#### 2.1. Preliminaries

Let us recall some basic information on group actions. For a more detailed description, see [

21,

22]. Let

G be a group. Recall that a

G-set is a pair

$(X,{\xi}_{X})$, where

X is a set and

${\xi}_{X}:G\times X\to X$ is the action of

G on

X, i.e., a map such that:

- (i)
${\xi}_{X}({g}_{1},\xi ({g}_{2},x))={\xi}_{X}({g}_{1}{g}_{2},x)$ for ${g}_{1},{g}_{2}\in G$ and $x\in X$;

- (ii)
${\xi}_{X}(e,x)=x$ for $x\in X$, where $e\in G$ is the group unit.

In the sequel, we write $gx$ instead of ${\xi}_{X}(g,x)$, $x\in X$, $g\in G$, unless it leads to ambiguity.

Given G-sets X and Y, a map $f:X\to Y$ is G-equivariant if $f(gx)=gf(x)$ for any $x\in X$ and $g\in G$. If the G-action on Y is trivial, then we say that f is G-invariant.

A subset $A\subset X$ of a G-set X is G-invariant if $gA:=\{gx\mid x\in A\}\subset A$ for all $g\in G$. The set $Gx:=\{gx\mid g\in G\}$ is called the orbit through $x\in X$, and $X/G$ denotes the set of all orbits. Observe that if $A\subset X$, then the set $GA:={\bigcup}_{x\in A}Gx={\bigcup}_{g\in G}gA$ is G-invariant.

If G is a topological group, X is a topological space and a G-set, then X is a G-space, provided the action ${\xi}_{X}$ is (jointly) continuous. We say that a real (resp. complex) Banach space $\mathbb{E}$ is a real (resp. complex) Banach representation of G if $\mathbb{E}$ is a G-space, and for each $g\in G$, the map ${\xi}_{\mathbb{E}}(g,\xb7):\mathbb{E}\ni x\mapsto gx$ is linear and bounded. Throughout the whole paper, we assume that G is a compact Lie group.

Set-valued maps are the main object of our studies. Recall that given sets X and Y, a set-valued map $\phi $ from X into Y (written $\phi :X\u22b8Y$) is a map that assigns to each $x\in X$ the value $\phi (x)$, being a non-empty subset of Y. If X and Y are topological spaces and, for any closed (resp. open) set $U\subset Y$, the preimage ${\phi}^{-1}(U):=\{x\in X\mid \phi (x)\cap U\ne \varnothing \}$ is closed (resp. open), then we say that $\phi $ is upper (resp. lower) semicontinuous; $\phi $ is continuous if it is upper and lower semicontinuous simultaneously.

If Y is a metric space, then $\phi :X\u22b8Y$ is lower semicontinuous if and only if for any $y\in Y$, the function $X\ni x\mapsto d(y,\phi (x)):={inf}_{z\in \phi (x)}d(y,z)$ is upper semicontinuous (as a real function) or, equivalently, given ${x}_{0}\in X$ and ${y}_{0}\in \phi ({x}_{0})$, ${lim}_{x\to {x}_{0}}d({y}_{0},\phi (x))=0$.

A similar characterization of upper semicontinuity is not true, i.e., the lower semicontinuity of

$d:X\ni x\mapsto d(y,\phi (x))\in \mathbb{R}$ does not imply in general that

$\phi $ is upper semicontinuous. However, if

$\phi $ has closed values, is locally compact, i.e., each point

$x\in X$ has a neighbourhood

U such that

$\overline{\phi (U)}$ is compact, and

d is lower semicontinuous, then

$\phi $ is upper semicontinuous (with compact values). The graph

$\mathrm{Gr}\phantom{\rule{0.166667em}{0ex}}(\phi ):=\{(x,y)\in X\times Y\mid y\in \phi (x)\}$ of an upper semicontinuous map

$\phi $ with closed values is closed;

$\phi :X\u22b8Y$ is upper semicontinuous with compact values if and only if the projection

$\mathrm{Gr}\phantom{\rule{0.166667em}{0ex}}(\phi )\to X$ is perfect (recall that a continuous map

$f:X\to Y$ is perfect if it is closed and

${f}^{-1}(y)$ is compact for any

$y\in Y$). We say that a map

$\phi $ is compact if it is upper semicontinuous and the closure of the image

$\phi (X):={\bigcup}_{x\in X}\phi (x)$ is compact. For other details on set-valued maps, see [

23] or [

18].

Let $X,Y$ be metric spaces and $\epsilon >0$. Recall the notion of graph approximations.

**Definition** **1.** A continuous map $f:X\to Y$ is an ε-approximation of $\phi :X\u22b8Y$ if for every x, there exists ${x}^{\prime}$ such that $d(x,{x}^{\prime})<\phi $ and $f(x)\in {B}_{\epsilon}(\phi ({x}^{\prime}))$.

One can formulate the above condition as follows:

$f(x)\in {B}_{\epsilon}(\phi ({B}_{\epsilon}(x)))$ for all

$x\in X$. It is also equivalent to the condition that the graph of

f is contained in the

$\epsilon $-neighbourhood of the graph of

$\phi $. It is well known that such approximations are a good tool for extending topological invariants, and they have been proven to exist for example for convex-valued u.s.c. maps; see [

5]. We are interested in equivariant versions of the results.

**Definition** **2.** Let X and Y be G-sets. A set-valued map $\phi :X\u22b8Y$ is G-equivariant (resp. G-invariant) if $\phi (gx)=g\phi (x)$ (resp. $\phi (gx)=\phi (x)$) for all $g\in G$ and $x\in X$.

Note that $\phi $ is G-equivariant if and only if $\phi (gx)\subset g\phi (x)$ for all $g\in G$ and $x\in X$ (or $g\phi (x)\subset \phi (gx)$ for all $g\in G$ and $x\in X$). Moreover, it is easy to see that $\phi $ is G-equivariant if and only if its graph $\mathrm{Gr}\phantom{\rule{0.166667em}{0ex}}(F)$ is a G-invariant subset of $X\times Y$ with a natural action $g(x,y):=(gx,gy)$, $x\in X$, $y\in Y$. Observe that if $\phi :X\u22b8Y$, where Y is a topological space, is G-equivariant, then so is its closure, i.e., the map $\overline{\phi}:X\u22b8Y$ given by $\overline{\phi}(x):=\overline{\phi (x)}$, $x\in X$.

Let us collect simple examples of G-equivariant set-valued maps.

**Example** **1.** (1) Let $\phi :[0,T]\times U\u22b8\mathbb{E}$, where $T>0$ and $U\subset \mathbb{E}$ is an open G-invariant subset of a Banach G-representation $\mathbb{E}$, be G-equivariant, i.e., $\phi (t,gx)=g\phi (t,x)$ for $0\le t\le T$ and $x\in U$. Consider a differential inclusion (under suitable assumptions assuring the existence of solutions): Consider the space $C([0,T],\mathbb{E})$ of continuous maps from $[0,T]$ to $\mathbb{E}$ with the G-action defined by $(g,x)\mapsto gx$, where $(gx)(t):=g(x(t))$ for $x\in C([0,T],\mathbb{E})$, $g\in G$ and $t\in [0,T]$. If $x:[0,T]\to \mathbb{E}$ is a solution to the above problem, i.e., there is an integrable function $y:[0,T]\to \mathbb{E}$ such that $x(t)={x}_{0}+{\int}_{0}^{t}y(s)\phantom{\rule{0.166667em}{0ex}}ds$, $t\in [0,T]$, then $gx$ is also a solution to this problem with the initial condition $g{x}_{0}$. Therefore, the solution map $P:U\u22b8C([0,T],\mathbb{E})$ that assigns to each initial value ${x}_{0}\in U$ the set of all solutions is G-equivariant, whenever well defined.

(2) Let $f:\mathbb{E}\to \mathbb{R}\cup \{\infty \}$, where $\mathbb{E}$ is a real Banach representation of G, be a convex function. For each ${x}_{0}\in dom(f):=\{x\in \mathbb{E}\mid f(x)<\infty \}$, the subdifferential:is defined. Let f be G-invariant. Then, the map $\partial f:dom(f)\u22b8{\mathbb{E}}^{*}$ is G-equivariant. Clearly, $dom(f)$ is invariant, and if $p\in \partial f({x}_{0})$, then for all $x\in \mathbb{E}$, $\langle p,x-{x}_{0}\rangle \le f(x)-f({x}_{0})$. Hence:which gives the assertion. In a similar manner, one shows that the Clarke generalized gradient $\partial f:U\to {\mathbb{E}}^{*}$, where U is a G-invariant open in $\mathbb{E}$ and $f:U\to \mathbb{R}$ is a G-invariant locally Lipschitz function, is G-equivariant. #### 2.2. Gradient approximations

In

${\mathbb{R}}^{n}$, we shall use various analytic characterizations of graph-approximations. For

$A\in {\mathbb{R}}^{n}$, we recall the notion of a support function

${\sigma}_{A}:{\mathbb{R}}^{n}\to \mathbb{R}\cup \{\infty \}$ given by:

One easily checks that ${\sigma}_{A}$ is finite and continuous if A is bounded. Moreover, $\overline{\mathrm{conv}}A=\{x\in {\mathbb{R}}^{n}\mid \forall v\in {\mathbb{R}}^{n}\langle x,v\rangle \le {\sigma}_{A}(v)\}$.

We also define:

or equivalently:

**Lemma** **1.** ([19], Lemma 2.2) Let $U\subset {\mathbb{R}}^{n}$, $\phi :U\u22b8{\mathbb{R}}^{n}$ with closed convex values, $g:U\to {\mathbb{R}}^{n}$, and $\epsilon :U\to (0,+\infty )$. The following conditions are equivalent: - (i)
g is an ε-approximation of φ;

- (ii)
for every $x\in U$, there is $\overline{x}\in U\cap {B}_{\epsilon (x)}(x)$ such that: - (iii)
for every $x\in U$, there is $\overline{x}\in U\cap {B}_{\epsilon (x)}(x)$ such that:

Let

$U\subset {\mathbb{R}}^{n}$ be an open subset, and consider a locally Lipschitz function

$f:U\to \mathbb{R}$. A generalized directional Clarke derivative:

is defined for all

$x\in U,v\in {\mathbb{R}}^{n}$. Clarkés generalized gradient at

x is the set:

It is well known that the map

$\partial f:U\u22b8{\mathbb{R}}^{n}$ is an u.s.c. mapping with compact convex values (see [

17] for this and more of the properties of the generalized gradient). One observes also that

${\sigma}_{\partial f(x)}(v)={f}^{o}(x;v)$ for

$v\in {\mathbb{R}}^{n}$. Recall only the local u.s.c. property:

**Definition** **3.** The map $\partial f$ is upper semicontinuous at $x\in X$ if: The following approximation result was proved in [

19]:

**Theorem** **1.** Let $f:U\to \mathbb{R}$ be locally Lipschitz and $\epsilon :U\to (0,\infty )$ a continuous function. Then, there exists a ${C}^{\infty}$-map $g:U\to \mathbb{R}$ such that:

- (i)
$\phantom{\rule{1.em}{0ex}}|f(x)-g(x)|<\epsilon (x)$ for all $x\in U$,

- (ii)
$\nabla g(x)\in {B}_{\epsilon (x)}(\partial f({B}_{\epsilon (x)}(x)))$ for all $x\in U$.

In other words, g is a uniform approximation of f, and $\nabla g$ is an $\epsilon $-approximation of $\partial f$. We shall call such pairs $(g,\nabla g)$ Whitney approximations.

**Definition** **4.** Suppose $f:X\to \mathbb{R}$, where $X\subset {\mathbb{R}}^{n}$ compact, is a locally Lipschitz function. If $g\in {C}^{\infty}(X,\mathbb{R})$ satisfies:

- 1.
$|f(x)-g(x)|\le \epsilon (x)$ for all $x\in {\mathbb{R}}^{n}$ (uniform ε-approximation),

- 2.
$\nabla g(x)\in {B}_{\epsilon (x)}(\partial f({B}_{\epsilon (x)}(x)))$ for all $x\in {\mathbb{R}}^{n}$ (ε-approximation (on the graph)),

for a continuous function $\epsilon :X\to (0,\infty )$, then we say that $(g,\nabla g)$ is the ε-WT-approximation (ε-Whitney-type-approximation) of $(f,\partial f)$.

Let us assume that V is an n-dimensional orthogonal representation of a compact Lie group G and $U\subset V$ is open and G-invariant. Assume that a locally Lipschitz function $f:U\to \mathbb{R}$ is G-invariant, i.e., $f(hx)=f(x)$ for all $h\in G$. It is easy to verify that $\partial f$ is then equivariant, i.e., $\partial (gx)=g(\partial (x))$ as sets.

We now modify the proof from [

19] in order to obtain an equivariant version of Theorem 1.

Let us start from the remark that since

G acts orthogonally, there exists an invariant

${C}^{\infty}$-mollifier, i.e., a nonnegative invariant function

$\omega :{\mathbb{R}}^{n}\to \mathbb{R}$ with the support

$\mathrm{supp}\phantom{\rule{0.166667em}{0ex}}\omega \subset \overline{{B}_{1}}(0)$ and such that:

We can take, e.g., a function given by:

and then, we normalize it:

Let

$f:{\mathbb{R}}^{n}\to \mathbb{R}$ be an invariant

L-Lipschitz map with a compact support, and let

$\lambda >0$. We define the

$\lambda $-regularization of

f by the formula:

The following is a straightforward consequence of the definition.

**Proposition** **1.** The regularization ${g}_{\lambda}$ satisfies the following properties:

- (i)
${g}_{\lambda}$ is ${C}^{\infty}$-smooth;

- (ii)
${g}_{\lambda}$ is G-invariant;

- (iii)
${g}_{\lambda}$ is a uniform $\lambda L$-approximation of f;

- (iv)
$\mathrm{supp}\phantom{\rule{0.166667em}{0ex}}{g}_{\lambda}\subset \overline{{B}_{\lambda}}(\mathrm{supp}\phantom{\rule{0.166667em}{0ex}}f)$.

The following technical lemma was proven in detail in [

19], Lemma 3.4.

**Lemma** **2.** Let ${f}_{1},\dots ,{f}_{k}:{\mathbb{R}}^{n}\to \mathbb{R}$ be Lipschitz functions of common rank L and with compact supports. For any $\epsilon >0$, there exists sufficiently small $\mu >0$ with the property that for every $x\in {\mathbb{R}}^{n}$, there exists $\overline{x}\in {B}_{\epsilon}(x)$ such that:for each $i=1,\dots ,k$. A local approximation result is a consequence of the above.

**Proposition** **2.** Let ${f}_{i}(i=1,\dots ,k)$ be G-invariant functions as in Lemma 2. Then, for any $\epsilon >0$, there exists ${\lambda}_{0}>0$ such that:

- (i)
for any $\lambda \le {\lambda}_{0}$, the regularization ${g}_{\lambda}^{i}$ of ${f}_{i}$ is a uniform ε-approximation of ${f}_{i}$, for each $i=1,\dots ,k$;

- (ii)
for every $x\in {\mathbb{R}}^{n}$, there exists $\overline{x}\in {B}_{\epsilon}(x)$ such that for all $\lambda \le {\lambda}_{0}$ and $i=1,\dots ,k$:

**Proof.** We put

${\lambda}_{0}:=min\{\mu ,\frac{\epsilon}{L}\}>0$, and let

$\lambda \le {\lambda}_{0}$. Then, Property (i) follows from Proposition 1. We prove (ii) by means of Property (iii) from Lemma 1. For this, we fix a point

x and

$u\in \overline{{B}_{1}(0)}$. There exists a

$\delta =\delta (\lambda ,x,u)<\mu $ such that, for all

i, we have:

On the other hand, we have:

and by the use of Lemma 2, we find

$\overline{x}$ such that:

Therefore, we have the inequality $\langle \nabla {g}_{\lambda}^{i}(x),u\rangle <{f}_{i}^{o}(\overline{x};u)+\epsilon $. This ends the proof because of the equality ${\sigma}_{\partial f(x)}(u)={f}^{o}(x;u)$. □

**Theorem** **2.** Let $f:U\to \mathbb{R}$ be a locally G-invariant map as in Theorem 1. Then, for every $\epsilon >0$, there exists a ${C}^{\infty}$-smooth G-invariant map $g:U\to \mathbb{R}$ such that $(g,\nabla g)$ is an ε-WT-approximation of $(f,\partial f)$.

**Proof.** Since

U is

G-invariant, there exists an increasing family of

G-invariant compact sets

${\left\{{K}_{k}\right\}}_{k=0}^{\infty}$ such that:

For each k = 1, the set ${P}_{k}:=\mathrm{int}\phantom{\rule{0.166667em}{0ex}}{K}_{k+1}\backslash {K}_{k-1}$ is also G-invariant. Therefore, the family $\left\{{P}_{k}\right\}$ is an open G-invariant covering of U. Let ${\left\{{\phi}_{k}\right\}}_{k=1}^{\infty}$ be a ${C}^{\infty}$-smooth and G-invariant partition of unity subordinated to $\left\{{P}_{k}\right\}$.

Thus, the functions ${\phi}_{k}\xb7f\phantom{\rule{4pt}{0ex}}(k\ge 1)$ are globally Lipschitz, G-invariant, and they can be considered as defined on ${\mathbb{R}}^{n}$.

We define a sequence of positive numbers:

Now, we can apply Proposition 2 for each

$k\ge 1$. We find

${\lambda}_{k}$ such that for any

$0<\lambda \le {\lambda}_{k}$ invariant

$\lambda $-regularizations:

${g}_{\lambda}^{k}$ of

${\phi}_{k}\xb7f$ and

${g}_{\lambda}^{k+1}$ of

${\phi}_{k+1}\xb7f$ are uniform

${\epsilon}_{k}/2$-approximations. Moreover, for every

$x\in {P}_{k}\cup {P}_{k+1}$, there is

$\overline{x}\in {B}_{{\epsilon}_{k}}(x)\cap ({P}_{k}\cup {P}_{k+1})$ such that:

for all

$0<\lambda \le {\lambda}_{n}$. We can assume that

$\mathrm{supp}\phantom{\rule{0.166667em}{0ex}}{g}_{\lambda}^{k}\subset {P}_{k}$, and the sequence

${\left\{{\lambda}_{k}\right\}}_{k=1}^{\infty}$ is nonincreasing.

The desired map

$g:U\to \mathbb{R}$ is defined by the formula:

It is obviously well defined, since at most two terms are different from zero at given x. By definition, it is ${C}^{\infty}$-smooth and G-invariant. We have to check the approximation conditions. For a fixed $x\in U$, there is a unique $k\ge 0$ such that $x\in {K}_{k+1}\setminus {K}_{k}$. If $k=0$, then $g(y)={g}_{{\lambda}_{1}}^{1}$ in some neighbourhood of x, and the claim is clear from the choice of ${\epsilon}_{1}$.

If

$k\ge 1$, then in some neighbourhood of

x contained in

${B}_{{\epsilon}_{k}}(x)$, we have

$g(y)={g}_{{\lambda}_{k}}^{k}(y)+{g}_{{\lambda}_{k+1}}^{k+1}(y)$ and

${\phi}_{k}(y)+{\phi}_{k+1}(y)=1$. Therefore:

By the properties of the generalized gradient, we obtain:

Because of Lemma 1, the last condition means that $\nabla g$ is a graph $\epsilon $-approximation of $\partial f$. This ends the proof. □

**Theorem** **3.** Let X be a compact G-invariant subset of V. Suppose that $f:X\to \mathbb{R}$ is locally Lipschitz and G-invariant. Then, for all $\epsilon >0$, there exists a $\delta >0$ with $\delta <\epsilon $ and such that, for every two $W{T}_{\frac{\delta}{2}}$-approximations $({g}_{1},\nabla {g}_{1}),({g}_{2},\nabla {g}_{2})$ of $(f,\partial f)$, the pair $({g}_{t},\nabla {g}_{t})$ is an $W{T}_{\epsilon}$-approximation of $(f,\partial f)$, where:is a linear homotopy. **Proof.** For all $x\in X$, $\partial f(\xb7)$ is upper semicontinuous (1) at x, so we can apply Definition 3 with a given $\epsilon $ and obtain an appropriate, small $\delta (x)$.

We can assume that

$\delta (x)<|x|$ for

$x\ne 0$, and

$\delta (0)={\delta}_{0}>0$. Then, we have

$x\to 0\Rightarrow \delta (x)\to 0$. For all

$x\in X$, we observe that

${x}^{\prime}\in B(x,\delta (x))$ implies

$\partial f({x}^{\prime})\subset {B}_{\epsilon}(\partial f(x))$. The family

${\left\{B(x,\delta (x))\right\}}_{x\in X}$ is an open covering of

X. We can choose a finite subcovering:

Let

$l>0$ be a Lebesgue number for the covering

$\eta $, i.e.,:

and let

$\delta =2min\{l,\frac{1}{2}\epsilon \}$.

Fix

$x\in X$, and suppose that

$({g}_{1},\nabla {g}_{1}),({g}_{2},\nabla {g}_{2})\in {C}^{\infty}({\mathbb{R}}^{n})$ are

$W{T}_{\frac{\delta}{2}}$-approximations of

$(f,\partial f)$. Then, we have:

and:

Then, from (2) and (3), we get:

From (2) and (3), we have:

so:

We can use the upper semicontinuity property of

$\partial f$ in

${x}_{i}$:

for points

$\tilde{x},\overline{x}$, which are close enough to

${x}_{i}$:

However,

${B}_{\epsilon}\left(\partial f({x}_{i})\right)$ is a convex set, and therefore:

Finally, from (2) and (4), we have:

This ends the proof. □

#### 2.3. Equivariant Gradient Degree

In this section, we outline an axiomatic approach to the equivariant gradient degree. To this end, some preliminaries are in order.

Recall that the Euler ring

$U(G)$ of a compact Lie group

G as an abelian group is equal to

$\mathbb{Z}(\mathsf{\Phi}(G))$, where

$\mathsf{\Phi}(G)$ is the set of all conjugacy classes of normal subgroups of

G. The multiplicative structure of

$U(G)$ is quite involved (see [

22] or [

4] for details on the ring structure).

Let V be an orthogonal G-representation. Denote by ${C}_{G}^{2}(V,\mathbb{R})$ the space of G-invariant real ${C}^{2}$-functions on V. Clearly, if $\phi \in {C}_{G}^{2}(V,\mathbb{R})$, then the gradient map $\nabla \phi $ is G-equivariant. Let $\Omega \subset V$ be an open, bounded and G-invariant subset of V and $f:V\to V$. A pair $(f,\Omega )$ is said to be a G-gradient $\Omega $-admissible pair if $f(x)\ne 0$ for all $x\in \mathrm{bd}\phantom{\rule{0.166667em}{0ex}}\Omega $, and there exists $\phi \in {C}_{G}^{2}(V,\mathbb{R})$ such that $\nabla \phi =f$. Denote by ${M}_{\nabla}^{G}(V,V)$ the set of all G-gradient $\Omega $-admissible pairs, and put ${M}_{\nabla}^{G}:={\cup}_{V}{M}_{\nabla}^{G}(V,V)$. In an obvious way, one can define a G-gradient $\Omega $-admissible homotopy between two G-gradient $\Omega $-admissible maps.

The notion of equivariant gradient degree we consider here was defined originally by K. Gȩba [

12]. We use an axiomatic approach here adapted from [

8].

**Theorem** **4.** There exists a unique map ${deg}_{G}^{\nabla}:{M}_{\nabla}^{G}\to U(G)$, which assigns to every $(\nabla \phi ,\Omega )\in {M}_{\nabla}^{G}$ an element ${deg}_{G}^{\nabla}(\nabla \phi ,\Omega )\in U(G)$:satisfying the following properties ($({H}_{i})$ are generators of the Euler ring, and ${n}_{{H}_{i}}$ are integer coefficients that may be nonzero only if the orbit type $({H}_{i})$ is present in the representation V): - 1.
(Existence) If ${deg}_{G}^{\nabla}(f,\Omega )\ne 0$, i.e., there is a nonzero coefficient ${n}_{{H}_{i}}$, then $\exists x\in \Omega $ such that $\nabla \phi (x)=0$ and $({G}_{x})\ge ({H}_{i})$.

- 2.
(Additivity) Let ${\Omega}_{1},{\Omega}_{2}$ be two disjoint, open, G-invariant subsets of Ω

and $\nabla {\phi}^{-1}(0)\cap (\overline{\Omega})\subset {\Omega}_{1}\cup {\Omega}_{2}$. Then: - 3.
(Homotopy) If ${\nabla}_{v}\mathsf{\Psi}:[0,1]\times \overline{\Omega}\to V$ is an admissible gradient G-equivariant homotopy, then: - 4.
(Normalization) Let $\phi \in {C}_{G}^{2}(V,\mathbb{R})$ be a special Ω

-Morse function such that ${(\nabla \phi )}^{-1}(0)\cap \Omega =G({v}_{0})$ and ${G}_{{v}_{0}}=H$. Then,where “${m}^{-}(\xb7)$” stands for the total dimension of eigenspaces for negative eigenvalues of a (symmetric Hessian) matrix. - 5.
(Multiplicativity) For all $(\nabla {\phi}_{1},{\Omega}_{1}),(\nabla {\phi}_{2},{\Omega}_{2})\in {M}_{\nabla}^{G}$,where the multiplication “⋆” is taken in the Euler ring $U(G)$. - 6.
(Suspension) If W is an orthogonal G-representation and $\mathcal{B}$ an open bounded invariant neighbourhood of $0\in W$, then: - 7.
(Hopf property) Assume that $B(V)$ is the unit ball of an orthogonal G-representation V, and for $(\nabla {\phi}_{1},\mathcal{B}(V)),(\nabla {\phi}_{2},\mathcal{B}(V))\in {M}_{\nabla}^{G}$, one has:Then, $\nabla {\phi}_{1}$ and $\nabla {\phi}_{2}$ are G-gradient $B(V)$-admissible homotopic.

Let us remark that in the case of the trivial action of

G, the above degree reduces to the classical Brouwer degree, as well as the usual non-gradient

G-equivariant degree (comp. [

8]). This follows from the celebrated A. Parusiński theorem (see [

24]), which says that every two gradient maps that are

$B(V)$-admissible homotopic are also gradient

$B(V)$-admissible homotopic. However, this is no longer true for equivariant maps.

**Example** **2.** Take $V=\mathbb{C}$ with the natural ${S}^{1}$-action. Clearly, $\mathsf{\Phi}=Id$ and $\mathsf{\Psi}=-Id$ are ${S}^{1}$-equivariant. Moreover, $h(t,z)={e}^{i\pi t}z$ is a $B(V)$-admissible ${S}^{1}$-equivariant homotopy between Φ

and Ψ

. Clearly, this homotopy is not a gradient. Moreover, it follows from [11] that such a gradient ${S}^{1}$-equivariant homotopy does not exist. Since the equivariant gradient degree satisfies the Hopf property with respect to gradient G-homotopies, as well as the ordinary equivariant degree with respect to equivariant homotopies, the gradient degree is more delicate homotopy invariant.

The ordinary equivariant degree for multivalued convex-valued u.s.c. maps was defined in [

7] by means of the equivariant version of the Cellina approximation theorem (see [

6]). The approximation results from the previous section enable us to develop the equivariant degree theory in quite an analogous way.

Let V be a finite dimensional orthogonal representation of a compact Lie group G, and let $\Omega $ be an open, bounded G-invariant subset of V. Let $f:V\to \mathbb{R}$ be a locally Lipschitz G-invariant map such that its Clarkés generalized gradient $\partial f:V\to V$ is $\Omega $-admissible, i.e., $0\notin \partial f(x)$ for $x\in \mathrm{bd}\phantom{\rule{0.166667em}{0ex}}\Omega $. Since the map $\partial f$ is u.s.c., the image of the boundary $\partial f(\mathrm{bd}\phantom{\rule{0.166667em}{0ex}}\Omega )$ is compact, and its distance from zero is positive. The same is true for a sufficiently fine neighbourhood of $\mathrm{bd}\phantom{\rule{0.166667em}{0ex}}\Omega $. Thus, there exists an ${\epsilon}_{0}>0$ such that for every $0<\epsilon <{\epsilon}_{0}$, each $\epsilon $-WT-approximation of $(f,\partial f)$ is $\Omega $-admissible. Moreover, Theorem 3 assures that there exists a smaller $\delta >0$ such that all $\delta $-approximations are $\Omega $-admissible homotopic by a linear homotopy. Take such a small $\delta $.

**Definition** **5.** We define the equivariant degree of Ω

-admissible pair $(f,\partial f)$ as follows:where $(g,\nabla g)$ is a δ-WT-approximation (with g being G-invariant) of $(f,\partial f)$. The properties of the degree are consequences of Theorem 4. The first three of them are the most important for applications. Properties 1 and 2 are quite straightforward. The homotopy property needs some comment. Recall that the generalized gradient of a locally Lipschitz map is a multivalued u.s.c. map with convex compact values. Considering homotopies as continuous families of maps, we need here a stronger regularity than the u.s.c. property, which may admit quite a rapid change of values.

**Definition** **6.** A gradient G-homotopy is a pair $({f}_{t},\partial {f}_{t})$, where ${f}_{t}:V\to \mathbb{R}$ are locally Lipschitz G-invariant maps and the family of maps $\partial {f}_{t}:V\to V$ is continuous with respect to $t\in [0,1]$, as well as the family ${f}_{t}$.

Observe that in the most popular case of linear homotopy

${f}_{t}(x)=t{f}_{1}(x)+(1-t){f}_{2}(x)$, the Clarke gradient with respect to the

x variable satisfies the condition:

In some cases, an equality holds, e.g., if at least one of the ${f}_{i}$ is of class ${C}^{1}$, or if both of them are convex functions.

**Theorem** **5.** Let the linear homotopy of locally Lipschitz maps ${f}_{t}(x)=t{f}_{1}(x)+(1-t){f}_{2}(x)$ be Ω

-admissible, and let the condition $\partial {f}_{t}(x)=t\partial {f}_{1}(x)+(1-t)\partial {f}_{2}(x)$ be satisfied for all x and $t\in [0,1]$. Then: **Proof.** The proof is similar to the proof of Theorem 3. For all $x\in X$, $\partial {f}_{i}(\xb7)$$(i=1,2)$ are upper semicontinuous (1) at x, so we can apply Definition 3 with a given $\epsilon $ and obtain an appropriate $\delta (x)$.

Let $\delta (x)<\left|x\right|$ and $\delta (0)={\delta}_{0}>0$. Then, $x\to 0\Rightarrow \delta (x)\to 0$.

For all

$x\in X$, we have that

${x}^{\prime}\in B(x,\delta (x))$ implies

$\partial {f}_{i}({x}^{\prime})\subset {B}_{\epsilon}(\partial {f}_{i}(x))$ and

${\left\{B(x,\delta (x))\right\}}_{x\in X}$ is an open covering of

X. We can choose a finite subcovering:

Let

$l>0$ be a Lebesgue number for the covering

$\eta $, i.e.,:

and let

$\delta =2min\{l,\frac{1}{2}\epsilon \}$.

Fix

$x\in X=\overline{\Omega}$, and suppose that

$({g}_{1},\nabla {g}_{1}),({g}_{2},\nabla {g}_{2})\in {C}^{\infty}({\mathbb{R}}^{n})$ are

$W{T}_{\frac{\delta}{2}}$-approximations of

$({f}_{1},\partial {f}_{1}),({f}_{2},\partial {f}_{2})$, respectively. Then, we have:

and:

Then, from (5) and (6), we get:

From (5) and (6), we have:

so:

We can use the upper semicontinuity of

$\partial {f}_{j}$ in

${x}_{i}$:

for points

$\tilde{x},\overline{x}$:

This means that there exist

${y}_{1}\in \partial {f}_{1}({x}_{i}),{y}_{2}\in \partial {f}_{2}({x}_{i})$ such that:

Therefore:

and

$t{y}_{1}+(1-t){y}_{2}\in \partial {f}_{t}$ by our assumption. Thus:

Finally, from (5) and (7), we have:

This ends the proof. □

The above theorem is satisfactory in many applications. In particular, the Hopf property follows from this weak version of the homotopy property.

**Theorem** **6.** Assume that $\Omega =B(V)$ is the unit ball of an orthogonal representation V and $({f}_{1},\partial {f}_{1}),({f}_{2},\partial {f}_{2}))$ are Ω

-admissible pairs such that: Then, the pairs are Ω-admissible homotopic.

**Proof.** It is enough to observe that if we choose a

$\delta $-WT-approximation

$(g,\nabla g)$ of

$(f,\partial f)$ defining the equivariant degree, then the linear formula:

defines an

$\Omega $-admissible homotopy satisfying the assumptions of Theorem 5. Therefore, we apply this to both pairs. Now, we can apply the Hopf property for smooth maps

$\nabla {g}_{i},$$i=1,2$. The transitivity property of the homotopy relation for single-valued maps ends the proof. □

We are ready to prove the full homotopy property now.

**Theorem** **7.** Let $({f}_{t},\partial {f}_{t}),t\in [0,1]$ be an Ω

-admissible gradient G-homotopy. Then: **Proof.** First, we observe that because of the continuity of the homotopy with respect to $t\in [0,1]$, for each $\epsilon >0$ and every fixed t, there exist a $\delta (t)>0$ and a Whitney approximation $({g}_{t},\nabla {g}_{t})$ of $({f}_{t},\partial {f}_{t})$, which is also an $\epsilon $-Whitney approximation of $({f}_{\tau},\partial {f}_{\tau})$ for $\tau \in [t-\delta (t),t+\delta (t)]$. Therefore, $({g}_{t},\nabla {g}_{t})$ is linearly homotopic to $({f}_{t-\delta},\partial {f}_{t-\delta})$, as well as to $({f}_{t+\delta},\partial {f}_{t+\delta})$. Thus, by Theorem 5, the gradient degree is constant in the interval $[t-\delta (t),t+\delta (t)]$. Now, we consider the open covering of $[0,1]$ by the intervals $(t-\delta (t),t+\delta (t))$ (for $t=0,1$, we put half-open intervals). We find a finite subcovering and apply the transitivity of the homotopy property to finish the proof. □