arXiv subjects

Helge Glockner

Publications and source records attributed to Helge Glockner.

At least 37 records · Page 2Linked to original sources

arXiv2017

Completeness of locally $k_ω$-groups and related infinite-dimensional Lie groups

Recall that a topological space is said to be a $k_ω$-space if it is the direct limit of an ascending sequence of compact Hausdorff topological spaces. If each point in a Hausdorff space $X$ has an open neighbourhood which is a $k_ω$-space, then $X$ is called locally $k_ω$. We show that a topological group is complete whenever the underlying topological space is locally $k_ω$. As a consequence, every infinite-dimensional Lie group modelled on a Silva space is complete.

math.GR↗

arXiv2017

Endomorphisms of Lie groups over local fields

Lie groups over local fields furnish prime examples of totally disconnected, locally compact groups. We discuss the scale, tidy subgroups and further subgroups (like contraction subgroups) for analytic endomorphisms of such groups. The text is both a research article and a worked out set of lecture notes for a mini-course held June 27-July 1, 2016 at the MATRIX research center in Creswick (Australia) as part of the "Winter of Disconnectedness". The text can be read in parallel to the earlier lecture notes arXiv:0804.2234 which are devoted to automorphisms, with sketches of proof. Complementary aspects are emphasized.

math.GR↗

arXiv2016

Lectures on Lie groups over local fields

These are the lecture notes of a 2-hour mini-course on Lie groups over local fields presented at the "Workshop on Totally Disconnected Groups, Graphs and Geometry" at the Heinrich-Fabri-Institut Blaubeuren in May 2007. The goal of the notes is to provide an introduction to p-adic Lie groups and Lie groups over fields of formal Laurent series, with an emphasis on relations to the structure theory of totally disconnected, locally compact groups. In particular, they contain a discussion of the scale, tidy subgroups and contraction groups for automorphisms of Lie groups over local fields. Special attention is paid to the case of Lie groups over local fields of positive characteristic.

math.GR↗

arXiv2016

Elementary p-adic Lie groups have finite construction rank

The class of elementary totally disconnected groups is the smallest class of totally disconnected, locally compact, second countable groups which contains all discrete countable groups, all metrizable pro-finite groups, and is closed under extensions and countable ascending unions. To each elementary group G, a (possibly infinite) ordinal number rk(G) can be associated, its construction rank. By a structure theorem of Phillip Wesolek, elementary p-padic Lie groups are among the basic building blocks for general sigma-compact p-adic Lie groups. We characterize elementary p-adic Lie groups in terms of the subquotients needed to describe them. The characterization implies that every elementary p-adic Lie group has finite construction rank. Results concerning general p-adic Lie groups are also obtained, concerning the isomorphism types of subquotients needed to build up the latter.

math.GR↗

arXiv2016

Fundamentals of submersions and immersions between infinite-dimensional manifolds

We define submersions f between manifolds M and N modelled on locally convex spaces. If the range N is finite-dimensional or a Banach manifold, then these coincide with the naive notion of a submersion. We study pre-images of submanifolds under submersions and pre-images under mappings whose differentials have dense image. An infinite-dimensional version of the constant rank theorem is provided. We also construct manifold structures on homogeneous spaces G/H of infinite-dimensional Lie groups. Some fundamentals of immersions between infinite-dimensional manifolds are developed as well.

math.DG↗

arXiv2016

Diffeomorphism groups of compact convex sets

For a compact convex subset K with non-empty interior in a finite-dimensional vector space, let G be the group of all smooth diffeomorphisms of K which fix the boundary of K pointwise. We show that G is a C^0-regular infinite-dimensional Lie group. As a byproduct, we obtain results concerning solutions to ordinary differential equations on compact convex sets.

math.GR↗

arXiv2016

Regularity properties of infinite-dimensional Lie groups, and semiregularity

Let G be a Lie group modelled on a locally convex space, with Lie algebra g, and k be a non-negative integer or infinity. We say that G is C^k-semiregular if each C^k-curve c in g admits a left evolution Evol(c) in G. If, moreover, the map taking c to evol(c):=Evol(c)(1) is smooth, then G is called C^k-regular. For G a C^k-semiregular Lie group and m an order of differentiability, we show that evol is C^m if and only if Evol is C^m. If evol is continuous at 0, then evol is continuous. If G is a C^0-semiregular Lie group, then continuity of evol implies its smoothness (so that G will be C^0-regular), if smooth homomorphisms from G to C^0-regular Lie groups separate points on G and g is (e.g.) sequentially complete. Further criteria for regularity properties are provided, and used to prove regularity for several important classes of Lie groups. Notably, we find that the Lie group Diff(M) of smooth diffeomorphisms of a paracompact finite-dimensional smooth manifold M (which need not be sigma-compact) is C^1-regular. We also provide tools which enable to show that the Lie group of analytic diffeomorphisms of a compact real analytic manifold is C^1-regular.

math.FA↗

arXiv2016

Complexifications of infinite-dimensional manifolds and new constructions of infinite-dimensional Lie groups

Let M be a real analytic manifold modeled on a locally convex space and K be a non-empty compact subset of M. We show that if an open neighborhood of K in M admits a complexification which is a regular topological space, then the germ of the latter (as a complex manifold) is uniquely determined. If M is regular and the complexified modeling space of M is normal, then a regular complexification exists for some neighborhood of K. For each (real or complex) analytic regular manifold M modeled on a metrizable locally convex space and Banach-Lie group H, this allows the group Germ(K,H) of germs of H-valued analytic maps around K in M to be turned into an analytic Lie group which is regular in Milnor's sense. A special case is a regular real analytic Lie group structure on the group of real analytic H-valued maps on a compact real analytic manifold M (which, previously, had only been treated in the convenient setting of analysis). Combining our results concerning groups of germs with an idea by Neeb and Wagemann, one can also obtain a regular Lie group structure on the group of all real analytic H-valued mappings on the real line.

math.DG↗

arXiv2015

Measurable regularity properties of infinite-dimensional Lie groups

We consider differential equations of the form y'(t)=f(t,y(t)) on a (possibly infinite-dimensional) Lie group G, for f : [0,1] x G -> TG a time-dependent left invariant vector field with measurable (but not necessarily continuous) dependence on t. If a solution Evol(c):=y on [0,1] starting at the neutral element e of G exists for each f corresponding to an L^1-curve c : [0,1] -> g in the Lie algebra g of G, and Evol is smooth as a map from L^1([0,1],g) to C([0,1],G), then G is called L^1-regular. We show that all Banach-Lie groups are L^1-regular, as well as all direct limits of finite-dimensional Lie groups and many further classical examples of infinite-dimensional Lie groups, like diffeomorphism groups of paracompact finite-dimensional smooth manifolds. If a Lie group G is L^1-regular, then the Trotter product formula and the commutator formula hold in G, in a strong sense. The same conclusion holds if G merely satisfies certain weaker measurable regularity properties (like L^p-regularity), which are discussed as well. As a tool, we study differentiability properties of certain non-linear mappings to vector-valued Lebesgue spaces and spaces of vector-valued absolutely continuous functions.

math.FA↗

arXiv2015

Expansive automorphisms of totally disconnected, locally compact groups

We study automorphisms $α$ of a totally disconnected, locally compact group $G$ which are expansive in the sense that, for some identity neighbourhood $U$, the sets $α^n(U)$ (for integers $n$) intersect in the trivial group. Notably, we prove that the automorphism induced by $α$ on $G/N$ for an $α$-stable closed normal subgroup $N$ of $G$ is always expansive. Further results involve the associated contraction groups $U_α$ consisting of all $x$ in $G$ such that $α^n(x) \to e$ as $n$ tends to infinity. If $α$ is expansive, then $W := U_αU_{α^{-1}}$ is an open identity neighbourhood in $G$. We give examples where $W$ fails to be a subgroup. However, $W$ is a nilpotent open subgroup whenever $G$ is a closed subgroup of a general linear group over the $p$-adic numbers. Further results are devoted to the divisible and torsion parts of $U_α$, and to the so-called "nub" $U_0$ of an expansive automorphism $α$ (the intersection of the closures of $U_α$ and $U_{α^{-1}}$).

math.DS↗

arXiv2015

Invariant manifolds for finite-dimensional non-archimedean dynamical systems

Let M be an analytic manifold modelled on an ultrametric Banach space over a complete ultrametric field. Let f be an analytic diffeomorphism from M onto itself and p be a fixed point of f. We discuss invariant manifolds around p, like stable manifolds, centre-stable manifolds and centre manifolds, with an emphasis on results specific to the case that M has finite dimension. The results have applications in the theory of Lie groups over totally disconnected local fields.

math.DS↗

arXiv2014

The kernel of the adjoint representation of a p-adic Lie group need not have an abelian open normal subgroup

Let G be a p-adic Lie group and Ad be the adjoint representation of G on its Lie algebra. It was claimed in the literature that the kernel K of Ad always has an abelian open normal subgroup. We show by means of a counterexample that this assertion is false; it can even happen that K=G but G has no abelian subnormal subgroup except for the trivial group. The arguments are based on auxiliary results on subgroups of free products with central amalgamation.

math.GR↗

arXiv2014

Bounded solutions of finite lifetime to differential equations in Banach spaces

Consider a smooth vector field $f\colon \mathbb{R}^n\to\mathbb{R}^n$ and a maximal solution $γ\colon \,]a,b[\,\to \mathbb{R}^n$ to the ordinary differential equation $x'=f(x)$. It is a well-known fact that, if $γ$ is bounded, then $γ$ is a global solution, i.e., $\,]a,b[\,=\mathbb{R}$. We show by example that this conclusion becomes invalid if $\mathbb{R}^n$ is replaced with an infinite-dimensional Banach space.

math.FA↗

arXiv2013

Differentiable mappings between spaces of sections

In this work, various versions of the so-called Omega-Lemma are provided, which ensure differentiability properties of pushforwrds between spaces of C^r-sections (or compactly supported C^r-sections) in vector bundles over finite-dimensional base manifolds whose fibres are (possibly infinite-dimensional) locally convex spaces. Applications are given, including the proof of continuity for some natural module multiplications on spaces of sections and the construction of certain infinite-dimensional Lie groups of Lie group-valued maps.

math.FA↗

arXiv2012

Grobman-Hartman theorems for diffeomorphisms of Banach spaces over valued fields

Consider a local diffeomorphism f of an ultrametric Banach space over an ultrametric field, around a hyperbolic fixed point x. We show that, locally, the system is topologically conjugate to the linearized system. An analogous result is obtained for local diffeomorphisms of real p-Banach spaces (like l^p) for 0 < p =< 1. More generally, we obtain a local linearization if f is merely a local homeomorphism which is strictly differentialble at a hyperbolic fixed point x. Also a new global version of the Grobman-Hartman theorem is provided. It applies to Lipschitz perturbations of hyperbolic automorphisms of Banach spaces over valued fields. The local conjugacies H constructed are not only homeomorphisms, but H and H^{-1} are Hoelder. We also study the dependence of H and H^{-1} on f (keeping x and f'(x) fixed).

math.DS↗

arXiv2012

Weighted inversion of general Dirichlet series

Inversion theorems of Wiener type are essential tools in analysis and number theory. We derive a weighted version of an inversion theorem of Wiener type for general Dirichlet series from that of Edwards from 1957, and we outline an alternative proof based on the duality theory of convex cones and extension techniques for characters of semigroups. Variants and arithmetical applications are described, including the case of multidimensional weighted generalized Dirichlet series.

math.FA↗

arXiv2012

Exponential laws for ultrametric partially differentiable functions and applications

We establish exponential laws for certain spaces of differentiable functions over a valued field K. For example, we show that the topological vector spaces C^{r,s}(U x V,E) and C^r(U,C^s(V,E)) are isomorphic if U and V are open subsets of K^n and K^m, respectively, E is a topological K-vector space, and r,s are degrees of differentiability. As a first application, we study the density of locally polynomial functions in spaces of partially differentiable functions over an ultrametric field (thus solving an open problem by E. Nagel), and also global approximations by polynomial functions. As a second application, we obtain a new proof for the characterization of C^r-functions on powers Z_p^n of the p-adic integers in terms of the decay of their Mahler expansions. In both applications, the exponential laws enable simple inductive proofs via a reduction to the one-dimensional, vector-valued case.

math.FA↗

arXiv2012

Continuity of LF-algebra representations associated to representations of Lie groups

Let G be a Lie group and E be a locally convex topological G-module. If E is sequentially complete, then E and its space of smooth vectors are modules for the algebra D(G) of compactly supported smooth functions on G. However, the module multiplication need not be continuous. The pathology can be ruled out if E is (or embeds into) a projective limit of Banach G-modules. Moreover, in this case the space of analytic vectors is a module for the algebra A(G) of superdecaying analytic functions introduced by Gimperlein, Kroetz and Schlichtkrull. We prove that the space of analytic vectors is a topological A(G)-module if E is a Banach space or, more generally, if every countable set of continuous seminorms on E has an upper bound. The same conclusion is obtained if G has a compact Lie algebra. The question of whether D(G) and A(G) are topological algebras is also addressed.

math.FA↗