arXiv subjects

Friedrich Martin Schneider

Publications and source records attributed to Friedrich Martin Schneider.

35 records · Page 2Linked to original sources

arXiv2020

Concentration of measure, classification of submeasures, and dynamics of $L_{0}$

Exhibiting a new type of measure concentration, we prove uniform concentration bounds for measurable Lipschitz functions on product spaces, where Lipschitz is taken with respect to the metric induced by a weighted covering of the index set of the product. Our proof combines the Herbst argument with an analogue of Shearer's lemma for differential entropy. We give a quantitative "geometric" classification of diffuse submeasures into elliptic, parabolic, and hyperbolic. We prove that any non-elliptic submeasure (for example, any measure, or any pathological submeasure) has a property that we call covering concentration. Our results have strong consequences for the dynamics of the corresponding topological $L_{0}$-groups.

math.PR↗

arXiv2020

The Liouville property and random walks on topological groups

We study harmonic functions and Poisson boundaries for Borel probability measures on general (i.e., not necessarily locally compact) topological groups, and we prove that a second-countable topological group is amenable if and only if it admits a fully supported, regular Borel probability measure with trivial Poisson boundary. This generalizes work of Kaimanovich--Vershik and Rosenblatt, confirms a general topological version of Furstenberg's conjecture, and entails a characterization of the amenability of isometry groups in terms of the Liouville property for induced actions. Moreover, our result has non-trivial consequences concerning Liouville actions of discrete groups on countable sets

math.FA↗

arXiv2018

Equivariant dissipation in non-archimedean groups

We prove that, if a topological group $G$ has an open subgroup of infinite index, then every net of tight Borel probability measures on $G$ UEB-converging to invariance dissipates in $G$ in the sense of Gromov. In particular, this solves a 2006 problem by Pestov: for every left-invariant (or right-invariant) metric $d$ on the infinite symmetric group $\mathrm{Sym}(\mathbb{N})$, compatible with the topology of pointwise convergence, the sequence of the finite symmetric groups $\left(\mathrm{Sym}(n),d\!\!\upharpoonright_{\mathrm{Sym}(n)},μ_{\mathrm{Sym}(n)}\right)_{n \in \mathbb{N}}$ equipped with the restricted metrics and their normalized counting measures dissipates, thus fails to admit a subsequence being Cauchy with respect to Gromov's observable distance.

math.MG↗

arXiv2018

MacWilliams' extension theorem for infinite rings

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle embeds into the semisimple quotient. Providing a topological perspective on the MacWilliams property, we also show that the finitary left socle of a left Artinian ring embeds into the semisimple quotient if and only if it admits a finitarily left torsion-free character, if and only if the Pontryagin dual of the regular left module is almost monothetic. In conclusion, an Artinian ring has the MacWilliams property if and only if it is finitarily Frobenius, i.e., it is quasi-Frobenius and its finitary socle embeds into the semisimple quotient.

math.RA↗

arXiv2018

Intrinsic dimension and its application to association rules

The curse of dimensionality in the realm of association rules is twofold. Firstly, we have the well known exponential increase in computational complexity with increasing item set size. Secondly, there is a \emph{related curse} concerned with the distribution of (spare) data itself in high dimension. The former problem is often coped with by projection, i.e., feature selection, whereas the best known strategy for the latter is avoidance. This work summarizes the first attempt to provide a computationally feasible method for measuring the extent of dimension curse present in a data set with respect to a particular class machine of learning procedures. This recent development enables the application of various other methods from geometric analysis to be investigated and applied in machine learning procedures in the presence of high dimension.

cs.AI↗

arXiv2018

Equivariant concentration in topological groups

We prove that, if $G$ is a second-countable topological group with a compatible right-invariant metric $d$ and $(μ_{n})_{n \in \mathbb{N}}$ is a sequence of compactly supported Borel probability measures on $G$ converging to invariance with respect to the mass transportation distance over $d$ and such that $\left(\mathrm{spt} \, μ_{n}, d\!\!\upharpoonright_{\mathrm{spt} \, μ_{n}}, μ_{n}\!\!\upharpoonright_{\mathrm{spt} \, μ_{n}}\right)_{n \in \mathbb{N}}$ concentrates to a fully supported, compact $mm$-space $\left(X,d_{X},μ_{X}\right)$, then $X$ is homeomorphic to a $G$-invariant subspace of the Samuel compactification of $G$. In particular, this confirms a conjecture by Pestov and generalizes a well-known result by Gromov and Milman on the extreme amenability of topological groups. Furthermore, we exhibit a connection between the average orbit diameter of a metrizable flow of an arbitrary amenable topological group and the limit of Gromov's observable diameters along any net of Borel probability measures UEB-converging to invariance over the group.

math.FA↗

arXiv2017

On amenability and groups of measurable maps

We show that if $G$ is an amenable topological group, then the topological group $L^{0}(G)$ of strongly measurable maps from $([0,1],λ)$ into $G$ endowed with the topology of convergence in measure is whirly amenable, hence extremely amenable. Conversely, we prove that a topological group $G$ is amenable if $L^{0}(G)$ is.

math.FA↗

arXiv2017

About von Neumann's problem for locally compact groups

We note a generalization of Whyte's geometric solution to the von Neumann problem for locally compact groups in terms of Borel and clopen piecewise translations. This strengthens a result of Paterson on the existence of Borel paradoxical decompositions for non-amenable locally compact groups. Along the way, we study the connection between some geometric properties of coarse spaces and certain algebraic characteristics of their wobbling groups.

math.GR↗

arXiv2016

Profinite algebras and affine boundedness

We prove a characterization of profinite algebras, i.e., topological algebras that are isomorphic to a projective limit of finite discrete algebras. In general profiniteness concerns both the topological and algebraic characteristics of a topological algebra, whereas for topological groups, rings, semigroups, and distributive lattices, profiniteness turns out to be a purely topological property as it is is equivalent to the underlying topological space being a Stone space. Condensing the core idea of those classical results, we introduce the concept of affine boundedness for an arbitrary universal algebra and show that for an affinely bounded topological algebra over a compact signature profiniteness is equivalent to the underlying topological space being a Stone space. Since groups, semigroups, rings, and distributive lattices are indeed affinely bounded algebras over finite signatures, all these known cases arise as special instances of our result. Furthermore, we present some additional applications concerning topological semirings and their modules, as well as distributive associative algebras. We also deduce that any affinely bounded simple compact algebra over a compact signature is either connected or finite. Towards proving the main result, we also establish that any topological algebra is profinite if and only if its underlying space is a Stone space and its translation monoid is equicontinuous.

math.LO↗

arXiv2016

On Følner sets in topological groups

We extend Følner's amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left translation action can be approximated in a uniform manner by amenable actions on the set $G$. As applications we obtain a topological version of Whyte's geometric solution to the von Neumann problem and provide an affirmative answer to a question posed by Rosendal.

math.GR↗

arXiv2016

Topological Entropy of Formal Languages

We introduce the notion of topological entropy of a formal languages as the topological entropy of the minimal topological automaton accepting it. Using a characterization of this notion in terms of approximations of the Myhill-Nerode congruence relation, we are able to compute the topological entropies of certain example languages. Those examples suggest that the notion of a "simple" formal language coincides with the language having zero entropy.

math.DS↗

arXiv2015

A topological characterisation of endomorphism monoids of countable structures

A topological monoid is isomorphic to an endomorphism monoid of a countable structure if and only if it is separable and has a compatible complete ultrametric such that composition from the left is non-expansive. We also give a topological characterisation of those topological monoids that are isomorphic to endomorphism monoids of countable omega-categorical structures. Finally we present analogous characterisations for polymorphism clones of countable structures and for polymorphism clones of countable omega-categorical structures.

math.LO↗

arXiv2015

A uniform Birkhoff theorem

Garret Birkhoff's HSP theorem characterizes the classes of models of algebraic theories as those being closed with respect to homomorphic images, subalgebras, and products. In particular, it implies that an algebra $\mathbf{B}$ satisfies all equations that hold in an algebra $\mathbf{A}$ of the same type if and only if $\mathbf{B}$ is a homomorphic image of a subalgebra of a (possibly infinite) direct power of $\mathbf{A}$. The former statement is equivalent to the existence of a natural map sending term functions of the algebra $\mathbf{A}$ to those of $\mathbf{B}$, and it is natural to wonder about continuity properties of this mapping. We show that this map is uniformly continuous if and only if every finitely generated subalgebra of $\mathbf{B}$ is a homomorphic image of a subalgebra of a finite power of $\mathbf{A}$ -- without any additional assumptions concerning the algebras $\mathbf{A}$ and $\mathbf{B}$. Moreover, provided that $\mathbf{A}$ is almost locally finite (for instance if $\mathbf{A}$ is locally oligomorphic or locally finite), the considered map is uniformly continuous if and only if it is Cauchy-continuous. In particular, our results extend a recent theorem by Bodirsky and Pinsker beyond the countable $ω$-categorical setting.

math.LO↗

arXiv2015

Every simple compact semiring is finite

A Hausdorff topological semiring is called simple if every non-zero continuous homomorphism into another Hausdorff topological semiring is injective. Classical work by Anzai and Kaplansky implies that any simple compact ring is finite. We generalize this result by proving that every simple compact semiring is finite, i.e., every infinite compact semiring admits a proper non-trivial quotient.

math.RA↗

arXiv2015

Topological matchings and amenability

We establish a characterization of amenability for general Hausdorff topological groups in terms of matchings with respect to finite uniform coverings. Furthermore, we prove that it suffices to just consider two-element uniform coverings. We also show that extremely amenable as well as compactly approximable topological groups satisfy a perfect matching property condition -- the latter even with regard to arbitrary uniform coverings. Finally, we prove that the automorphism group of a Fraïssé limit of a metric Fraïssé class is amenable if and only if the considered metric Fraïssé class has a certain Ramsey-type matching property.

math.GR↗

arXiv2015

Asymptotic uniform complexity and amenability

We introduce a novel quantity for general dynamical systems, which we call the asymptotic uniform complexity. We prove an inequality relating the asymptotic uniform complexity of a dynamical system to its mean topological matching number. Furthermore, we show that the established inequality yields an exact equation for perfect Hausdorff dynamical systems. Utilizing these results, we conclude that a dynamical system is amenable if its asymptotic uniform complexity equals one, and that the converse is true for perfect Hausdorff dynamical systems, such as non-discrete Hausdorff topological groups acting on themselves. Moreover, we establish a characterization of amenability for topologically free dynamical systems on perfect Hausdorff uniform spaces by means of a similar invariant. Furthermore, we provide a sufficient criterion for amenability concerning groups of Lipschitz-automorphisms of precompact pseudo-metric spaces in terms of entropic dimension. Finally, we show that vanishing topological entropy implies the asymptotic uniform complexity to equal one.

math.GR↗

arXiv2015

Topological entropy of continuous actions of compactly generated groups

We introduce a notion of topological entropy for continuous actions of compactly generated topological groups on compact Hausdorff spaces. It is shown that any continuous action of a compactly generated topological group on a compact Hausdorff space with vanishing topological entropy is amenable. Given an arbitrary compactly generated locally compact Hausdorff topological group $G$, we consider the canonical action of $G$ on the closed unit ball of $L^{1}(G)' \cong L^{\infty}(G)$ endowed with the corresponding weak-$^{\ast}$ topology. We prove that this action has vanishing topological entropy if and only if $G$ is compact. Furthermore, we show that the considered action has infinite topological entropy if $G$ is almost connected and non-compact.

math.GR↗