The Beer Barrel Theorem

A new proof of the asymptotic

conjecture in fixed point theory

by A. J. Tromba

In a seminar in Bonn during the summer of 1977, H. Peitgen

challenged the author to give an easy proof of the differentiable

case of the classic Schauder conjecture; namely (simply stated)

let T : E~ be a C i map with T 2 compact then T has a fixed point.

The prize was a barrel of beer. This theorem had already been

proved by Nussbaum [21 but involved rather long, technical and

sophisticated methods in fixed point theory. The purpose of this

note is to present a rather simple proof of the Nussbaum result

which uses only the mod 2 degree of Smale introduced in ~31 , and a

transversality result [5~ .

§ I The Mod 2 degree of Smale

Let WcE be an open subset of a Banach space. Let f:W ÷ F be a

continuous proper map (inverse image of compact sets is compact)

such that f:W + F is Fredholm of index zero; i.e. df(x):E + F is

linear Fredholm of index zero. Assume further that O#f(~W). Let

0 be a component of F-f(~W) containing O. By Smale's version of the

Sard theorem there is a regular value yeO for flW. Then the number

of points in f-1(y) will be finite and this number mod 2 is called e

the mod 2 degree of f relative to W and O, and is denoted by

deg2(f,W,O). Smale shows, using cobordism arguments going back to

Pontryagin that this number is independent of the choice of regular

value y60 and is invariant under proper homotopies ft such that for

all t O~ft(~W).

§ 2 Fredholm families and a transversalit~ theorem

Let f:AxW+~ be C i where A is a smooth Banach manifold. We say

that f is a Fredholm famil~, of index zero if for each aeA, fa=f(a, ")

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is Fredholm of index zero on W. We say that f is a zero proper

family if whenever an+a and fa (Xn)+O, then x n has a convergent n

subsequence.

Definition. If g:W+F is Fredholm of index zero, a zero x0 of g

(g(x0)=O) is said to be non-degenerate if dg(x0) is an isomorphism.

A proof of the following transversality result in case A is

Hilbert can be found in [51 .

Theorem I. Suppose f:A×W+F is a zero proper Fredholm family of -I

index zero and such that for each x£f (O), x=(a,e), the total

derivative df(x):TaA×E÷F is surjecture. Then there exists an open

and dense set AcA such that whenever a¢~ all the zeros of fa are

non-degenerate.

§ 3 Proof of the Schauder conjecture

Theorem 2. Let ~cE be open and convex and let T:~+~ be C l with T 2

compact. Then T has a fixed point.

Remark. By the classic Schauder theorem, T 2 has a compact set of

fixed points .-5;. Nussbaum only had to assume that T is C I on some

neighbourhood of ~_~. The same will be true for our proof, however,

for pedagodical reasons, we shall make no reduction in our hypothesis.

The basic idea of the proof is quite simple. Suppose the fixed

point set ~of T 2 is non-degenerate; i.e. id-d(T2) (x) is an

isomorphism for each xe~. Since T 2 is compact, the map g(x)=x-T2x

is proper Fredholm of index zero and therefore has a finite number

of zeros (which are of course the fixed points of T). Using the

mod 2 degree with W=E and considering the homotopy (Ix)÷x-IT2x for

le[0,1], we see that the mod 2 degree of g is I and consequently T 2

must have an odd number of fixed points. But T acts as a permutation

on this finite set, and each element is of order 2 under T. Since

there are an odd number of fixed points, one of them must have order

I. This is a fixed point of T, and the Schauder conjecture is

proved in this case. In the remainder of the paper, we will show

how to apply theorem I to perturb T so that the fixed points of the

perturbed mapping are non-degenerate.

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We shall now have the standing assumption that T has no fixed

points~ also without loss of ~enerality we can assume that Oe~.

Let~be the Banach space of C ~ compact maps on E with bounded

differentials; i.e. ~/if sup lldk(x) Ii<~ and let ~be the set of xeE

su lldk(x) il< 1 C I diffeomorphisms G:E~ of the form G=id+k, kc~and x¢{

Then ~ is clearly a C Banach manifold modelled on ~'with the

global chart G ÷ k. Moreover each Ge~{ is properly homotopic to the

identity though diffeomorphisms via the homotopy t ÷ id+tk.

As before, let ,~ denote the compact fixed point set of T 2 and

hence the zero set of fi(x)=x-T2x. -Since ~c~ is compact, there

exist neighbourhoods V • of ~, and U • of I~id in ~ such that both

V"c~ and (GT) (V~)c~ for all G£U ". Thus the double composition (GT) 2

is defined on V ~. Consider the family of mappings f:U~×V~+E defined

by f(G,x)=x-(GT)2x, f need not necessarily be a Fredholm family

but fi=f(I, .) is Fredholm of index zero. It therefore follows from

the implicit function theorem that for each xc~there exists a

splitting of E =El (x)eEz (x), dim E2 (x)< ~, and bounded neighbourhoods

U(x) of IeU ~ and V(x)~V ~ of x such that for GeU(x) there is a local

diffeomorphism YG of a neighbourhood of OEE to V(x) with

(I) fGO~G(Xl,X2) = (XI,0G(Xi,X2))

where (G,XI,X2) + ~G(Xl,X2) and (G,XI,X~) + @G(Xl,X2) are C I

Remark. This normal form has been used by several authors (see e.g.

[13, and [3~).

Since _~is compact, we can find a neighbourhood U of IEU ~ and a

finite number VI,...,V 1 of neighbourhoods covering ~so that for

GCU and xcVjcV • representation (I) holds.

Recall that we are assuming that T has no fixed points from which

it follows that [lT2x-Tx!! ~6>O for all xe~ Therefore we can find

neighbourhoods U of I in ~ and V of ~ such that

(2) IITGTx - Txll > 6/2

for all G£U and x£V.

Consequently, we can assume that neighbourhoods U&U ~ of I and

V--qJVi~V" have been chosen so that with respect to these neighbourhoods

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representation (I) and inequality (2) holds. Let W be a neighbour-

hood of ~with WcV. From representation (I) it follows that there is

a neighbourhood AcU of I so that for all GeA, O%fG(~W).

By the normal form (I), f:A×W+E is a zero proper Fredholm family of

index zero. Moreover for each GeA, fG is proper and hence has a

mod 2 degree. Since, by construction, each GeA is properly homotopic

to the identity, one can immediately conclude that

deg2 (fG,W,O) = deg2 (fi,W,O). But fI is proper Fredholm of index

zero on all of Q and the zero set of fI is contained in W. Therefore

deg2 (fi,W,O) = deg2 (fi,~,O). Since ~ is convex, the homotopy

(l,x) + x-IT2x is a proper homotopy of fI to the identity introducing

no zeros on 3~. Thus deg2 (fi,~,O) = I = deg2 (fi,W,O).

The next lemma is the main step in the proof of the Schauder conjec-

ture.

Lemma. The family f:AxW÷E has the property that whenever fG(X)=O

the total derivative df(G,x) is surjecture

Proof. By direct computation

df(G,x) [H,h] : h - DG o DT " DG o DTx[h ] TGT (x) GT(x) T(x)

- H(TGTx) - DG o DTeH(Tx) TGT (x)

We know that xEV and that on V

TGTx % Tx for GeA.

So in order to show surjectivity given weE, we must produce an

[H,hj so that df(G,x) [H,h~ = w. Choose h=w and H so that H(Tx)=O

and H(TGTx) = -DGoDToDGODTx(W ) .

This concludes the lemma.

By theorem I, we can conclude that there exists a sequence GnCA,

G n ÷ I such that (GnOT)2 has non-degenerate fixed points in W. By

degree there must be an odd number and by parity GnT has a fixed

point x n. However GnTXn=X n implies that fG (Xn)=O" Since the n

family f is zero proper, we can conclude that x n has a subsequence

converging to zeW. Clearly Tz=z which concludes the proof of the

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Schauder conjecture.

Remarks. Nussbaum's original proof worked for the case T n compact,

n>2. A slightly more technical argument than that presented here

should also work for this case. It is also clear that these methods

yield a clean proof of the Steinlein mod (p) result [4].

References

[1]

[2]

[3]

[4]

[5]

K. D. Elworthy and A. J. Tromba; Differential structures and

Fredholm maps on Banach manifolds; Prac. Symp. pure math.,

AMS vol XV, 45-94

R. D. Nussbaum; Some asymptotic fixed point theorems, Trans.

Amer. Math. Soc. vol 171 (1972)', 349-375

S. Smale; An infinite dimensional version of Sard's theorem,

Amer. J. Math., 87 (1965), 861-866

H. Steinlein; A new proof of the (mod p) theorem in asymptotic

fixed point theory, Prac. Symp. on problems in nonlinear

functional analysis, Universit~t Bonn, July 22-26, 1974.

Ber. Geso Math. Datenverarbeitung Bonn, No. 103, 29-42 (1975)

A. J. Tromba; Fredholm vector fields and a transversality

theorem, J. Funct. Anal. vol 23, No. 4 (1976)

A. J. Tromba

Mathematisches Institut

Universit~t Bonn, SFB 72

and

University of California

at

Santa Cruz