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We study the entanglement entropy in confining theories with gravity duals using the holographic prescription of Ryu and Takayanagi. The entanglement entropy between a region and its complement is proportional to the minimal area of a bulk…
Topological entropy or spatial entropy is a way to measure the complexity of shift spaces. This study investigates the relationships between the spatial entropy and the various periodic entropies which are computed by skew-coordinated…
We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.
Magnitude and (co)weightings are quite general constructions in enriched categories, yet they have been developed almost exclusively in the context of Lawvere metric spaces. We construct a meaningful notion of magnitude for flow graphs…
A class of finite-dimensional Hopf algebras which generalise the notion of Taft algebras is studied. We give necessary and sufficient conditions for these Hopf algebras to omit a pair in involution, that is, to not have a group-like and a…
We prove exponential correlation decay in dispersing billiard flows on the 2-torus assuming finite horizon and lack of corner points. With applications aimed at describing heat conduction, the highly singular initial measures are…
Convergence spaces are a generalization of topological spaces. The category of convergence spaces is well-suited for Algebraic Topology, one of the reasons is the existence of exponential objects provided by continuous convergence. In this…
Combining two existing rigorous computational methods, for verifying hyperbolicity (due to Arai) and for computing topological entropy bounds (due to Day et al.), we prove lower bounds on topological entropy for 43 hyperbolic plateaus of…
Given a closed, orientable surface of constant negative curvature and genus $g \ge 2$, we study the topological entropy and measure-theoretic entropy (with respect to a smooth invariant measure) of generalized Bowen--Series boundary maps.…
The chaotical dynamics is studied in different Friedmann-Robertson- Walker cosmological models with scalar (inflaton) field and hydrodynamical matter. The topological entropy is calculated for some particular cases. Suggested scheme can be…
Trigonometric and trigonometric-algebraic entropies are introduced. Regularity increases the entropy and the maximal entropy is shown to result when a regular $n$-gon is inscribed in a circle. A regular $n$-gon circumscribing a circle gives…
Consider two $k$-gons $P$ and $Q$. We say that the billiard flows in $P$ and $Q$ are homotopically equivalent if the set of conjugacy classes in the fundamental group of $P$ which contain a periodic billiard orbit agrees with the analogous…
In this note we study some properties of topological entropy for non-compact non-metrizable spaces. We prove that if a uniformly continuous self-map $f$ of a uniform space has topological shadowing property then the map $f$ has positive…
We define a hierarchy of systems with topological completely positive entropy in the context of continuous countable amenable group actions on compact metric spaces. For each countable ordinal we construct a dynamical system on the…
We study Tonelli Lagrangian systems on the 2-torus in energy levels above Ma\~n\'e's strict critical value and analyize the structure of global minimizers in the spirit of Morse, Hedlund and Bangert. In the case where the topological…
We consider a natural Hamiltonian system with two degrees of freedom and Hamiltonian $H=\|p\|^2/2+V(q)$. The configuration space $M$ is a closed surface (for noncompact $M$ certain conditions at infinity are required). It is well known that…
This article deals with topological assumptions under which the minimal volume entropy of a closed manifold $M$, and more generally of a finite simplicial complex $X$, vanishes or is positive. These topological conditions are expressed in…
We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the…
The topological entropy of piecewise affine maps is studied. It is shown that singularities may contribute to the entropy only if there is angular expansion and we bound the entropy via the expansion rates of the map. As a corollary we…
We consider billiard systems within compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We derive conditions for elliptic periodicity for such billiards. We describe the topology of those…