Invariants of the bi-Lipschitz contact equivalence of continuous definable function germs

TL;DR

This paper introduces an invariant for classifying continuous definable function germs under bi-Lipschitz contact equivalence, using asymptotic expansion coefficients along tangency components.

Contribution

It constructs a new invariant based on asymptotic expansions for definable function germs, aiding in their classification under bi-Lipschitz contact equivalence.

Findings

Invariant effectively distinguishes bi-Lipschitz contact classes.

Applicable to polynomially bounded o-minimal structures, including semialgebraic functions.

Provides a method to compute invariants from asymptotic data.

Abstract

We construct an invariant of the bi-Lipschitz contact equivalence of continuous function germs definable in a polynomially bounded o-minimal structure, such as semialgebraic functions. For a germ $f,$ the invariant is given in terms of the leading coefficients of the asymptotic expansions of $f$ along the connected components of the tangency variety of $f.$

Equations

$$\begin{array}{ccccc}({\mathbb{R}}^n,0)& \stackrel{(id,f)}{\to }& ({\mathbb{R}}^n\times {\mathbb{R}}^p,0)& \stackrel{{\pi }_n}{\to }& ({\mathbb{R}}^n,0)\\ h\downarrow & & \downarrow H& & h\downarrow \\ ({\mathbb{R}}^n,0)& \stackrel{(id,g)}{\to }& ({\mathbb{R}}^n\times {\mathbb{R}}^p,0)& \stackrel{{\pi }_n}{\to }& ({\mathbb{R}}^n,0)\end{array}$$

$$c_{1}f(x)$$

$$⟨v,x^{\prime }-x⟩$$

$$\mathrm{epi}f$$

$$\underset{x^{\prime }\to x}{\mathrm{liminf}}f(x^{\prime })$$

$$\partial f(x)$$

$${\partial }^{\infty }f(x)$$

$${\partial }^{\infty }f(\bar{x})\cap N_{\bar{x}}\Omega$$

$$v\in \partial f(\varphi (t))$$

$$v\in {\partial }^{\infty }f(\varphi (t))$$

$$M$$

$$(v,-1)$$

$$⟨v,{\varphi }^{\prime }(t)⟩$$

$$\Gamma (f)$$

$$\partial f(x)$$

$$\Gamma (f)$$

$$\Gamma (f)\to \mathbb{R},x\mapsto \Vert x\Vert ,$$

$$f_k:(0,ϵ)\to \mathbb{R},t\mapsto f_k(t),$$

$$v\in \partial f(\varphi (\tau ))$$

$$(f\circ \varphi {)}^{\prime }(\tau )$$

$$\psi :(0,ϵ)\to \mathbb{R},t\mapsto \psi (t):=\underset{x\in {\mathbb{S}}_t}{\min }f(x),$$

$$\overline{\psi }:(0,ϵ)\to \mathbb{R},t\mapsto \overline{\psi }(t):=\underset{x\in {\mathbb{S}}_t}{\max }f(x),$$

$$\psi (t)$$

$$\Vert \varphi (t)\Vert$$

$$v\in {\partial }^{\infty }f(\varphi (t))$$

$$⟨\varphi (t),{\varphi }^{\prime }(t)⟩$$

$${\partial }^{\infty }f(\varphi (t))\cap N_{\varphi (t)}{\mathbb{S}}_t=\{0\}$$

$$\psi (t)$$

$$\overline{\psi }(t)$$

Full text

The authors are partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED), grant 101.04-2016.05

1. Introduction

Lipschitz geometry of maps is a rapidly growing subject in contemporary Singularity Theory. Recent progress in this area is due to the tameness theorems proved by several authors (see, for example, [1, 6, 8, 9, 10, 17]). However the description of a set of invariants is barely developed (see also [2]). This paper presents a numerical invariant of continuous function germs definable in a polynomially bounded o-minimal structure (e.g., semialgebraic functions) with respect to the bi-Lipschitz contact equivalence. The most important ingredient of the invariant constructed here is the so-called tangency variety. More precisely, let $f\colon(\mathbb{R}^{n},0)\rightarrow({\mathbb{R}},0)$ be a continuous function germ, which is definable in a polynomially bounded o-minimal structure. The tangency variety $\Gamma(f)$ of $f$ consists of all points $x$ in some neighborhood of the origin $0\in\mathbb{R}^{n}$ such that the fiber $f^{-1}(f(x))$ is tangent to the sphere in $\mathbb{R}^{n}$ centered at [math] with radius $\|x\|.$ The restriction of $f$ on each connected component of $\Gamma(f)\setminus\{0\}$ defines a definable function $f_{k}$ of a single variable. Then the invariant of $f$ is given in terms of the leading coefficients of the asymptotic expansions of these functions $f_{k}.$

The rest of the paper is organized as follows. In Section 2, we present some preliminaries which will be used later. The definition and some properties of tangency varieties are given in Section 3. The main result is provided in Section 4.

2. Preliminaries

Throughout this work we shall consider the Euclidean vector space ${\mathbb{R}}^{n}$ endowed with its canonical scalar product $\langle\cdot,\cdot\rangle$ and we shall denote its associated norm $\|\cdot\|.$ The closed ball (resp., the sphere) centered at the origin $0\in\mathbb{R}^{n}$ of radius $\epsilon$ will be denoted by $\mathbb{B}_{\epsilon}$ (resp., $\mathbb{S}_{\epsilon}$).

2.1. The bi-Lipschitz contact equivalence

The contact equivalence between (smooth) mappings was introduced by J. Mather [14]. The natural extension of Mather’s definition to the Lipschitz setting in the function case appeared in [1], and to the general case in [17]. Let us start with the following definition.

Definition 2.1**.**

Two map germs $f,g\colon(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0)$ are called bi-Lipschitz contact equivalent (or $\mathcal{K}$-bi-Lipschitz equivalent) if there exist two germs of bi-Lipschitz homeomorphisms $h\colon(\mathbb{R}^{n},0)\to(\mathbb{R}^{n},0)$ and $H\colon(\mathbb{R}^{n}\times\mathbb{R}^{p},0)\to(\mathbb{R}^{n}\times\mathbb{R}^{p},0)$ such that $H(\mathbb{R}^{n}\times\{0\})=\mathbb{R}^{n}\times\{0\}$ and the following diagram is commutative:

[TABLE]

where $id\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the identity map and $\pi_{n}\colon\mathbb{R}^{n}\times\mathbb{R}^{p}\to\mathbb{R}^{n}$ is the canonical projection.

In this paper we consider the case $p=1,$ thus the maps $f,g$ are functions. There is a more convenient way to work with the bi-Lipschitz contact equivalence of functions, due to the following result:

Theorem 2.1** (see [1, Theorem 2.1]).**

Let $f,g\colon(\mathbb{R}^{n},0)\to(\mathbb{R},0)$ be two continuous function germs. If $f$ and $g$ are bi-Lipschitz contact equivalent, then there exists a bi-Lipschitz homeomorphism germ $h\colon(\mathbb{R}^{n},0)\to(\mathbb{R}^{n},0),$ there exist positive constants $c_{1},c_{2}$ and a sign $\sigma\in\{-1,1\}$ such that in a neighbourhood of the origin $0\in\mathbb{R}^{n}$ the following inequalities hold true

[TABLE]

2.2. O-minimal structures

The notion of o-minimality was developed in the late 1980s after it was noticed that many proofs of analytic and geometric properties of semi-algebraic sets and maps could be carried over verbatim for sub-analytic sets and maps. We refer the reader to [4, 12, 13, 18, 19] for the basic properties of o-minimal structures used in this paper.

Definition 2.2**.**

An o-minimal structure on the real field $\mathbb{R}$ is a sequence $\mathcal{S}:=(\mathcal{S}_{n})_{n\in\mathbb{N}}$ such that for each $n\in\mathbb{N}$:

• (a)

$\mathcal{S}_{n}$ is a Boolean algebra of subsets of $\mathbb{R}^{n}$.

• (b)

If $A\in\mathcal{S}_{m}$ and $B\in\mathcal{S}_{n}$, then $A\times B\in\mathcal{S}_{m+n}.$

• (c)

If $A\in\mathcal{S}_{n+1},$ then $p(A)\in\mathcal{S}_{n},$ where $p\colon\mathbb{R}^{n+1}\to\mathbb{R}^{n}$ is the projection on the first $n$ coordinates.

• (d)

$\mathcal{S}_{n}$ contains all algebraic subsets of $\mathbb{R}^{n}.$

• (e)

Each set belonging to $\mathcal{S}_{1}$ is a finite union of points and intervals.

A set $A\subset\mathbb{R}^{n}$ is said to be a definable set if $A\in\mathcal{S}_{n}.$ A map $f\colon A\rightarrow\mathbb{R}^{m}$ is said to be a definable map if its graph is definable.

The structure $\mathcal{S}$ is said to be polynomially bounded if for every definable function $f\colon\mathbb{R}\rightarrow\mathbb{R},$ there exist $d\in\mathbb{N}$ and $R>0$ (depending on $f$) such that $|f(x)|\leq x^{d}$ for all $x>R,$

Examples of (polynomially bounded) o-minimal structures are

• •

the semi-linear sets,

• •

the semi-algebraic sets (by the Tarski–Seidenberg theorem),

• •

the globally sub-analytic sets, i.e., the sub-analytic sets of $\mathbb{R}^{n}$ whose (compact) closures in $\mathbb{R}\mathbb{P}^{n}$ are sub-analytic (using Gabrielov’s complement theorem).

2.3. Normals and subdifferentials

Here we recall the notions of the normal cones to sets and the subdifferentials of real-valued functions used in this paper. For more details we refer the reader to [15, 16].

Definition 2.3**.**

Consider a set $\Omega\subset\mathbb{R}^{n}$ and a point ${x}\in\Omega.$

1. (i)

The regular normal cone (known also as the prenormal or Fréchet normal cone) $\widehat{N}_{x}\Omega$ to $\Omega$ at ${x}$ consists of all vectors $v\in\mathbb{R}^{n}$ satisfying

[TABLE] 2. (ii)

The limiting normal cone (known also as the basic or Mordukhovich normal cone) $N_{x}\Omega$ to $\Omega$ at ${x}$ consists of all vectors $v\in\mathbb{R}^{n}$ such that there are sequences $x^{k}\to{x}$ with $x^{k}\in\Omega$ and $v^{k}\rightarrow v$ with $v^{k}\in\widehat{N}_{x^{k}}\Omega.$

If $\Omega$ is a manifold of class $C^{1},$ then for every point $x\in\Omega,$ the normal cones $\widehat{N}_{x}\Omega$ and $N_{x}\Omega$ are equal to the normal space to $\Omega$ at ${x}$ in the sense of differential geometry, i.e., $\widehat{N}_{x}\Omega={N}_{x}\Omega$ and $v\perp T_{x}\Omega$ for all $v\in\widehat{N}_{x}\Omega,$ where $T_{x}\Omega$ stands for the tangent space of $\Omega$ at $x;$ see [16, Example 6.8].

For a function $f\colon\mathbb{R}^{n}\rightarrow{\mathbb{R}},$ we define the epigraph of $f$ to be

[TABLE]

A function $f\colon\mathbb{R}^{n}\rightarrow{\mathbb{R}}$ is said to be lower semi-continuous at ${x}$ if it holds that

[TABLE]

Functional counterparts of normal cones are subdifferentials.

Definition 2.4**.**

Consider a function $f\colon\mathbb{R}^{n}\to{\mathbb{R}}$ and a point ${x}\in\mathbb{R}^{n}.$ The limiting and horizon subdifferentials of $f$ at ${x}$ are defined respectively by

[TABLE]

The limiting subdifferential $\partial f({x})$ generalizes the classical notion of gradient. In particular, for $C^{1}$-smooth functions $f$ on $\mathbb{R}^{n},$ the subdifferential consists only of the gradient $\nabla f(x)$ for each $x\in\mathbb{R}^{n}.$ The horizon subdifferential $\partial^{\infty}f({x})$ plays an entirely different role–it detects horizontal “normal” to the epigraph–and it plays a decisive role in subdifferential calculus; see [16, Corollary 10.9] for more details.

Theorem 2.2** (Fermat rule).**

Consider a lower semi-continuous function $f\colon\mathbb{R}^{n}\to{\mathbb{R}}$ and a closed set $\Omega\subset\mathbb{R}^{n}.$ If $\bar{x}\in\Omega$ is a local minimizer of $f$ on $\Omega$ and the qualification condition

[TABLE]

is valid, then the inclusion $0\in\partial f(\bar{x})+N_{\bar{x}}\Omega$ holds.

We will also need the following lemma.

Lemma 2.1**.**

Consider a lower semi-continuous definable function $f\colon\mathbb{R}^{n}\rightarrow{\mathbb{R}}$ and a definable curve $\phi\colon[a,b]\rightarrow\mathbb{R}^{n}.$ Then for all but finitely many ${{t}}\in[a,b],$ the mappings $\phi$ and $f\circ\phi$ are $C^{1}$-smooth at ${{t}}$ and satisfy

[TABLE]

Proof.

(cf. [3, Proposition 4] and [5, Lemma 2.10]). Without loss of generality, assume that the curve $\phi$ is non-constant. In light of the monotonicity theorem [19, Theorem 4.1], there exists a real number $\epsilon\in(0,1)$ such that on the open interval $(0,\epsilon)$ we have the mappings $\phi$ and $f\circ\phi$ are $C^{1}$-smooth and ${\phi}^{\prime}$ is nonzero. Let

[TABLE]

which is a subset of the epigraph of $f.$ Clearly, $M$ is a connected definable $C^{1}$-manifold of dimension $1.$ Taking if necessary a smaller $\epsilon,$ we can be sure that there exists a Whitney $C^{1}$-stratification $\mathscr{W}$ of $\mathrm{epi}f$ such that $M$ is a stratum of $\mathscr{W};$ see [19, Theorem 4.8], for example.

Take arbitrary (but fixed) ${t}\in(0,\epsilon)$ and $v\in\partial f(\phi({t})).$ By definition, there exist sequences $\{x^{k}\}\subset U$ and $\{(v^{k},t^{k})\}\subset\widehat{N}_{(x^{k},f(x^{k}))}\mathrm{epi}f\subset\mathbb{R}^{n}\times\mathbb{R},$ such that $x^{k}\to x:=\phi({t})$ and $(v^{k},t^{k})\to(v,-1)$ as $k\to\infty.$ Due to the finiteness property of $\mathscr{W},$ we may suppose that the sequence $\{(x^{k},f(x^{k}))\}$ lies entirely in some stratum $S\in\mathscr{W}$ of dimension $d.$ Using the compactness of the Grassmannian manifold of $d$-dimensional subspaces of $\mathbb{R}^{n},$ we may assume that the sequence of tangent spaces $T_{(x^{k},f(x^{k}))}S$ converges to some vector space $T$ of dimension $d.$ Then the Whitney-(a) property yields that $T_{(x,f(x))}M\subset T.$ By definition, for each $k\geq 1$ we have that the vector $(v^{k},t^{k})$ is Fréchet normal to the epigraph $\mathrm{epi}f$ of $f$ at $(x^{k},f(x^{k}));$ hence, it is also normal (in the classical sense) to the tangent space $T_{(x^{k},f(x^{k}))}S.$ By a standard continuity argument, the vector

[TABLE]

must be normal to $T$ and a fortiori to $T_{(x,f(x))}M.$ On the other hand, $T_{(x,f(x))}M$ is the vector space generated by the vector $({\phi}^{\prime}({t}),(f\circ\phi)^{\prime}({t}))\in\mathbb{R}^{n}\times\mathbb{R}.$ Consequently, we obtain

[TABLE]

A similar argument also shows

[TABLE]

for all ${t}\in(0,\epsilon)$ and all $v\in\partial^{\infty}(\phi({t})).$

Finally, let $c$ be the supremum of real numbers $T\in[0,1]$ such that for all but finitely many ${t}\in[0,T),$ we have for all $v\in\partial f(\phi({t}))$ and all $w\in\partial^{\infty}(\phi({t})),$

[TABLE]

Then $c\geq\epsilon.$ We must prove that $c=1.$ Suppose that this is not the case. Replacing the interval $[0,1)$ by the interval $[c,1)$ and repeating the previous argument, we find a small real number $\epsilon^{\prime}>0$ such that for all ${t}\in(c,c+\epsilon^{\prime}),$ all $v\in\partial f(\phi({t}))$ and all $w\in\partial^{\infty}(\phi({t})),$

[TABLE]

thus contradicting the definition of $c.$ The proof is complete. ∎

3. Tangencies

Let $f\colon(\mathbb{R}^{n},0)\rightarrow({\mathbb{R}},0)$ be a continuous definable function germ. Let us begin with the following definition (see also [7]).

Definition 3.1**.**

The tangency variety of $f$ (at [math]) is defined as follows:

[TABLE]

Remark 3.1**.**

When $f$ is of class $C^{1}$ one has

[TABLE]

and so

[TABLE]

By definition, it is not hard to check that $\Gamma(f)$ is a definable set. Moreover, thanks to the Fermat rule (Theorem 2.2), we can see that for any $t>0,$ the tangency variety $\Gamma(f)$ contains the set of minimizers (and minimizers) of $f$ on the sphere ${\mathbb{S}}_{t};$ in particular, [math] is a cluster point of $\Gamma(f).$

Applying the Hardt triviality theorem (see [19, Theorem 4.11]) for the definable function

[TABLE]

we find a constant $\epsilon>0$ such that the restriction of this function on $\Gamma(f)\cap\mathbb{B}_{\epsilon}\setminus\{0\}$ is a topological trivial fibration. Let $p$ be the number of connected components of a fiber of this restriction. Then $\Gamma(f)\cap\mathbb{B}_{\epsilon}\setminus\{0\}$ has exactly $p$ connected components, say $\Gamma_{1},\ldots,\Gamma_{p},$ and each such component is a definable set. Moreover, for all $t\in(0,\epsilon)$ and all $k=1,\ldots,p,$ the sets $\Gamma_{k}\cap\mathbb{S}_{t}$ are connected. Corresponding to each $\Gamma_{k},$ let

[TABLE]

be the function defined by $f_{k}(t):=f(x),$ where $x\in\Gamma_{k}\cap\mathbb{S}_{t}.$

Lemma 3.1**.**

For each $\epsilon>0$ small enough, all the functions $f_{k}$ are well-defined and definable.

Proof.

Fix $k\in\{1,\ldots,p\}$ and take any $t\in(0,\epsilon).$ We will show that the restriction of $f$ on $\Gamma_{k}\cap\mathbb{S}_{t}$ is constant. To see this, let $\phi\colon[0,1]\rightarrow{\mathbb{R}}^{n}$ be a definable $C^{1}$-curve such that $\phi(\tau)\in\Gamma_{k}\cap\mathbb{S}_{t}$ for all $\tau\in[0,1].$ By definition, we have $\|\phi(\tau)\|=t$ and either $\lambda(\tau)\phi(\tau)\in\partial f(\phi(\tau))$ or $\lambda(\tau)\phi(\tau)\in\partial(-f)(\phi(\tau))$ for some $\lambda(\tau)\in\mathbb{R}.$ By replacing $f$ by $-f,$ if necessary, we may assume that $\lambda(\tau)\phi(\tau)\in\partial f(\phi(\tau)).$ In view of Lemma 2.1, for all but finitely many $\tau\in[a,b],$ the mappings $\phi$ and $f\circ\phi$ are $C^{1}$-smooth at $\tau$ and satisfy

[TABLE]

Therefore

[TABLE]

So $f$ is constant on the curve $\phi.$

On the other hand, since the set $\Gamma_{k}\cap\mathbb{S}_{t}$ is connected definable, it is path connected. Hence, any two points in $\Gamma_{k}\cap\mathbb{S}_{t}$ can be joined by a piecewise $C^{1}$-smooth definable curve. It follows that the restriction of $f$ on $\Gamma_{k}\cap\mathbb{S}_{t}$ is constant and so the function $f_{k}$ is well-defined. Finally, by definition, $f_{k}$ is definable. ∎

For each $t\in(0,\epsilon),$ the sphere $\mathbb{S}_{t}$ is a nonempty compact definable set. Hence, the functions

[TABLE]

are well-defined and definable. The following lemma is simple but useful.

Lemma 3.2**.**

For $\epsilon>0$ small enough, the following equalities

[TABLE]

hold for all $t\in(0,\epsilon).$

Proof.

Applying the Curve Selection Lemma (see [19, Property 1.17]) and shrinking $\epsilon$ (if necessary), we find a definable $C^{1}$-curve $\phi\colon(0,\epsilon)\rightarrow\mathbb{R}^{n}$ such that for all $t\in(0,\epsilon),$

[TABLE]

By Lemma 2.1, then we have for any $t\in(0,\epsilon),$

[TABLE]

Observe

[TABLE]

and hence the qualification condition

[TABLE]

holds for all $t\in(0,\epsilon).$ Consequently, since $\phi(t)$ minimizes $f$ subject to $\|x\|=t,$ applying the Fermat rule (Theorem 2.2), we deduce that $\phi(t)$ belongs to $\Gamma(f).$ Therefore,

[TABLE]

Using the same argument, we also have

[TABLE]

The lemma is proved. ∎

4. The main result

In this section, we fix a polynomially bounded o-minimal structure on $\mathbb{R}.$ The word “definable” will mean definable in this structure.

Let $f\colon(\mathbb{R}^{n},0)\rightarrow({\mathbb{R}},0)$ be a continuous definable function germ. As in the previous section, we associate to the function $f$ a finite number of (definable) functions $f_{1},\ldots,f_{p}$ of a single variable. Let

[TABLE]

By the Growth Dichotomy Lemma (see [19, Theorem 4.12]), we can write for each $k\in\{1,\ldots,p\}\setminus K_{0},$

[TABLE]

where $a_{k}\in\mathbb{R},a_{k}\neq 0,$ and $\alpha_{k}\in\mathbb{R},\alpha_{k}>0.$ Put

[TABLE]

Finally we let

[TABLE]

If $\mathrm{Inv}(f)=(a,b),$ we follow the convention that $-\mathrm{Inv}(f):=\mathrm{Inv}(-f)=(-b,-a).$

We now arrive to the main result of this paper.

Theorem 4.1**.**

Let $f,g\colon(\mathbb{R}^{n},0)\to(\mathbb{R},0)$ be two continuous definable function germs. If $f$ and $g$ are bi-Lipschitz contact equivalent then

[TABLE]

Proof.

Since $f$ and $g$ are bi-Lipschitz contact equivalent, it follows from Theorem 2.1 that there exist a bi-Lipschitz homeomorphism germ $h\colon(\mathbb{R}^{n},0)\to(\mathbb{R}^{n},0)$ and some positive constants $c_{1},c_{2}$ and a sign $\sigma\in\{\pm 1\}$ such that

[TABLE]

Assume that $\sigma=1.$ (The case $\sigma=-1$ is proved similarly.) Consider the definable functions

[TABLE]

where $\epsilon$ is a positive number and small enough so that these functions are either constant or strictly monotone. Assume that we have proved the following relations:

[TABLE]

where $A\simeq B$ means that $A/B$ lies between two positive constants. These, together with Lemma 3.2, imply easily that $\mathrm{Inv}(f)=\mathrm{Inv}(g),$ which is the desired conclusion.

So we are left with showing (2). We will prove the first relation; the second one is proved similarly. Indeed, if $\psi_{f}\equiv 0,$ then $\psi_{g}\equiv 0$ because of (1) and there is nothing to prove. So assume that $\psi_{f}\not\equiv 0.$ Since $h$ is a bi-Lipschitz homeomorphism germ, there exists a positive constant $L$ such that

[TABLE]

In particular, we get

[TABLE]

This, together with (1), implies that for all sufficiently small $t\geq 0,$

[TABLE]

Let $\phi\colon[0,\epsilon)\rightarrow{\mathbb{R}}^{n}$ be a definable curve such that

[TABLE]

Reducing $\epsilon$ if necessary, we may assume that $\phi$ is of class $C^{1}$ and that either $L^{-1}t<\|\phi(t)\|<Lt$ or $\|\phi(t)\|=L^{-1}t$ or $\|\phi(t)\|=Lt$ for all $t\in[0,\epsilon).$

If $L^{-1}t<\|\phi(t)\|<Lt,$ then $\phi(t)$ is a local minimizer of the function $g$ on the open set $\{y\in\mathbb{R}^{n}\,|\,L^{-1}t<\|y\|<Lt\}.$ By the Fermat rule (Theorem 2.2), we get $0\in\partial g(\phi(t))$. This, together with Lemma 2.1, implies that for all but finitely many $t\in[0,\epsilon),$

[TABLE]

Consequently, $(g\circ\phi)(t)=(g\circ\phi)(0)=0$ for all $t\in[0,\epsilon),$ which is a contradiction.

Therefore, we have $\|\phi(t)\|\equiv rt,$ where either $r=L^{-1}$ or $r=L.$ Moreover, it holds that

[TABLE]

Combining this with (3) and (4), we can find a constant $c>0$ such that

[TABLE]

Applying the above argument again and using the first inequality in (1), we also obtain

[TABLE]

for some $c^{\prime}>0.$ Therefore, $\psi_{f}\simeq\psi_{g}.$ ∎

Remark 4.1**.**

(i) Notice that, in the above proof, we do not assume that the homeomorphism $h$ is definable.

(ii) When $f$ is of class $C^{1},$ it is not hard to see that the exponents $\alpha_{k}$ belong to the set of characteristic exponents defined by Kurdyka, Mostowski, and Parusiński [11], and moreover, the latter set is preserved by bi-Lipschitz homeomorphisms (see [9]). On the other hand, we do not know whether the set of the exponents $\alpha_{k}$ is an invariant of the bi-Lipschitz contact equivalence or not.

We conclude the paper with some examples illustrating our results. For simplicity we consider the case where $f$ is a $C^{1}$-function in two variables $(x,y)\in\mathbb{R}^{2}.$ By definition, then

[TABLE]

In view of Theorem 4.1, the four functions given below are not bi-Lipschitz contact equivalent to each other.

Example 4.1**.**

(i) Let $f(x,y):=x^{3}+y^{6}.$ The tangency variety $\Gamma(f)$ is given by the equation:

[TABLE]

Hence, for $\epsilon>0$ the set $(\Gamma(f)\cap\mathbb{B}_{\epsilon})\setminus\{0\}$ has six connected components:

[TABLE]

Consequently,

[TABLE]

It follows that $K_{0}=\emptyset,K_{-}=\{-3\},K_{+}=\{\pm 1,\pm 2,3\}$ and $\mathrm{Inv}(f)=\{-3,3\}.$

(ii) Let $f(x,y):=(x^{2}-y^{3})^{2}.$ The tangency variety $\Gamma(f)$ is given by the equation:

[TABLE]

Hence, for $0<\epsilon<\frac{2}{3},$ the set $(\Gamma(f)\cap\mathbb{B}_{\epsilon})\setminus\{0\}$ has six connected components:

[TABLE]

Consequently,

[TABLE]

It follows that $K_{0}=\{\pm 2\},K_{-}=\emptyset,K_{+}=\{\pm 1,\pm 3\}$ and $\mathrm{Inv}(f)=\{0,4\}.$

(iii) Let $f(x,y):=x^{2}+y^{4}.$ The tangency variety $\Gamma(f)$ is given by the equation:

[TABLE]

Hence, for $0<\epsilon<\sqrt{\frac{1}{2}},$ the set $(\Gamma(f)\cap\mathbb{B}_{\epsilon})\setminus\{0\}$ has four connected components:

[TABLE]

Consequently,

[TABLE]

It follows that $K_{0}=K_{-}=\emptyset,K_{+}=\{\pm 1,\pm 2\}$ and $\mathrm{Inv}(f)=\{2,4\}.$

(iv) Let $f(x,y):=-x^{2}-2y^{6}.$ The tangency variety $\Gamma(f)$ is given by the equation:

[TABLE]

Hence, for $0<\epsilon<\sqrt[4]{\frac{1}{6}},$ the set $(\Gamma(f)\cap\mathbb{B}_{\epsilon})\setminus\{0\}$ has four connected components:

[TABLE]

Consequently,

[TABLE]

It follows that $K_{0}=K_{+}=\emptyset,K_{-}=\{\pm 1,\pm 2\}$ and $\mathrm{Inv}(f)=\{-2,-6\}.$