math. Soc., Providence, RI, 1998, 75-81. Zbl0929.54028
  3. [D] T. Downarowicz, Weakly almost periodic flows and hidden eigenvalues, in: Contemp. Math. 215, Amer. Math. Soc., 1998, 101-120. Zbl0899.28004
  4. [EN] R. Ellis and M. Nerurkar, Weakly almost periodic flows, Trans. Amer. Math. Soc. 313 (1989), 103-119. Zbl0674.54026
  5. [F1] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation, Math. Systems Theory 1 (1967), 1-55. Zbl0146.28502
  6. [F2] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton, NJ, 1981. Zbl0459.28023
  7. [GM] S. Glasner and D. Maon, Rigidity in topological dynamics, Ergodic Theory Dynam. Systems 9 (1989), 309-320. Zbl0661.58027
  8. [GW] E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity 6 (1993), 1067-1075. Zbl0790.58025
  9. [HR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer, Berlin, 1963. 

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