bi-Lipschitz Equivalence Research Articles

Local quasi-isometries and tangent cones of definable germs

Abstract In this paper, we introduce the notion of local quasi-isometry for metric germs and prove that two definable germs are quasi-isometric if and only if their tangent cones are bi-Lipschitz homeomorphic. Since bi-Lipschitz equivalence is a particular case of local quasi-isometric equivalence, we obtain Sampaio’s tangent cone theorem as a corollary. As an application, we provide a different proof of the theorem by Fernandes-Sampaio, which states that the tangent cone of a Lipschitz normally embedded germ is also Lipschitz normally embedded.

Sublinear bilipschitz equivalence and sublinearly Morse boundaries

Sublinear bilipschitz equivalence and sublinearly Morse boundaries

Divergence of separated nets with respect to displacement equivalence

We introduce a hierarchy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions ϕ:(0,∞)→(0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi :(0,\\infty )\\rightarrow (0,\\infty )$$\\end{document}. Two separated nets are called ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-displacement equivalent if, roughly speaking, there is a bijection between them which, for large radii R, displaces points of norm at most R by something of order at most ϕ(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi (R)$$\\end{document}. We show that the spectrum of ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}, to the indiscrete equivalence relation, corresponding to ϕ(R)∈Ω(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi (R)\\in \\varOmega (R)$$\\end{document}, in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of ϕ(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi (R)$$\\end{document} for R→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R\\rightarrow \\infty $$\\end{document}. We further undertake a comparison of our notion of ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}-displacement equivalence with that of bilipschitz equivalence.

Global-in-time well-posedness of the one-dimensional hydrodynamic Gross–Pitaevskii equations without vacuum

We establish global-in-time well-posedness of the one-dimensional hydrodynamic Gross–Pitaevskii equations in the absence of vacuum in (1 + H^s) times H^{s-1} with s ge 1. We achieve this by a reduction via the Madelung transform to the previous global-in-time well-posedness result for the Gross–Pitaevskii equation in Koch and Liao (Adv Math 377, 2021; Adv Math 420, 2023). Our core result is a local bilipschitz equivalence of the relevant function spaces, which enables the transfer of results between the two equations.

Geometries of topological groups

The paper provides an overarching framework for the study of some of the intrinsic geometries that a topological group may carry. An initial analysis is based on geometric nonlinear functional analysis, that is, the study of Banach spaces as metric spaces up to various notions of isomorphism, such as bi-Lipschitz equivalence, uniform homeomorphism, and coarse equivalence. This motivates the introduction of the various geometric categories applicable to all topological groups, namely, their uniform and coarse structure, along with those applicable to a more select class, that is, (local) Lipschitz and quasimetric structure. Our study touches on Lie theory, geometric group theory, and geometric nonlinear functional analysis and makes evident that these can all be seen as instances of a single coherent theory.

Highly irregular separated nets

In 1998 Burago and Kleiner and (independently) McMullen gave examples of separated nets in Euclidean space which are bilipschitz non-equivalent to the integer lattice. We study weaker notions than bilipschitz equivalence and demonstrate that such notions also distinguish between separated nets. Put differently, we find occurrences of particularly strong divergence of separated nets from the integer lattice. Our approach generalises that of Burago and Kleiner and McMullen which takes place largely in a continuous setting. Existence of irregular separated nets is verified via the existence of non-realisable density functions ρ: [0,1]d → (0, ∞). In the present work we obtain stronger types of non-realisable densities.

Lipschitz geometry of pairs of normally embedded Hölder triangles

We consider a special case of the outer bi-Lipschitz classification of real semialgebraic (or, more general, definable in a polynomially bounded o-minimal structure) surface germs, obtained as a union of two normally embedded Hölder triangles. We define a combinatorial invariant of an equivalence class of such surface germs, called sigma tau -pizza, and conjecture that, in this special case, it is a complete combinatorial invariant of outer bi-Lipschitz equivalence.

On the logarithmic coarse structures of Lie groups and hyperbolic spaces

We characterize the Lie groups with finitely many connected components that are O(u)-bilipschitz equivalent (almost quasiisometric in the sense that the sublinear function u replaces the additive bounds of quasiisometry) to the real hyperbolic space, or to the complex hyperbolic plane. The characterizations are expressed in terms of deformations of Lie algebras and in terms of pinching of sectional curvature of left-invariant Riemannian metrics in the real case. We also compare sublinear bilipschitz equivalence and coarse equivalence, and prove that every coarse equivalence between the logarithmic coarse structures of geodesic spaces is a \(O(\log )\)-bilipschitz equivalence. The Lie groups characterized are exactly those whose logarithmic coarse structure is equivalent to that of a real hyperbolic space or the complex hyperbolic plane. Finally we point out that a conjecture made by Tyson about the conformal dimensions of the boundaries of certain hyperbolic buildings holds conditionally to the four exponentials conjecture.

Quasiconformal flows on non-conformally flat spheres

We study integral curvature conditions for a Riemannian metric g on S4 that quantify the best bilipschitz constant between (S4,g) and the standard metric on S4. Our results show that the best bilipschitz constant is controlled by the L2-norm of the Weyl tensor and the L1-norm of the Q-curvature, under the conditions that those quantities are sufficiently small, g has a positive Yamabe constant and the Q-curvature is mean-positive. The proof of the result is achieved in two steps. Firstly, we construct a quasiconformal map between two conformally related metrics in a positive Yamabe class. Secondly, we apply the Ricci flow to establish the bilipschitz equivalence from such a conformal class to the standard conformal class on S4.

On the Moduli Space of Quasi-Homogeneous Functions

We relate the moduli space of analytic equivalent germs of reduced quasi-homogeneous functions at $$({\mathbb {C}}^{2},0)$$ with their bi-Lipschitz equivalence classes. We show that any non-degenerate continuous family of (reduced) quasi-homogeneous (but not homogeneous) functions with constant Henry–Parusiński invariant is analytically trivial. Further, we show that there are only a finite number of distinct bi-Lipschitz classes among quasi-homogeneous functions with the same Henry–Parusiński invariant providing a maximum quota for this number. Finally, we conclude that the moduli space of bi-Lipschitz equivalent quasi-homogeneous function-germs admits an analytic structure.

Multiplicity, regularity and blow-spherical equivalence of real analytic sets

This article is devoted to studying multiplicity and regularity of analytic sets. We present an equivalence for analytic sets, named blow-spherical equivalence, which generalizes differential equivalence and subanalytic bi-Lipschitz equivalence and, with this approach, we obtain several applications on analytic sets. On multiplicity, we present a generalization for Gau–Lipman’s Theorem about differential invariance of the multiplicity in the complex and real cases, and we show that the multiplicity \(\mathrm{mod}\,2\) is invariant under blow-spherical homeomorphisms in the case of real analytic curves and surfaces and also for a class of real analytic foliations and is invariant by (image) arc-analytic blow-spherical homeomorphisms in the case of real analytic hypersurfaces, generalizing some results proved by G. Valette. On regularity, we show that blow-spherical regularity of real analytic sets implies \(C^1\) smoothness only in the case of real analytic curves. We present also a complete classification of the germs of real analytic curves.

Conditional bi-Lipschitz equivalence of self-similar sets

Let A,B be two non-empty sets in R. If there exists a bi-Lipschitz map ϕ between A and B, then we say A and B are bi-Lipschitz equivalent. If the function ϕ satisfies some initial extra conditions, for instance, some point in A should be mapped to a particular point in B, then we call ϕ a conditional bi-Lipschitz map. In this paper, we shall consider a class of self-similar sets with touching and overlapping structure, and construct a conditional bi-Lipschitz map between two self-similar sets from this class. Our methodology is different from the graph-directed self-similar sets. As an application, we strengthen the main result of [9].

Gromov hyperbolic graphs arising from iterations

For a contractive iterated function system (IFS), it is known that there is a natural hyperbolic graph structure (augmented tree) on the symbolic space of the IFS that reflects the relationship among neighboring cells, and its hyperbolic boundary with the Gromov metric is Hölder equivalent to the attractor K[14,26,28]. This setup was taken up to study the probabilistic potential theory on K[21,20], and the bi-Lipschitz equivalence on K[29]. In this paper, we formulate a broad class of hyperbolic graphs, called expansive hyperbolic graphs, to capture the most essential properties from the augmented trees and the hyperbolic boundaries (e.g., the special geodesics, bounded degree property, metric doubling property, and Hölder equivalence). We also study a new setup of “weighted” IFS and investigate its connection with the self-similar energy form in the analysis of fractals.

Bilipschitz equivalence of polynomials

AbstractWe study a family of two variables polynomials having moduli up to bilipschitz equivalence: two distinct polynomials of this family are not bilipschitz equivalent. However any level curve of the first polynomial is bilipschitz equivalent to a level curve of the second.

Toward a quasi-Möbius characterization of invertible homogeneous metric spaces

We study locally compact metric spaces that enjoy various forms of homogeneity with respect to Möbius self-homeomorphisms. We investigate connections between such homogeneity and the combination of isometric homogeneity with invertibility. In particular, we provide a new characterization of snowflakes of boundaries of rank-one symmetric spaces of non-compact type among locally compact and connected metric spaces. Furthermore, we investigate the metric implications of homogeneity with respect to uniformly strongly quasi-Möbius self-homeomorphisms, connecting such homogeneity with the combination of uniform bi-Lipschitz homogeneity and quasi-invertibility. In this context we characterize spaces containing a cut point and provide several metric properties of spaces containing no cut points. These results are motivated by a desire to characterize the snowflakes of boundaries of rank-one symmetric spaces up to bi-Lipschitz equivalence.

Sublinear quasiconformality and the large-scale geometry of Heintze groups

This article analyzes sublinearly quasisymmetric homeomorphisms (generalized quasisymmetric mappings), and draws applications to the sublinear large-scale geometry of negatively curved groups and spaces. It is proven that those homeomorphisms lack analytical properties but preserve a conformal dimension and appropriate function spaces, distinguishing certain (nonsymmetric) Riemannian negatively curved homogeneous spaces, and Fuchsian buildings, up to sublinearly biLipschitz equivalence (generalized quasi-isometry).

Isomorphism and bi-Lipschitz equivalence between the univoque sets

In this paper, we consider a class of self-similar sets, denoted by \begin{document}$ \mathcal{A} $\end{document} , and investigate the set of points in the self-similar sets having unique codings. We call such set the univoque set and denote it by \begin{document}$ U_1 $\end{document} . We analyze the isomorphism and bi-Lipschitz equivalence between the univoque sets. The main result of this paper, in terms of the dimension of \begin{document}$ U_1 $\end{document} , is to give several equivalent conditions which describe that the closure of two univoque sets, under the lazy maps, are measure theoretically isomorphic with respect to the unique measure of maximal entropy. Moreover, we prove, under the condition \begin{document}$ U_1 $\end{document} is closed, that isomorphism and bi-Lipschitz equivalence between the univoque sets have resonant phenomenon.

Differentiable points of Sierpinski-like sponges

For the Sierpinski-like sponge in Rn, we obtain the Hausdorff dimension of the set of differentiable points such that there exists an (n−1)-rectifiable subsurface which has a tangent hyperplane at the point, using the techniques such as multi dynamical systems, geometric types of slices and restricted Frobenius problem. Furthermore, we can use the dimension of the set of differentiable points, as a bilipschitz invariant, to classify Sierpinski-like sponges in terms of bilipschitz equivalence.

On sublinear bilipschitz equivalence of groups

We discuss the notion of sublinearly bilipschitz equivalences (SBE), which generalize quasi-isometries, allowing some additional terms that behave sublinearly with respect to the distance from the origin. Such maps were originally motivated by the fact they induce bilipschitz homeomorphisms between asymptotic cones. We prove here that for hyperbolic groups, they also induce Holder homeomorphisms between the boundaries. This yields many basic examples of hyperbolic groups that are pairwise non-SBE. Besides, we check that subexponential growth is an SBE-invariant. The central part of the paper addresses nilpotent groups. While classification up to sublinearly bilipschitz equivalence is known in this case as a consequence of Pansu's theorems, its quantitative version is not. We introduce a computable algebraic invariant $e=e_G<1$ for every such group $G$, and check that $G$ is $O(r^e)$-bilipschitz equivalent to its associated Carnot group. Here $r\mapsto r^e$ is a quantitive sublinear bound. Finally, we define the notion of large-scale contractable and large-scale homothetic metric spaces. We check that these notions imply polynomial growth under general hypotheses, and formulate conjectures about groups with these properties.

Bilipschitz equivalence of trees and hyperbolic fillings

We show that quasi-isometries between uniformly discrete bounded geometry spaces that satisfy linear isoperimetric inequalities are within bounded distance to bilipschitz equivalences. We apply this result to regularly branching trees and hyperbolic fillings of compact, Ahlfors regular metric spaces.