arXiv subjects

Michael Francis

Publications and source records attributed to Michael Francis.

8 recordsLinked to original sources

arXiv2026

On the spectrum of non-ergodic measures

Consider a topological dynamical system where the group is abelian and the topologies are locally compact and second-countable. Given an invariant measure for this system, we show that if its dynamical spectrum is contained in some Borel subset of the dual group then the same holds almost surely for all ergodic measures arising via the Choquet theorem. In particular, if the invariant measure has pure point dynamical spectrum, so do almost all the ergodic measures. As an application, we show that given any mean almost periodic measure, in its hull there exists a Besicovitch almost periodic measure.

math.DS↗

arXiv2023

On singular foliations tangent to a given hypersurface

We consider a class of singular foliations in the sense of Androulidakis and Skandalis that we call transverse order $k$ foliations. These have a finite number of leaves: one hypersurface (the singular leaf) together with the components of its complement (open leaves). The positive integer parameter $k$ encodes the "order of tangency" of the leafwise vector fields to $L$. We show that a loop in the singular leaf induces a well-defined holonomy transformation at the level of $(k-1)$-jets. The resulting holonomy invariant can be used to give a complete classification of these foliations and obtain concrete descriptions of their associated groupoids and algebras.

math.OA↗

arXiv2023

The Newlander-Nirenberg theorem for complex $b$-manifolds

Melrose defined the b-tangent bundle of a smooth manifold M with boundary as the vector bundle whose sections are vector fields on M tangent to the boundary. Mendoza defined a complex b-manifold as a manifold with boundary together with an involutive splitting of the complexified b-tangent bundle into complex conjugate factors. We prove complex b-manifolds have a single local model depending only on dimension. This can be thought of as the Newlander-Nirenberg theorem for complex b-manifolds. Our proof uses Mendoza's result that complex b-manifolds have no "formal local invariants" and a singular coordinate change to leverage the classical Newlander-Nirenberg theorem and Catlin's generalization for complex manifolds with pseudoconvex boundary.

math.DG↗

arXiv2023

On Automorphisms of Complex $b^k$-Manifolds

The $b$-calculus of Melrose is a tool for studying structures on a smooth manifold with a first order degeneracy at a given hypersurface. In this framework, Mendoza defined complex $b$-manifolds. In the spirit of work of Scott, we extend Mendoza's definition to the case of higher-order degeneracies, introducing the notion of a complex $b^k$-manifold for $k$ a positive integer. We then investigate the local and global automorphisms of complex $b^k$-manifolds. We also propose $b^k$-analogues for some classical spaces of holomorphic functions.

math.DG↗

arXiv2023

H-Unitality of Smooth Groupoid Algebras

We show that the convolution algebra of smooth, compactly-supported functions on a Lie groupoid is H-unital in the sense of Wodzicki. We also prove H-unitality of infinite order vanishing ideals associated to invariant, closed subsets of the unit space. This furthermore gives H-unitality for the quotients by such ideals, which are noncommutative algebras of Whitney functions. These results lead immediately to excision properties in discrete Hochschild and cyclic homology around invariant, closed subsets. This work extends previous work of the author establishing the Dixmier-Malliavin theorem in this setting.

math.OA↗

arXiv2020

The smooth algebra of a one-dimensional singular foliation

Androulidakis and Skandalis showed how to associate a holonomy groupoid, a smooth convolution algebra and a C*-algebra to any singular foliation. In this note, we consider the singular foliations of a one-dimensional manifold given by vector fields that vanish to order k at a point. We show that, whereas the C*-algebras of these foliations are divided into two isomorphism classes according to the parity of k, the smooth algebras are pairwise nonisomorphic. This is accomplished by analyzing certain natural ideals in the smooth algebras. Issues of factorization with respect to convolution arise and are resolved using a context-appropriate version of the Diximier-Malliavin theorem.

math.OA↗

arXiv2020

A Dixmier-Malliavin theorem for Lie groupoids

A famous theorem of Dixmier-Malliavin asserts that every smooth, compactly-supported function on a Lie group can be expressed as a finite sum in which each term is the convolution, with respect to Haar measure, of two such functions. We establish that the same holds for a Lie groupoid. Most of the heavy lifting is done by a lemma in the original work of Dixmier-Malliavin. We also need the technology of Lie algebroids and the corresponding notion of exponential map. As an application, we obtain a result on the arithmetic of ideals in the smooth convolution algebra of a Lie groupoid arising from functions vanishing to given order on an invariant submanifold of the unit space.

math.OA↗

arXiv2012

Two Topological Uniqueness Theorems for Spaces of Real Numbers

A 1910 theorem of Brouwer characterizes the Cantor set as the unique totally disconnected, compact metric space without isolated points. A 1920 theorem of Sierpinski characterizes the rationals as the unique countable metric space without isolated points. The purpose of this exposition is to give an accessible overview of this celebrated pair of uniqueness results. It is illuminating to treat the problems simultaneously because of commonalities in their proofs. Some of the more counterintuitive implications of these results are explored through examples. Additionally, near-examples are provided which thwart various attempts to relax hypotheses.

math.GN↗