Topological entropy for local processes

The aim of this article is to formalize definition of chaos (in terms of topological entropy) for dynamics of processes described by nonautonomous differential equations. We state a formal definition of topological entropy in this setting and provide tools for estimation of its value (its upper or lower bounds) in terms of Poincaré sections.     

Entropy and the variational principle for actions of sofic groups

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group C ^∗-algebra.     

Topological free entropy dimension in unital C-algebras

The notion of topological free entropy dimension of n-tuple of elements in a unital C^∗ algebra was introduced by Voiculescu. In the paper, we compute topological free entropy dimension of one self-adjoint element and topological free orbit dimension of one self-adjoint element in a unital C^∗ algebra. We also calculate the values of topological free entropy dimensions of any families of self-adjoint generators of some unital C^∗ algebras, including irrational rotation C^∗ algebra, UHF algebra, and minimal tensor product of two reduced C^∗ algebras of free groups.     

Density-equicontinuity and Density-sensitivity

In this paper we introduce the notions of (Banach) density-equicontinuity and densitysensitivity. On the equicontinuity side, it is shown that a topological dynamical system is densityequicontinuous if and only if it is Banach density-equicontinuous. On the sensitivity side, we introduce the notion of density-sensitive tuple to characterize the multi-variant version of density-sensitivity. We further look into the relation of sequence entropy tuple and density-sensitive tuple both in measuretheoretical and topological setting, and it turns out that every sequence entropy tuple for some ergodic measure on an invertible dynamical system is density-sensitive for this measure; and every topological sequence entropy tuple in a dynamical system having an ergodic measure with full support is densitysensitive for this measure.     

Topological entropy for discontinuous semiflows

We study two variations of Bowen's definitions of topological entropy based on separated and spanning sets which can be applied to the study of discontinuous semiflows on compact metric spaces. We prove that these definitions reduce to Bowen's ones in the case of continuous semiflows. As a second result, we prove that our entropies give a lower bound for the τ-entropy defined by Alves, Carvalho and Vásquez (2015). Finally, we prove that for impulsive semiflows satisfying certain regularity condition, there exists a continuous semiflow defined on another compact metric space which is related to the first one by a semiconjugation, and whose topological entropy equals our extended notion of topological entropy by using separated sets for the original semiflow.     

Local variational principle concerning entropy of a sofic group action

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of countable sofic groups admitting a generating measurable partition with finite entropy; and then David Kerr and Hanfeng Li developed an operator-algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space, and established the global variational principle concerning measure-theoretic and topological entropy in this sofic context. By localizing these two kinds of entropy, in this paper we prove a local version of the global variational principle for any finite open cover of the space, and show that these local measure-theoretic and topological entropies coincide with their classical counterparts when the acting group is an infinite amenable group.     

Independence entropy of \mathbb{Z}^{d}-shift spaces

We introduce a concept of independence entropy for symbolic dynamical systems. This notion of entropy measures the extent to which one can freely insert symbols in positions without violating the constraint defined by the shift space. We show that for a certain class of one-dimensional shift spaces X, the independence entropy coincides with the limiting, as d tends to infinity, topological entropy of the dimensional shift defined by imposing the constraints of X in each of the d cardinal directions. This is of interest because for these shift spaces independence entropy is easy to compute. Thus, while in these cases, the topological entropy of the d-dimensional shift (d≥2) is difficult to compute, the limiting topological entropy is easy to compute. In some cases, we also compute the rate of convergence of the sequence of d-dimensional entropies. This work generalizes earlier work on constrained systems with unconstrained positions.     

Upper bound for the topological entropy of a meromorphic correspondence

Let f be a meromorphic correspondence on a compact Kähler manifold. We show that the topological entropy of f is bounded from above by the logarithm of its maximal dynamical degree. An analogous estimate for the entropy on subvarieties is given. We also discuss a notion of Julia and Fatou sets.     

Entropy of geometric structures

We give a notion of entropy for general gemetric structures, which generalizes well-known notions of topological entropy of vector fields and geometric entropy of foliations, and which can also be applied to singular objects, e.g. singular foliations, singular distributions, and Poisson structures. We show some basic properties for this entropy, including the additivity property, analogous to the additivity of Clausius-Boltzmann entropy in physics. In the case of Poisson structures, entropy is a new invariant of dynamical nature, which is related to the transverse structure of the characteristic foliation by symplectic leaves.     

Examples of entropy generating sequence

We introduce the notion of entropy generating sequence for infinite words and define its dimension when it exists. We construct an entropy generating sequence for each symbolic example constructed by Cassaigne such that the dimension of the sequence is the same as its topological entropy dimension. Hence the complexity can be measured via the dimension of an entropy generating sequence. Moreover, we construct a weakly mixing example with subexponential growth rate.     

Dynamical intricacy and average sample complexity of amenable group actions

In 2018, Petersen and Wilson introduced the notion of dynamical intricacy and average sample complexity for dynamical systems of ?-action, based on the past works on the notion of intricacy in the research of brain network and probability theory. If one wants to take into account underlying system geometry in applications, more general group actions may need to be taken into consideration. In this paper, we consider this notion in the case of amenable group actions. We show that many basic properties in the ?-action case remain true. We also show that their suprema over covers or partitions are equal to the amenable topological entropy and the measure entropy, using the quasitiling technique in the theory of the amenable group.

     

Relative Entropy, Asymptotic Pairs and Chaos

In this paper, we prove that positive conditional entropy impliesthe existence of asymptotic pairs and scrambled sets on fibers.Moreover, we introduce the notion of conditional topologicalentropy for a subset using Bowen's definition of separated andspanning sets, and prove that the conditional topological entropyof a system relative to a factor is the supremum of conditionaltopological entropy of its scrambled sets on fibers.     

Universality of mod-p cohomology

Let p be an odd prime number. We relate the algebraic notion of a mod-p formal group law and the topological notion of a mod-p oriented ring spectrum. It is shown that there exists a universal mod-p oriented ring spectrum MOp which splits as a wedge sum of Eilenberg-MacLane spectra. The mod-p Eilenberg-MacLane spectrum is shown to be the universal mod-p oriented ring spectrum with an additive mod-p formal group law.     

A note on distributional chaos with respect to a sequence

The aim of this note is to use methods developed by Kuratowski and Mycielski to prove that some more common notions in topological dynamics imply distributional chaos with respect to a sequence. In particular, we show that the notion of distributional chaos with respect to a sequence is only slightly stronger than the definition of chaos due to Li and Yorke. Namely, positive topological entropy and weak mixing both imply distributional chaos with respect to a sequence, which is not the case for distributional chaos as introduced by Schweizer and Smítal.     

Nonarchimedean coalgebras and coadmissible modules

We show that the basic notions of the locally analytic representation theory can be reformulated in the language of topological coalgebras (Hopf algebras) and comodules. We introduce the notion of admissible comodule and show that it corresponds to the notion of admissible representation in the case of compact p-adic group.     

Formal power series with cyclically ordered exponents

We define and study a notion of ring of formal power series with exponents in a cyclically ordered group. Such a ring is a quotient of various subrings of classical formal power series rings. It carries a two variable valuation function. In the particular case where the cyclically ordered group is actually totally ordered, our notion of formal power series is equivalent to the classical one in a language enriched with a predicate interpreted by the set of all monomials.Received: 24 February 2003     

Measure-theoretical sensitivity and equicontinuity

For an invariant measure μ in a topological dynamics, notions of μ-sensitivity, μ-complexity and μ-equicontinuity are introduced and investigated. It turns out that μ-sensitivity defined here is equivalent to pairwise sensitivity defined by Cadre and Jacob. For an ergodic μ, μ-equicontinuity, no μ-complexity pair and non-μ-sensitivity are equivalent, which implies minimality and equicontinuity when restricted to the support. Moreover, the notion of μ-sensitive set is introduced, it is shown that a transitive system with an ergodic measure of full support has zero topological entropy if there is no uncountable μ-sensitive set, and a non-minimal transitive system with dense minimal points has infinite sequence entropy for some sequence.     

Entropy for semigroup actions on general topological spaces

This paper introduces both notions of topological entropy and invariance entropy for semigroup actions on general topological spaces. We use the concept of admissible family of open coverings to extending and studying the notions of Adler–Konheim–McAndrew topological entropy, Bowen topological entropy, and invariance entropy to the general theory of topological dynamics.     

The enriched Vietoris monad on representable spaces

Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual(?) notion of completeness does not behave as the mirror image of the one of cocompleteness; and in this paper we have a closer look at complete spaces. In particular, we construct the “up-set monad” on representable spaces (in the sense of L. Nachbin for topological spaces, respectively C. Hermida for multicategories); we show that this monad is of Kock–Zöberlein type; we introduce and study a notion of weighted limit similar to the classical notion for enriched categories; and we describe the Kleisli category of our “up-set monad”. We emphasise that these generic categorical notions and results can be indeed connected to more “classical” topology: for topological spaces, the “up-set monad” becomes the lower Vietoris monad, and the statement “X   is totally cocomplete if and only if X^op$X^{\mathrm{op}}$ is totally complete” specialises to O. Wyler's characterisation of the algebras of the Vietoris monad on compact Hausdorff spaces as precisely the continuous lattices.     

Asymptotics of Weil�CPetersson Geodesics II: Bounded Geometry and Unbounded Entropy

We use ending laminations for Weil–Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil–Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equivalent condition for Weil–Petersson geodesics. As an application, we show theWeil–Petersson geodesic flow has compact invariant subsets with arbitrarily large topological entropy.