Weak forms of topological and measure-theoretical equicontinuity: relationships with discrete spectrum and sequence entropy

By Felipe Garc'ia-Ramos. Ergodic Theory and Dynamical Systems, 2014

91 citations recorded by the source corpus.

We define weaker forms of topological and measure-theoretical equicontinuity for topological dynamical systems, and we study their relationships with sequence entropy and systems with discrete spectrum. We show that for topological systems equipped with ergodic measures having discrete spectrum is equivalent to $\unicode[STIX]{x1D707}$ -mean equicontinuity. In the purely topological category we show that minimal subshifts with zero topological sequence entropy are strictly contained in diam-mean equicontinuous systems; and that transitive almost automorphic subshifts are diam-mean equicontinuous if and only if they are regular (i.e. the maximal equicontinuous factor map is one–one on a set of full Haar measure). For both categories we find characterizations using stronger versions of the classical notion of sensitivity. As a consequence, we obtain a dichotomy between discrete spectrum and a strong form of measure-theoretical sensitivity.

Subfields: Cell Biology, Computer Networks and Communications, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/af8f14a064d130a13d6e09377ed2e2aa9514051b

How researchers see this paper

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In Cell Biology

1. Wen Huang
2. Xiangdong Ye
3. Leiye Xu
4. J. Thouvenot
5. Siming Tu
6. Jian Li
7. Guohua Zhang

In Computer Networks and Communications

1. Leiye Xu
2. Xiangdong Ye
3. J. Thouvenot
4. Siming Tu
5. Wen Huang
6. Jian Li

In Mathematical Physics

1. Jian Li
2. Xiangdong Ye
3. Wen Huang
4. Siming Tu
5. J. Thouvenot
6. Leiye Xu
7. Felipe Garc'ia-Ramos
8. Eli Glasner

Citations in the corpus

• Bounded complexity, mean equicontinuity and discrete spectrum (2018, methodology): “Motivated by this open question and inspired by the discussions in [9, 18, 2, 8, 21, 11, 12, 14], in this paper, we study topological and measure-theoretic complexity via a sequence of metrics induced by a metric d, namely the metrics dn, d̂n and d̄n.”
• Mean equicontinuity and mean sensitivity (2013, methodology): “Using the same idea, Garcı́a-Ramos in a recent preprint [10] defines the notion of μ-mean equicontinuous, and shows that (X, T, μ) is μ-mean equicontinuous if and only if μ has discrete spectrum.”