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In this paper, we study flows and semiflows defined on any given finite topological $T_0$-space $X$. We show that there exist non-trivial semiflows on $X$, unless $X$ is a minimal finite space. Specifically, non-trivial semiflows exist if…
Motivated by the classical theory of spin structures, we develop a theory for lifting free C$^*$-dynamical systems, a.k.a. noncommutative principal bundles, along central extensions. This theory extends the bundle-theoretic notion of spin…
Using McMahon pseudo-metrics, for any minimal semiflow admitting an invariant measure, we study the relationships between its equicontinuous structure relation, regionally proximal relation and Veech's relations; and characterize its…
We classify certain sofic shifts (the irreducible Point Extension Type, or PET, sofic shifts) up to flow equivalence, using invariants of the canonical Fischer cover. There are two main ingredients: (1) An extension theorem, for extending…
For subsets X,Y of a finite group G, we write Pr(X,Y) for the probability that two random elements x in X and y in Y commute. This paper addresses the relation between the structure of an approximate subgroup A of G and the probabilities…
Evolution of systems in which Hamiltonians are generators of gauge transformations is a notion that requires more structure than the canonical theory provides. We identify and study this additional structure in the framework of relational…
We study projectivity in the category of $G$-flows and affine $G$-flows for Polish groups $G$. We also introduce the notion of \emph{proximally irreducible} extensions between affine $G$-flows. Using this we provide a characterization of…
A classic family in topological dynamics is that of minimal rotations. One natural extension of this family is the class of nilsystems and their inverse limits. These systems have arisen in recent applications in ergodic theory and in…
A quantum-mechanical framework is set up to describe the full counting statistics of particles flowing between reservoirs in an open system under time-dependent driving. A symmetry relation is obtained which is the consequence of…
Let $T\times X\rightarrow X, (t,x)\mapsto tx$, be a topological semiflow on a topological space $X$ with phase semigroup $T$. We introduce and discuss in this paper various transitivity dynamics of $(T,X)$.
Consider a renormalization group flow preserving a pre-modular fusion category $\mathcal S_1$. If it flows to a rational conformal field theory, the surviving symmetry $\mathcal S_1$ flows to a pre-modular fusion category $\mathcal S_2$…
Given Hilbert space operators $T, S\in\B$, let $\triangle$ and $\delta\in B(\B)$ denote the elementary operators $\triangle_{T,S}(X)=(L_TR_S-I)(X)=TXS-X$ and $\delta_{T,S}(X)=(L_T-R_S)(X)=TX-XS$. Let $d=\triangle$ or $\delta$. Assuming $T$…
At variance from the cases of relative equilibria and relative periodic orbits of dynamical systems with symmetry, the dynamics in relative quasi-periodic tori (namely, subsets of the phase space that project to an invariant torus of the…
We establish a correspondence among simple objects of the relative commutant of a full fusion subcategory in a larger fusion category in the sense of Drinfeld, irreducible half-braidings of objects in the larger fusion category with respect…
We study the massless flows described by the staircase model introduced by Al.B. Zamolodchikov through the analytic continuation of the sinh-Gordon S-matrix, focusing on the renormalisation group flow from the tricritical to the critical…
For $G$ a Polish group, we consider $G$-flows which either contain a comeager orbit or have all orbits meager. We single out a class of flows, the maximally highly proximal (MHP) flows, for which this analysis is particularly nice. In the…
We investigate the structural organization of the point-to-point electric, diffusive or hydraulic transport in complex scale-free networks. The random choice of two nodes, a source and a drain, to which a potential difference is applied,…
A flow is homotopy continuous if it is indefinitely divisible up to S-homotopy. The full subcategory of cofibrant homotopy continuous flows has nice features. Not only it is big enough to contain all dihomotopy types, but also a morphism…
Let $K$ be an algebraically closed and complete non-archimedean and non-trivially valued field, and let $G$ be a reductive group scheme acting on a flat projective scheme $X$ defined over the base ring of $K$-integers. For every $K$-point…
In this paper it is proved that near a compact, invariant, proper subset of a continuous flow on a compact, connected metric space, at least one, out of twenty eight relevant dynamical phenomena, will necessarily occur. This result shows…