Topological Dynamics

By Ethan Akin. Encyclopedia of Complexity and Systems Science, 2009

444 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/808c7ada2722733781db376f51b4e98b0f21ac1a

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Citations in the corpus

Three distance theorems and combinatorics on words (2007, background): “Note that it is equivalent (see [30]) to the minimality of the dynamical system ( O ( u ) , T ), i.e., the orbit of every element of O ( u ) is dense, or equivalently every sequence in the orbit closure of u has the same set of factors as u .”
Injective linear cellular automata and sofic groups (2007, methodology): “A topological proof of this result immediately follows from compacity of subshifts and Hedlund’s characterization [GoH] of cellular automata as being exactly the continuous maps commuting with the shift action (see Theorem 1.”
Infinite-step nilsystems, independence and complexity (2011, background): “We refer to [1, 8, 30, 31] for the details of this construction.”
Topological Transitivity and Mixing Notions for Group Actions (2007, background): “The above notions of transitive, and strongly and weakly mixing actions are classical; see [20,19,24].”
Square-mean almost automorphic solutions for some stochastic differential equations (2010, background): “[16] W. A. Veech, Topological dynamics.”
Nil Bohr-sets and almost automorphy of higher order (2014, background): “To show the following result we need several well known facts (related to distality) from the Ellis enveloping semigroup theory, see [2, 53].”
Analogues of Auslander–Yorke theorems for multi-sensitivity (2015, background): “Recall that the concept of almost automorphy, as a generalization of almost periodicity, was introduced first by Bochner in 1955 (in the context of differential geometry [8]) and has been studied by many authors including [9, 24, 26].”
Recent development of chaos theory in topological dynamics (2015, methodology): “The idea of using families of subsets ofZ+ when dealing with dynamical properties of maps was first used by Gottschalk andHedlund [34], and then further developed by Furstenberg [26].”
Frequently recurrent operators (2020, background): “Such points have been called regularly recurrent (or regularly almost periodic) in non-linear dynamics, see [24], [11].”
Combinatorial algebra : syntax and semantics (2014, unknown): “2 (George Birkhoff recurrence theorem [238, 117]).”
The structure of tame minimal dynamical systems (2006, background): “The structure theorem for the general minimal system is proved in [7] and [20] (see also [21]) and it asserts that every minimal system admits a canonically defined proximal extension which is a weakly mixing RIC extension of a strictly PI-system.”