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Let $G$ be the group $mathbb{Z}^d$ or the monoid $mathbb{N}^d$ where $d$ is a positive integer. Let $X$ be a subshift over $G$, i.e., a closed and shift-invariant subset of $A^G$ where $A$ is a finite alphabet. We prove that the topological entropy of $X$ is equal to the Hausdorff dimension of $X$ and has a sharp characterization in terms of the Kolmogorov complexity of finite pieces of the orbits of $X$. In the version of this paper that has been published in Theory of Computing Systems, the proof of Lemma 4.3 contains a confusing typographical error. This version of the paper corrects that error.
Visibility algorithms are a family of geometric and ordering criteria by which a real-valued time series of N data is mapped into a graph of N nodes. This graph has been shown to often inherit in its topology non-trivial properties of the series stru
For a group $G$ definable in a first order structure $M$ we develop basic topological dynamics in the category of definable $G$-flows. In particular, we give a description of the universal definable $G$-ambit and of the semigroup operation on it. We
We develop topological dynamics for the group of automorphisms of a monster model of any given theory. In particular, we find strong relationships between objects from topological dynamics (such as the generalized Bohr compactification introduced by
We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks we focus our attention on the generic homeomorphism of the Cantor set, as construc
We initiate the study of p-adic algebraic groups G from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of G on its space of types over Q_p in the language of fields. We consid