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Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topological dynamics. We describe relations between recurrence in different sorts of systems, study ways to formulate…

We show that for a certain class of dynamics at the nodes the response of a network of any topology to arbitrary inputs is defined in a simple way by its response to a monotone input. The nodes may have either a discrete or continuous set…

We introduce the notions of topological entropy of a formal language and of a topological automaton. We show that the entropy function is surjective and bound the entropy of languages accepted by deterministic {\epsilon}-free push-down…

Thouless pumping provides one of the simplest manifestations of topology in quantum systems, and has attracted a lot of recent interest, both theoretically and experimentally. Since the seminal works by Thouless and Niu in 1983 and 1984, it…

The interplay between unitary dynamics and quantum measurements induces diverse phenomena in open quantum systems with no counterparts in closed quantum systems at equilibrium. Here, we generally classify Kraus operators and their effective…

Let $\boldsymbol{X}=\{X_k\}_{k=0}^\infty$ be a sequence of compact metric spaces $X_{k}$ and $\boldsymbol{T}=\{T_k\}_{k=0}^\infty$ a sequence of continuous mappings $T_{k}: X_{k} \to X_{k+1}$. The pair $(\boldsymbol{X},\boldsymbol{T})$ is…

In this paper we study the dynamics of a general non-autonomous dynamical system generated by a family of continuous self maps on a compact space $X$. We derive necessary and sufficient conditions for the system to exhibit complex dynamical…

A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of the Stone-Cech compactification of the natural numbers, or it…

Probabilistic automata are an extension of nondeterministic finite automata in which transitions are annotated with probabilities. Despite its simplicity, this model is very expressive and many of the associated algorithmic questions are…

We study the stabilized automorphism group of minimal and, more generally, certain transitive dynamical systems. Our approach involves developing new algebraic tools to extract information about the rational eigenvalues of these systems…

We investigate indeterminate points in discrete integrable system. They appear in singularity confinement phenomenon naturally. We develop a method to analyse indeterminate points of dynamical maps and using this method we clarify behaviour…

Let $(X,\rho,G)$ be a $G-$action topological system, where $G$ is a countable infinite discrete amenable group and $X$ a compact metric space. We prove a variational principle for topological entropy of saturated sets for systems which have…

We begin development of a method for studying dynamical systems using concepts from computational complexity theory. We associate families of decision problems, called telic problems, to dynamical systems of a certain class. These decision…

A minimal Cantor system is said to be self-induced whenever it is conjugate to one of its induced systems. Substitution subshifts and some odometers are classical examples, and we show that these are the only examples in the equicontinuous…

Knots naturally appear in continuous dynamical systems as flow periodic trajectories. However, discrete dynamical systems are also closely connected with the theory of knots and links. For example, for Pixton diffeomorphisms, the…

This paper explores the concept of topological transitivity in nonautonomous dynamical systems, which are defined as sequences of continuous maps from a compact metric space to itself. It investigates various conditions (including…

We first partly develop a mathematical notion of stable consistency intended to reflect the actual consistency property of human beings. Then we give a generalization of the first and second G\"odel incompleteness theorem to stably…

We introduce a generalized notion of finiteness that provides a structural principle for the set of effective theories that can be consistently coupled to quantum gravity. More concretely, we propose a Tameness Conjecture that states that…

In the theory of renormalization for classical dynamical systems, e.g. unimodal maps and critical circle maps, topological conjugacy classes are stable manifolds of renormalization. Physically more realistic systems on the other hand may…

The dynamics of symbolic systems, such as multidimensional subshifts of finite type or cellular automata, are known to be closely related to computability theory. In particular, the appropriate tools to describe and classify topological…