I enjoy pursuing mathematical discovery in the area of Geometric Function Theory and/or Metric Space Geometry. In particular, I am interested in characterizing spaces which are bi-Lipschitz homogeneous. That is, spaces which admit a transitive action by a collection of uniformly bi-Lipschitz self-homeomorphisms. Note that we do not assume this collection of mappings forms a group under composition. There are many examples of such spaces. For example, any isometrically homogeneous space (such as Euclidean spaces and Carnot groups) are clearly bi-Lipschitz homogeneous. Stepping beyond isometric homogeneity, any space admitting a quasi-homogeneous parametrization is bi-Lipschitz homogeneous. Without getting into the technicalities, a quasi-homogeneous mapping quasi-preserves distances between pairs of points at a fixed scale. However, the dilation/expansion of distances may change at different scales. Generally speaking, a quasi-homogeneous mapping lies somewhere between a quasi-symmetric mapping and a bi-Lipschitz mappings. The Von Koch snowflake curve is an example of a space admitting such a parametrization. Note that such curves are not isometrically homogeneous.
I also find it interesting to consider the implications of metric invertibility. A metric space is said to be inversion invariant provided there exists a self-mapping of the space (minus a point) that behaves in a manner analogous to classical Möbius inversion in Euclidean space. Some of my work has focused on spaces that are assumed to be both bi-Lipschitz homogeneous and inversion invariant. Examples of such spaces include the boundaries at infinity of the classical real, complex, quaternionic, and octonionic hyperbolic spaces (equipped with appropriate distances).
Here is a list of my publications about bi-Lipschitz homogeneity and metric invertibility along with their abstracts:
David M. Freeman, Invertible Carnot groups, Analysis and Geometry in Metric Spaces 2 (2014), 248-257.
Abstract: We characterize Carnot groups admitting a 1-quasiconformal metric inversion as the Lie groups of Heisenberg type whose Lie algebras satisfy the J^2-condition, thus characterizing a special case of inversion invariant bi-Lipschitz homogeneity. A more general characterization of inversion invariant bi-Lipschitz homogeneity for certain non-fractal metric spaces is also provided.
David M. Freeman, Transitive bi-Lipschitz group actions and bi-Lipschitz parameterizations, Indiana University Mathematics Journal 62 (2013), no. 1, 311-331.
Abstract: We prove that Ahlfors 2-regular quasisymmetric images of are bi-Lipschitz images of if and only if they are uniformly bi-Lipschitz homogeneous with respect to a group. We also prove that certain geodesic spaces are bi-Lipschitz images of Carnot groups if they are inversion invariant bi-Lipschitz homogeneous with respect to a group.
David M. Freeman, Inversion invariant bilipschitz homogeneity, Michigan Mathematical Journal 61 (2012), no. 2, 415-430.
Abstract: We show that, for a proper, connected, locally connected doubling space with no cut points, inversion invariant bilipschitz homogeneity implies Ahlfors Q-regularity and linear local connectivity.
David M. Freeman, Unbounded bilipschitz homogeneous Jordan curves, Annales Academiæ Scientiarum Fennicæ Mathematica, 36 (2011), no. 1, 81-99.
Abstract: We prove that an unbounded L-bilipschitz homogeneous Jordan curve in the plane is of B-bounded turning, where B depends only on L. Using this result, we construct a catalogue of snowflake-type curves that includes all unbounded bilipschitz homogeneous Jordan curves, up to bilipschitz self maps of the plane. This catalogue yields characterizations of such curves in terms of certain quasiconformal maps.
David M. Freeman, Bilipschitz homogeneous Jordan curves, Möbius maps, and dimension, Illinois Journal of Mathematics 54 (2010), no. 2, 753-770.
Abstract: We characterize fractal chordarc curves in Euclidean space by the fact that they remain bilipschitz homogeneous under inversion. We illustrate this result by constructing two examples. The techniques used in these constructions provide a means of calculating various dimensions of bilipschitz homogeneous Jordan curves.
David M. Freeman and David A. Herron, Bilipschitz homogeneity and inner diameter distance, Journal d’Analyse Mathématique 111 (2010), no. 1, 1-46.
Abstract: We prove that a Jordan plane domain whose boundary is bilipschitz homogeneous with respect to its inner diameter distance is a John disk. This opens the door to an abundance of equivalent conditions. We characterize such domains in terms of quasiconformal mappings as well as their Riemann maps. We introduce the notion of an inner diameter distance Jordan disk and present related results for these spaces.
More recently, I have been spending some time learning about intersections between the theories of continued fractions and transcendental numbers. In particular, I have been reading work of Boris Adamczewski and Yann Bugeaud (this paper, for example) about symmetric and repetitive patterns in infinite continued fractions. I discovered a relatively simple (but, in my opinion, interesting) generalization of their work concerning palindromic continued fractions. I wrote up my results in the following preprint:
David M. Freeman, Generalized Palindromic Continued Fractions, submitted.
Abstract: In this paper we introduce a generalization of palindromic continued fractions as studied by Adamczewski and Bugeaud. We refer to these generalized palindromes as m-palindromes, where m ranges over the positive integers. We provide a simple transcendence criterion for m-palindromes, extending and slightly refining an analogous result of Adamczewski and Bugeaud. We also provide methods for constructing examples of m-palindromes. Such examples allow us to illustrate our transcendence criterion and to explore the relationship between m-palindromes and stammering continued fractions, another concept introduced by Adamczewski and Bugeaud.
In fact, the above preprint was originally motivated by my attempts to illustrate continued fractions by way of certain sequences of cycloid curves. I’ve posted a few examples of such imagery here. I’ve also written up the following article about cycloid curves and continued fractions:
David M. Freeman, Cycloid Curves and Continued Fractions, submitted.
Abstract: In this paper we devise a means of visualizing a given continued fraction via a sequence of cycloid curves possessing a common center of rotational symmetry. As the continued fraction grows in length, the corresponding cycloids grow in diameter. The geometric relationships between subsequent cycloids in this expanding sequence reveal number-theoretic properties of the continued fraction, and vice-versa. Moreover, we find the visual images produced according to our methods to be an aesthetically pleasing and versatile artistic device. While we do not attempt to fully develop the aesthetic potential of the cycloid sequences defined in this paper, our goal is to provide a mathematical framework in which further artistic exploration can occur.