Finitness theorem for Multi-K-Bi-Lipschtiz equivalence of map-germs

TL;DR

This paper proves that, for polynomial map germs of bounded degree, the set of equivalence classes under multi-K-bi-Lipschitz equivalence is finite, extending understanding of classification in singularity theory.

Contribution

It establishes a finiteness theorem for the classification of polynomial map germs under multi-K-bi-Lipschitz equivalence, a new result in singularity theory.

Findings

Finiteness of equivalence classes for polynomial map germs of bounded degree.

Extension of classification results to multi-K-bi-Lipschitz equivalence.

Provides a foundational result for further studies in Lipschitz geometry of singularities.

Abstract

Let $P^{k}(n,p) $ be the set of all real polynomial map germs $f = ( f_1 , ..., f_p ) : (\mathbb{R}^{n},0) \rightarrow (\mathbb{R}^{p},0)$ with degree of $f_1 , ...,f_p$ less than or equal to $ k \in \mathbb{N}$. The main result of this paper shows that the set of equivalence classes of $ P^{k}(n,p)$, with respect to multi-$\mathcal{K}$-bi-Lipschitz equivalence, is finite.

Equations

$$(f,g):({\mathbb{R}}^n,0)\to ({\mathbb{R}}^p\times {\mathbb{R}}^q,0).$$

$$({\mathbb{R}}^q,0)\stackrel{g}{⟵}({\mathbb{R}}^n,0)\stackrel{f}{⟶}({\mathbb{R}}^p,0).$$

$$\begin{array}[]{lllll}(\mathbb{R}^{n},0)&\stackrel{{\scriptstyle(id_{n},f)}}{{\longrightarrow}}&(\mathbb{R}^{n}\times\mathbb{R}^{p},0)&\stackrel{{\scriptstyle\pi_{n}}}{{\longrightarrow}}&(\mathbb{R}^{n},0)\\ \,\,\,h\,\downarrow&&\,\,\,\,H\,\downarrow&&\,\,\,h\,\downarrow\\ (\mathbb{R}^{n},0)&\stackrel{{\scriptstyle(id_{n},g)}}{{\longrightarrow}}&(\mathbb{R}^{n}\times\mathbb{R}^{p},0)&\stackrel{{\scriptstyle\pi_{n}}}{{\longrightarrow}}&(\mathbb{R}^{n},0)\\ \end{array}$$

$$\mathrm{graph}(f)=\{(x,y)\in {\mathbb{R}}^n\times \mathbb{R}|y=f(x)\}.$$

$$c_{1}f(x)\le g(x)\le c_{2}f(x).$$

$$\left.\begin{array}[]{lll}\displaystyle\frac{\partial{\theta}}{\partial x_{i}}&=&\displaystyle\frac{\left({\sum}_{{}_{j=1}}^{n}\frac{\partial g}{\partial x_{j}}(h(x))\frac{\partial h_{j}}{\partial x_{i}}(x)f(x)-\frac{\partial f}{\partial x_{i}}(x)g\circ h(x)\right)y}{{(f(x))}^{2}}\\ &&\\ &=&\displaystyle{\sum}_{{}_{j=1}}^{n}\frac{\partial g}{\partial x_{j}}(h(x))\frac{\partial h_{j}}{\partial x_{i}}(x)\frac{y}{f(x)}-\frac{\partial f}{\partial x_{i}}(x)\frac{g\circ h(x)}{f(x)}\frac{y}{f(x)}.\\ &&\\ \end{array}\right.$$

$$\parallel H(x,y)-H(u,v)\parallel \le k\parallel (x,y)-(u,v)\parallel \forall (x,y),(u,v)\in V\setminus U,$$

$$\parallel H(x_n,y_n)-H(u_m,v_m)\parallel \le k\parallel (x_n,y_n)-(u_m,v_m)\parallel ,\forall n,m\in \mathbb{N}.$$

$$\parallel H(x_0,y_0)-H(u_0,v_0)\parallel \le k\parallel (x_0,y_0)-(u_0,v_0)\parallel .$$

$$\mathcal{H}:({\mathbb{R}}^n\times {\mathbb{R}}^{q_1}\times ...\times {\mathbb{R}}^{q_p},0)\to ({\mathbb{R}}^n\times {\mathbb{R}}^{q_1}\times ...\times {\mathbb{R}}^{q_p},0),$$

$$\mathcal{H}(x,y_1,...,y_p)=(h(x),\widetilde{H_1}(x,y_1),...,\widetilde{H_p}(x,y_p))$$

$$({\mathbb{R}}^n,0)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$c_1|f_1(x)|\le \parallel H(x,f_1(x),...,f_p(x))-H(x,0,f_2(x),...,f_p(x))\parallel \le c_2|f_1(x)|.$$

$$\parallel H(x,f_1(x),f_2(x),...,f_p(x))-H(x,0,f_2(x),...,f_p(x))\parallel =$$

$$=\parallel (h(x),g_1\circ h(x),...,g_p\circ h(x))-H(x,0,f_2(x),...,f_p(x))\parallel \ge |g_1\circ h(x)|.$$

$$|g_1\circ h(x)|\ge {\tilde{c}}_1|f_1(x)|,$$

$$sign(f_1(x))=sign(g_1\circ h(x))\mathrm{or}sign(f_1(x))=-sinal(g_1\circ h(x)).$$

$$f_i(x)\approx g_i\circ h(x)\text{mbox}or{f}_i(x)\approx -g_i\circ h(x).$$

$$\mathcal{H}(x,y_1,...,y_p)=(h(x),{\tilde{H}}_1(x,y_1),...,{\tilde{H}}_p(x,y_p)).$$

$$X_F=\{{\mathbb{R}}^n\times {\{0\}}^p,{\mathbb{R}}^n\times {\mathbb{R}}^{p-1}\times \{0\},{\mathbb{R}}^n\times {\mathbb{R}}^{p-2}\times \{0\}\times \mathbb{R},\dots ,{\mathbb{R}}^n\times \{0\}\times {\mathbb{R}}^{p-1},graf(F)\}.$$

$$\mathcal{F}=\{X_{F_1},X_{F_2},\dots ,X_{F_i},\dots \}.$$

$$H=(h,\tilde{H}),$$

Full text

*Research supported under CNPq 302655/2014-0 grant and by Capes-Cofecub

**Research supported by FAPESP and CAPES

***Research supported by Capes

1. Introduction

Given an equivalence relation to classify map germs in the context of Singularity theory, one initial problem is the following:

Problem. To decide if the classification under investigation has or not countable number of equivalence classes, becoming finite under some restrictions.

If the answer is affirmative, we say that the classification is tame or has the finiteness property. The finiteness property is an initial step in any possible attempt of the understanding the classification of singularities.

Let $P^{k}(n,p)$ be the set of all real polynomial map germs $f=(f_{1},...,f_{p}):(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{p},0)$ with degree of $f_{1},...,f_{p}$ less than or equal to $k\in\mathbb{N}$. For some equivalence relations the finitness property does not hold. Henry and Parusinski [5] showed existence of moduli for $\mathcal{R}$-bi-Lipschitz equivalence of analytic functions. By other hand, in [1] the authors showed that with respect to $\mathcal{K}$-bi-Lipschitz equivalence, the finiteness property holds for the set of all real polynomial map germs with bounded degree. Later on, Ruas and Valette [7] showed a more general result about the finiteness of Lipschitz types with respect to $\mathcal{K}$-bi-Lipschitz equivalence of map germs.

In this paper we introduce the notion of multi-$\mathcal{K}$-bi-Lipschitz equivalence to investigate the finiteness property in the set $P^{k}(n,p)$. This equivalence is closed related with the notion of contact equivalence for $q$-tuple of map germs introduced by Sitta in [8]. The approach of [8] was motivated by Dufour’s work [4].

A pair (or couple) of map-germs can be defined as follows:

[TABLE]

It can be seen also as a divergent diagram

[TABLE]

Divergent diagrams appear in several geometrical problems and it have many applications.

Recently, in [2] the authors showed the finiteness property for the called bi-$C^{0}$-$\mathcal{K}$-equivalence of pairs of map germs. This equivalence relation is the topological version of topological contact equivalence adopted to a pair of map germs. An overview of the theory involving classical equivalence relations of pairs of map germs can be found in [3].

The main result of this paper shows that the set of equivalence classes of $P^{k}(n,p)$, with respect to multi-$\mathcal{K}$-bi-Lipschitz equivalence, is finite. As a consequence, we obtain the finiteness property with respect to $\mathcal{K}$-bi-Lipschitz equivalence.

2. Preliminaries and notations

Definition 2.1**.**

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

[TABLE]

where $id_{n}:(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{n},0)$ is the identity map germ of $\mathbb{R}^{n}$, $\pi_{n}:(\mathbb{R}^{n}\times\mathbb{R}^{p},0)\rightarrow(\mathbb{R}^{n},0)$ is the canonical projection germ and $\{0\}^{p}=(0,\dots,0)\in\mathbb{R}^{p}$.

When $h=id_{n}$, $f$ and $g$ are said to be $\mathcal{C}$-bi-Lipschitz equivalent.

In other words, two map germs $f$ and $g$ are $\mathcal{K}$-bi-Lipschitz equivalent if there exists a germ of bi-Lipschitz map $H:(\mathbb{R}^{n}\times\mathbb{R}^{p},0)\rightarrow(\mathbb{R}^{n}\times\mathbb{R}^{p},0)$ such that $H(x,y)$ can be written in the form $H(x,y)=(h(x),\theta(x,y))$, $x\in\mathbb{R}^{n},y\in\mathbb{R}^{p}$, where $h$ is also a bi-Lipschitz map germ, such that $\theta(x,0)=0$ and $H$ maps the germ of the graph$(f)$ onto the graph$(g)$. Recall that graph$(f)$ is the set defined as follows:

[TABLE]

Definition 2.2**.**

Two functions $f,g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ are called of the same contact at a point $x_{0}\in\mathbb{R}^{n}$ if there exist a neighborhood $U_{x_{0}}$ of $x_{0}$ in $\mathbb{R}^{n}$ and two positive numbers $c_{1}$ and $c_{2}$ such that, for all $x\in U_{x_{0}}$, we have

[TABLE]

We use the notation: $f\approx g$.

The next Lemma is an adaptation of Theorem 2.4 given in [1]:

Lemma 2.3**.**

Let $f,g:(\mathbb{R}^{n},0)\rightarrow(\mathbb{R},0)$ be two germs of Lipschitz functions. Suppose that there exists a germ of bi-Lipschitz homeomorphism $h:(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{n},0)$ such that one of the following two conditions is true:

i) $f\approx g\circ h$ or

ii) $f\approx-g\circ h$.

Then, $f$ and $g$ are $\mathcal{K}$-bi-Lipschitz equivalent.

Proof.

Suppose $f\approx g\circ h$ (the case $f\approx-g\circ h$ is analogous). Let $H:(\mathbb{R}^{n}\times\mathbb{R},0)\rightarrow(\mathbb{R}^{n}\times\mathbb{R},0)$ given by

$H(x,y)=$ \left\{\begin{array}[]{ll}(h(x),0)&{\rm if}\,\,\,y=0\\ (h(x),\frac{{g}\circ h(x)y}{f(x)})&{\rm if}\,\,\,0<|y|\leq|f(x)|\\ (h(x),y-f(x)+g\circ h(x))&{\rm if}\,\,\,0<|f(x)|\leq|y|\,\,\,\,\,\mbox{and}\,\,\,\,\,sign(y)=sign(f(x))\\ (h(x),y+f(x)-g\circ h(x))&{\rm if}\,\,\,0<|f(x)|\leq|y|\,\,\,\,\,\mbox{and}\,\,\,\,\,sign(y)=-sign(f(x))\\ (h(x),y)&{\rm if}\,\,\,f(x)=0.\\ \end{array}\right.

The map $H(x,y)=(h(x),\theta(x,y))$ defined above is bi-Lipschitz. In fact, $H$ is injective because, for any fixed $x^{*}$, we can show that $\theta(x^{*},y)$ is a continuous and monotone function. Moreover, $H$ is Lipschitz if $0<|f(x)|\leq|y|.$ Let us show that $H$ is Lipschitz if $0<|y|\leq|f(x)|.$ Hence it is sufficient to show that all the partial derivatives of $H$ exist and are bounded in their domain up to a set of measure zero. Since $h$ is a bi-Lipschitz homeomorphism, follows that all its partial derivatives $\displaystyle\frac{\partial{h}}{\partial x_{i}}$ exist and are bounded in almost every $x$ near $0\in\mathbb{R}^{n}$, $i=1,\dots,n$. Thus, it is necessary to check only the partial derivatives of $\theta$. Then,

[TABLE]

Observe that,

i) $\displaystyle\frac{\partial h_{j}}{\partial x_{i}}(x)$ is bounded up to a set of measure zero, for all $\ i,j=1,...,n$;

ii) since $0<|y|\leq|f(x)|$ then $\displaystyle\frac{y}{f(x)}$ is bounded;

iii) since $f\approx g\circ h$, the expression $\frac{g\circ h(x)}{f(x)}$ is bounded;

iv) $\displaystyle\frac{\partial g}{\partial x_{j}},\frac{\partial f}{\partial x_{i}}$ are bounded for all $i,j=1,...,n,$ because $f$ and $g$ are Lipschitz.

From (i)-(iv) we can conclude the $\displaystyle\frac{\partial{\theta}}{\partial x_{i}}$ exist and it is bounded up to a set of measure zero, for all $i=1,\dots,n$. Observe that the partial derivative of $\theta$ with respect to $y\in\mathbb{R}$ is also bounded.

Since the map $H$ is Lipschitz outside of the set of the measure zero, the following expression holds:

[TABLE]

where $V$ is an open neighborhood of the origin in $(\mathbb{R}^{n}\times\mathbb{R},0)$, $U$ is a set of measure zero and $k$ is a real constant positive. We need to show that the last inequality remains valid for all $(x,y),(u,v)\in V$. In fact, take $(x_{0},y_{0}),(u_{0},v_{0})\in V\cap U.$ Since $U$ is a set of measure zero, there exist sequences $(x_{n},y_{n}),(u_{m},v_{m})\in V\setminus U$ such that $(x_{n},y_{n})\rightarrow(x_{0},y_{0})$ e $(u_{m},v_{m})\rightarrow(u_{0},v_{0})$.

Moreover,

[TABLE]

Since $H$ is a continuous map, taking $n\rightarrow\infty$ and then $m\rightarrow\infty$ follows that

[TABLE]

Hence, $H$ is Lipschitz in all $V$.

Since $H^{-1}$ can be constructed in the same form as $H$, we conclude that $H^{-1}$ is also Lipschitz. Then, $H$ is bi-Lipschitz. Moreover, by construction of $H$, follows that:

i) $H(x,f(x))=(h(x),g\circ h(x))$ and

ii) $H(\mathbb{R}^{n}\times\{0\})=\mathbb{R}^{n}\times\{0\}$.

Hence $f$ and $g$ are $\mathcal{K}$-bi-Lipschitz equivalent. ∎

Definition 2.4**.**

Let $q=(q_{1},q_{2},...,q_{p})\in\mathbb{N}^{p}$. A $(p,q)$-multi pair of map germs is a family of $p$ maps $\{F_{1},\dots,F_{p}\}$, where $F_{i}:(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{q_{i}},0)$, $i=1,\dots,p$ is a map germ. A $(p,q)$-multi pair of map germs can be considered as a map germ $F=(F_{1},...,F_{p}):(\mathbb{R}^{n},0)\to(\mathbb{R}^{q_{1}}\times\mathbb{R}^{q_{2}}\times\dots\times\mathbb{R}^{q_{p}},0)$, where $q$ is called multi index of a $(p,q)$-multi pair of map germs.

Remark 2.5**.**

A map germ $f:(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0)$ can be considered as a $(p,q)$-multi pair of function germs, i.e., take $q=(1,\dots,1)$ $p$-times.

Definition 2.6**.**

A family of $p+1$ germs of bi-Lipschitz homeomorphisms of type $(h,H_{1},...,H_{p}),$ where $h:(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{n},0)$ and $H_{i}:(\mathbb{R}^{n}\times\mathbb{R}^{q_{i}},0)\rightarrow(\mathbb{R}^{n}\times\mathbb{R}^{q_{i}},0)$ is called a contact family if $H_{i}(x,y_{i})=(h(x),\tilde{H_{i}}(x,y_{i}))$ for all $i=1,...,p$, $x\in\mathbb{R}^{n}$, $y_{i}\in\mathbb{R}^{q_{i}}$.

Definition 2.7**.**

Let $q=(q_{1},...,q_{p})$ be a multi index. A $(p,q)$-multi pair of bi-Lipschitz homeomorphisms is a germ of a bi-Lipschitz homeomorphism, generated by a contact family of $p+1$ bi-Lipschitz homeomorphisms in the form

[TABLE]

given by

[TABLE]

where $\tilde{H_{i}}$, $i=1,\dots,p$, are as in Definition 2.6.

We call the bi-Lipschitz map germ $h$ of a common factor of the $(p,q)$-multi pair of bi-Lipschitz homeomorphisms.

Definition 2.8**.**

Let $q=(q_{1},...,q_{p})$ be a multi index. Two $(p,q)$-multi pairs of map germs $F,G:(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{q_{1}}\times...\times\mathbb{R}^{q_{p}},0)$ are said to be multi-$\mathcal{K}$-bi-Lipschitz equivalent if there exists a $(p,q)$-multi pair of bi-Lipschitz homeomorphism $\mathcal{H}:(\mathbb{R}^{n}\times\mathbb{R}^{q_{1}}\times...\times\mathbb{R}^{q_{p}},0)\rightarrow\linebreak(\mathbb{R}^{n}\times\mathbb{R}^{q_{1}}\times...\times\mathbb{R}^{q_{p}},0)$ as in Definition 2.7, such that the following diagram is commutative:

[TABLE]

where $id_{n}$ is the identity map germ of $\mathbb{R}^{n}$; $\pi_{n}$ is the usual projection in $\mathbb{R}^{n}$; $h$ is the common factor of $(p,q)$-multi pair of bi-Lipschitz homeomorphisms and for all $i=1,...,p$ we have $\tilde{H}_{i}(\mathbb{R}^{n}\times{\{0\}}^{q_{i}})=\mathbb{R}^{n}\times{\{0\}}^{q_{i}},$

Remark 2.9**.**

Let $q=(q_{1},...,q_{p})$ be the multi index and let $F=(F_{1},...,F_{p}),G=(G_{1},...,G_{p}):(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{q_{1}}\times...\times\mathbb{R}^{q_{p}},0)$ be two $(p,q)$-multi pairs of map germs. Then:

i) When $q=q_{1}\in\mathbb{N}$, the definition of multi-$\mathcal{K}$-bi-Lipschitz equivalence coincides with the definition of $\mathcal{K}$-bi-Lipschitz equivalence.

ii) If $F$ and $G$ are multi-$\mathcal{K}$-bi-Lipschitz equivalent (as multi pair of map germs) then $F$ and $G$ are $\mathcal{K}$-bi-Lipschitz equivalent (as map germs).

2. If $F$ and $G$ are multi-$\mathcal{K}$-bi-Lipschitz equivalent, then the map germs $F_{i}$ and $G_{i}$ are $\mathcal{K}$-bi-Lipschitz equivalent, for all $i=1,...,p.$

3. Main results and finiteness property

Theorem 3.1**.**

Let $f=(f_{1},...,f_{p}),g=(g_{1},...,g_{p}):(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{p},0)$ be two germs of Lipschitz maps. Suppose there exists a bi-Lipschitz homeomorphism $H:(\mathbb{R}^{n}\times\mathbb{R}^{p},0)\rightarrow(\mathbb{R}^{n}\times\mathbb{R}^{p},0)$ such that:

i) The sets  $\mathbb{R}^{n}\times{\{0\}}^{p},\mathbb{R}^{n}\times\mathbb{R}^{p-1}\times\{0\},\mathbb{R}^{n}\times\mathbb{R}^{p-2}\times\{0\}\times\mathbb{R},...,\mathbb{R}^{n}\times\{0\}\times\mathbb{R}^{p-1}$ are invariant under $H$. In other words, $H$ satisfies

$H(\mathbb{R}^{n}\times{\{0\}}^{p})=\mathbb{R}^{n}\times{\{0\}}^{p},H(\mathbb{R}^{n}\times\mathbb{R}^{p-1}\times\{0\})=\mathbb{R}^{n}\times\mathbb{R}^{p-1}\times\{0\},\dots,\linebreak H(\mathbb{R}^{n}\times\{0\}\times\mathbb{R}^{p-1})=\mathbb{R}^{n}\times\{0\}\times\mathbb{R}^{p-1}.$**

ii) $H(graph(f))=graph(g)$.

Then, the germs $f$ and $g$ are multi-$\mathcal{K}$-bi-Lipschitz-equivalent.

To prove the Theorem 3.1 we need some preliminary results.

Let $f=(f_{1},...,f_{p}),g=(g_{1},...,g_{p}):(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{p},0)$ be two germs of Lipschitz maps and suppose there exists a germ of bi-Lipschitz homeomorphism $H:(\mathbb{R}^{n}\times\mathbb{R}^{p},0)\rightarrow(\mathbb{R}^{n}\times\mathbb{R}^{p},0)$, satisfying the conditions of Theorem 3.1. Define the following subsets of $\mathbb{R}^{n}\times\mathbb{R}^{p}$:

$V_{k}^{+}=\{(x,y_{1},...,y_{p})\in\mathbb{R}^{n}\times\mathbb{R}^{p}\mid y_{k}>0\}$   and

$V_{k}^{-}=\{(x,y_{1},...,y_{p})\in\mathbb{R}^{n}\times\mathbb{R}^{p}\mid y_{k}<0\}$, $k=1,\dots,p$.

Assertion 1. For each $k=1,\dots,p$, one of the following conditions holds:

i) $H(V_{k}^{+})=V_{k}^{+}$   and   $H(V_{k}^{-})=V_{k}^{-}$ or

ii) $H(V_{k}^{+})=V_{k}^{-}$   and   $H(V_{k}^{-})=V_{k}^{+}$,

Proof.

If the condition does not hold, there are points $a,b,c,d\in\mathbb{R}^{n}\times\mathbb{R}^{p}$, such that $a\in V_{k}^{+}$, $b,c,d\in V_{k}^{-}$ with $H(a)=b,H(c)=d$. Consider a path $\lambda:[0,1]\rightarrow V_{k}^{-}$ connecting the points $b$ and $d$. Therefore, $H^{-1}\circ\lambda$ is a path in $\mathbb{R}^{n}\times\mathbb{R}^{p}$, connecting the points $a$ and $c$ passing by $\mathbb{R}^{n}\times\mathbb{R}^{k-1}\times\{0\}\times\mathbb{R}^{p-k}$. However, this is not possible because $H(\mathbb{R}^{n}\times\mathbb{R}^{k-1}\times\{0\}\times\mathbb{R}^{p-k})=\mathbb{R}^{n}\times\mathbb{R}^{k-1}\times\{0\}\times\mathbb{R}^{p-k}$ and $H(H^{-1}\circ\lambda)=\lambda\subset V_{k}^{-}$. ∎

Assertion 2. Let $h:(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{n},0)$ be defined by $h(x)={\pi}_{n}(H(x,f(x)))$. Then $h$ is a bi-Lipschitz map germ.

Proof.

Since $g$ is a Lipschitz map, the projection ${{\pi}_{n}}_{{\mid}_{graf(g)}}$ is a bi-Lipschitz map. By the same argument, a map $x\mapsto(x,f(x))$ is bi-Lipschiz. By hypothesis, the map $H$ is bi-Lipschitz. Hence $h$ is bi-Lipschitz. ∎

Assertion 3. One of the following assertions is true:

i) $f_{i}(x)\approx\ g_{i}\circ h(x)$ or

ii) $f_{i}(x)\approx-\ g_{i}\circ h(x)$, for all $i=1,...,p.$

Proof.

Since $H$ is bi-Lipschitz, there exists two positives numbers $c_{1}$ and $c_{2}$, such that

[TABLE]

By above construction,

[TABLE]

[TABLE]

Therefore, $|g_{1}\circ h(x)|\leq c_{2}|f_{1}(x)|.$ Using the same procedure for the map $H^{-1}$, we obtain that

[TABLE]

where $\tilde{c}_{1}$ is a a real positive number.

Then, $\tilde{c}_{1}|f_{1}(x)\mid\ \leq\ |g_{1}\circ h(x)|\ \leq\ c_{2}|f_{1}(x)|.$

By Assertion 3 we have that for all $x\in\mathbb{R}^{n}$,

[TABLE]

Therefore,  $f_{1}(x)\approx\ g_{1}\circ h(x)\ \mbox{ or }\ f_{1}(x)\approx\ -g_{1}\circ h(x)$.

Repeating the same process for all $i=1,\dots,p$, we obtain

[TABLE]

∎

Proof of Theorem 3.1.

By Assertion 3 and Lemma 2.3, follows that $f_{i}$ and $g_{i}$ are $\mathcal{K}$-bi-Lipschitz equivalent, for all $i=1,...,p.$ Then, for each $i=1,\dots,p$, there exist a germ of bi-Lipschitz homeomorphisms $H_{i}:(\mathbb{R}^{n}\times\mathbb{R},0)\rightarrow(\mathbb{R}^{n}\times\mathbb{R},0)$, such that

$i)\ H_{i}(x,y_{i})=(h(x),\tilde{H}_{i}(x,y_{i})),$ with $\tilde{H}_{i}:(\mathbb{R}^{n}\times\mathbb{R},0)\to(\mathbb{R},0)$, $i=1,...,p.$

$ii)\ H_{i}(x,f_{i}(x))=(h(x),g_{i}\circ h(x))$, $i=1,...,p.$

$iii)\ H_{i}(\mathbb{R}^{n}\times\{0\})=\mathbb{R}^{n}\times\{0\}$, $i=1,...,p.$

Define the map $\mathcal{H}:(\mathbb{R}^{n}\times\mathbb{R}^{p},0)\rightarrow(\mathbb{R}^{n}\times\mathbb{R}^{p},0)$ given by

[TABLE]

Considering $q=(1,\dots,1)$ $p$-times, we obtain that $\mathcal{H}$ is a $(p,q)$-multi pair of bi-Lipschitz homomorphism, generated by the contact family of bi-Lipschitz homeomorphisms, $\{h,H_{1},...,H_{p}\}$. Moreover,

$i)\ \mathcal{H}(x,f_{1}(x),...,f_{p}(x))=(h(x),g_{1}\circ h(x),...,g_{p}\circ h(x))$ and

$ii)\ \mathcal{H}(\mathbb{R}^{n}\times{\{0\}}^{p})=\mathbb{R}^{n}\times{\{0\}}^{p}$.

Hence, $f$ and $g$ are multi-$\mathcal{K}$-bi-Lipschitz equivalent. ∎

Theorem 3.2**.**

(Finiteness theorem) Let $P^{k}(n,p)$ be the set of all real polynomial map germs $f=(f_{1},...,f_{p}):(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{p},0),$ with degree of $f_{1},...,f_{p}$ less than or equal to $k\in\mathbb{N}$. Then the set of the equivalence classes of $P^{k}(n,p)$, with respect to multi-$\mathcal{K}$-bi-Lipschitz-equivalence is finite.

Proof.

Let $F:(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0)\in P^{k}(n,p)$. We associate to $F$ the following family of algebraic subsets:

[TABLE]

Consider the set of all the families of algebraic subsets defined as above, that is,

[TABLE]

We define the following equivalence relation in $\mathcal{F}$:

$X_{F_{i}}$ and $X_{F_{j}}$ are called multi-$\mathcal{V}$-bi-Lipschitz equivalent if there exists a bi-Lipschitz homeomorphism $H:(\mathbb{R}^{n}\times\mathbb{R}^{p},0)\rightarrow(\mathbb{R}^{n}\times\mathbb{R}^{p},0)$ satisfying the conditions i) and ii) of Theorem 3.1.

By Vallete Lipschitz Triviality Theorem (cf. [6]), the number of equivalence classes with respect to multi-$\mathcal{V}$-bi-Lipschitz equivalence, is finite. Applying the Theorem 3.1 follows that $F_{i}$ and $F_{j}$ are multi-$\mathcal{K}$-bi-Lipschitz equivalent and so we conclude the proof of theorem. ∎

Corollary 3.3**.**

The set of the equivalence classes of $P^{k}(n,p)$, with respect to $\mathcal{K}$-bi-Lipschitz equivalence, is finite.

Proof.

Let $F,G\in P^{k}(n,p)$. By Remark 2.5, $F$ and $G$ can be considered a $(p,q)$-multi pair of function germs (through its coordinate functions). Since multi-$\mathcal{K}$-bi-Lipschitz equivalence admits the finiteness property, we can suppose that the corresponding families of coordinate functions are multi-$\mathcal{K}$-bi-Lipschitz equivalent. Then one can consider a map $H:(\mathbb{R}^{n}\times\mathbb{R}^{p},0)\rightarrow(\mathbb{R}^{n}\times\mathbb{R}^{p},0)$ defined as follows:

[TABLE]

where $h$ and $\tilde{H}=(\tilde{H}_{1},\dots,\tilde{H}_{p})$ such that $h$ and $\tilde{H}_{i}$ are the maps obtained from the corresponding multi-$\mathcal{K}$-bi-Lipschitz equivalence of the coordinate functions. Then $h$ and $H$ are bi-Lipschitz homeomorphisms and satisfy the conditions of Definition 2.1 for $F$ and $G$. Hence, $F$ and $G$ are $\mathcal{K}$-bi-Lipschitz equivalent.

Since in the proof we use the finiteness property of the multi-$\mathcal{K}$-bi-Lipschitz equivalence we have that the number of $\mathcal{K}$-bi-Lipschitz classes is also finite. ∎