bi-Lipschitz Equivalent Research Articles

Lipschitz geometry of surface germs in {mathbb {R}}^4: metric knots

A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in {mathbb {R}}^4 is a topological knot (or link) in S^3. We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in {mathbb {R}}^4 and knot theory. Namely, for any knot K, we construct a surface X_K in {mathbb {R}}^4 such that: the link at the origin of X_{K} is a trivial knot; the germs X_K are outer bi-Lipschitz equivalent for all K; two germs X_{K} and X_{K'} are ambient semialgebraic bi-Lipschitz equivalent only if the knots K and K' are isotopic. We show that the Jones polynomial can be used to recognize ambient bi-Lipschitz non-equivalent surface germs in {mathbb {R}}^4, even when they are topologically trivial and outer bi-Lipschitz equivalent.

Topology automaton of self-similar sets and its applications to metrical classifications

The topological and metrical classifications of fractal sets are important topics in analysis. The goal of the present paper is to carry out such studies by using a finite state automaton. Firstly, we introduce Σ-automaton for self-similar sets, and we define topology automaton for fractal gaskets. Next, we show that a fractal gasket is always bi-Hölder equivalent to the pseudo-metric space induced by its topology automaton. Thirdly, we investigate when the pseudo-metric spaces induced by different automata can be bi-Lipschitz equivalent. As an application, we obtain a rather general sufficient condition for two fractal gaskets to be bi-Hölder or bi-Lipschitz equivalent.

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.

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.

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].

RESONANCE BETWEEN SELF-SIMILAR SETS AND THEIR UNIVOQUE SETS

Let [Formula: see text] be a self-similar set in [Formula: see text]. Generally, if the iterated function system (IFS) of [Formula: see text] has some overlaps, then some points in [Formula: see text] may have multiple codings. If an [Formula: see text] has a unique coding, then we call [Formula: see text] a univoque point. We denote by [Formula: see text] (univoque set) the set of points in [Formula: see text] having unique codings. In this paper, we shall consider the following natural question: if two self-similar sets are bi-Lipschitz equivalent, then are their associated univoque sets also bi-Lipschitz equivalent. We give a class of self-similar sets with overlaps, and answer the above question affirmatively.

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.

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.

Rigidity theorem of graph-directed fractals

In this paper, we identify two fractals if and only if they are biLipschitz equivalent. Fix ratio $r,$ for dust-like graph-directed sets with ratio $r$ and integer characteristics, we show that they are rigid in the sense that they are uniquely determined by their Hausdorff dimensions. Using this rigidity theorem, we show that in some class of self-similar sets, two totally disconnected self-similar sets without complete overlaps are biLipschitz equivalent.

Some properties of median metric spaces

We describe a number of properties of a median metric space. In particular, we show that a complete connected median metric space of finite rank admits a canonical bi-lipschitz equivalent CAT(0) metric. Metric spaces of this sort arise, up to bi-lipschitz equivalence, as asymptotic cones of certain classes of finitely generated groups, and the existence of such a structure has various consequences for the large scale geometry of the group.

Symmetric products of the Euclidean spaces and the spheres

By $F_n(X)$, $n \geq 1$, we denote the $n$-th symmetric product of a metric space $(X,d)$ as the space of the non-empty finite subsets of $X$ with at most $n$ elements endowed with the Hausdorff metric $d_H$. In this paper we shall describe that every isometry from the $n$-th symmetric product $F_n(X)$ into itself is induced by some isometry from $X$ into itself, where $X$ is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the $n$-th symmetric product of the Euclidean space up to bi-Lipschitz equivalence and present that the $2$nd symmetric product of the plane is bi-Lipschitz equivalent to the 4-dimensional Euclidean space.

Equivalence relations on separated nets arising from linear toral flows

In 1998, Burago-Kleiner and McMullen independently proved the existence of separated nets in $\mathbb{R}^d$ which are not bi-Lipschitz equivalent (BL) to a lattice. A finer equivalence relation than BL is bounded displacement (BD). Separated nets arise naturally as return times to a section for minimal $\mathbb{R}^d$-actions. We analyze the separated nets which arise via these constructions, focusing particularly on nets arising from linear $\mathbb{R}^d$-actions on tori. We show that generically these nets are BL to a lattice, and for some choices of dimensions and sections, they are generically BD to a lattice. We also show the existence of such nets which are not BD to a lattice.

Equivalence of Gromov-Prohorov- and Gromov's $\underline\Box_\lambda$-metric on the space of metric measure spaces

The space of metric measure spaces (complete separable metric spaces with a probability measure) is becoming more and more important as state space for stochastic processes. Of particular interest is the subspace of (continuum) metric measure trees. Greven, Pfaffelhuber and Winter introduced the Gromov-Prohorov metric d_{GPW} on the space of metric measure spaces and showed that it induces the Gromov-weak topology. They also conjectured that this topology coincides with the topology induced by Gromov's Box_1 metric. Here, we show that this is indeed true, and the metrics are even bi-Lipschitz equivalent. More precisely, d_{GPW}= 1/2 Box_{1/2}, and hence d_{GPW} <= Box_1 <= 2d_{GPW}. The fact that different approaches lead to equivalent metrics underlines their importance and also that of the induced Gromov-weak topology. As an application, we give an easy proof of the known fact that the map associating to a lower semi-continuous excursion the coded R-tree is Lipschitz continuous when the excursions are endowed with the (non-separable) uniform metric. We also introduce a new, weaker, metric topology on excursions, which has the advantage of being separable and making the space of bounded excursions a Lusin space. We obtain continuity also for this new topology.

Rigidity of bi-Lipschitz equivalence of weighted homogeneous function-germs in the plane

The main goal of this work is to show that if two weighted homogeneous (but not homogeneous) function-germs ( C 2 , 0 ) → ( C , 0 ) (\mathbb {C}^2,0)\rightarrow (\mathbb {C},0) are bi-Lipschitz equivalent, in the sense that these function-germs can be included in a strongly bi-Lipschitz trivial family of weighted homogeneous function-germs, then they are analytically equivalent.

Quasicircles and bounded turning circles modulo bi-Lipschitz maps

We construct a catalog, of snowflake type metric circles, that describes all metric quasicircles up to bi-Lipschitz equivalence. This is a metric space analog of a result due to Rohde. Our construction also works for all bounded turning metric circles; these need not be doubling. As a byproduct, we show that a metric quasicircle with Assouad dimension strictly less than two is bi-Lipschitz equivalent to a planar quasicircle.

Bilipschitz equivalence is not equivalent to quasi-isometric equivalence for finitely generated groups

We show that certain lamplighter groups that are quasi-isometric to each other are not bilipschitz equivalent. This gives a positive answer to a question in Topics in Geometric Group Theory by Pierre de la Harpe (page 107).

On asymptotic Teichmüller space

In this article we prove that for any hyperbolic Riemann surface M M of infinite analytic type, the little Bers space Q 0 ( M ) Q_{0}(M) is isomorphic to c 0 c_{0} . As a consequence of this result, if M M is such a Riemann surface, then its asymptotic Teichmüller space A T ( M ) AT(M) is bi-Lipschitz equivalent to a bounded open subset of the Banach space l ∞ / c 0 l^{\infty }/c_{0} . Further, if M M and N N are two such Riemann surfaces, their asymptotic Teichmüller spaces, A T ( M ) AT(M) and A T ( N ) AT(N) , are locally bi-Lipschitz equivalent.

Bi-Lipschitz geometry of weighted homogeneous surface singularities

We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz equivalent.

LOCAL RIGIDITY OF INFINITE-DIMENSIONAL TEICHMÜLLER SPACES

This paper presents a rigidity theorem for infinite-dimensional Bergman spaces of hyperbolic Riemann surfaces, which states that the Bergman space A1(M), for such a Riemann surface M, is isomorphic to the Banach space of summable sequence, l1. This implies that whenever M and N are Riemann surfaces that are not analytically finite, and in particular are not necessarily homeomorphic, then A1(M) is isomorphic to A1(N). It is known from V. Markovic that if there is a linear isometry between A1(M) and A1(N), for two Riemann surfaces M and N of non-exceptional type, then this isometry is induced by a conformal mapping between M and N. As a corollary to this rigidity theorem presented here, taking the Banach duals of A1(M) and l1 shows that the space of holomorphic quadratic differentials on M, Q(M), is isomorphic to the Banach space of bounded sequences, l∞. As a consequence of this theorem and the Bers embedding, the Teichmüller spaces of such Riemann surfaces are locally bi-Lipschitz equivalent.

Infinite commensurable hyperbolic groups are bi-Lipschitz equivalent

It is proved that commensurable hyperbolic groups are bi-Lipschitz equivalent. Therefore, subgroups of finite index in an arbitrary hyperbolic group also share this property. In addition, it is shown that any two separated nets Γ1 and Γ2 in the hyperbolic space Hn of dimension n≥2 are bi-Lipschitz-equivalent. These results answer the questions posed in [1].