Interior fixed points of unit-sphere-preserving Euclidean maps

R E S E A R C H

Open Access

Interior fixed points of

unit-sphere-preserving Euclidean maps

Nirattaya Khamsemanan

_{, Robert F Brown}

_{, Catherine Lee}

_{and Sompong Dhompongsa}

*_{Correspondence:}

[email protected]

1_{School of Information, Computer,}

and Communication Technology, Sirindhorn International Institute of Technology (SIIT), Thammasat University, Prathum Thani, Thailand Full list of author information is available at the end of the article

Abstract

Schirmer proved that there is a class of smooth self-maps of the unit sphere in Euclideann-space with the property that any smooth self-map of the unit ball that extends a map of that class must have at least one fixed point in the interior of the ball. We generalize Schirmer’s result by proving that a smooth self-map of Euclidean n-space that extends a self-map of the unit sphere of that class must have at least one fixed point in the interior of the unit ball.

MSC: 55M20; 54C20

Keywords: interior fixed point theory

1 Introduction

For spacesX,Y and subsetsV⊆X,W⊆Y, a mapf:X→Y is anextensionof a map

φ: V →W iff(x) =φ(x) for allx∈V. We denote byBn_{the unit ball inR}n_{, byS}n–_its

boundary and byint(Bn_{) its interior.}

Iff: B_→B_{is an extension ofφ: S}_→S_{={–, }andφhas no fixed points, thenf}

must have aninteriorfixed point, that is, a fixed point inint(B_{). However, ifφhas a fixed}

point, then there need not be any interior fixed points.

Ifn= , the situation is more complicated. Of course the Brouwer fixed point theorem implies that a mapf:B_→B_{must have at least one interior fixed point if it is an extension}

of a mapφ:S_→S_{that has no fixed points. But it was proved in [] (see also []) that if}

the extensionf is smooth, it may still be required to have interior fixed points for certain mapsφthat have many fixed points. Representing the points ofS_{by complex numbers,}

letφ=φd:S→S, for an integerd, be thepower mapdefined byφd(z) =zd. Ifd≥ and

f:B_→B_{is a smooth extension ofφ}

d, thenf has at least one interior fixed point. It is also

demonstrated in [] that interior fixed points of extensions need not exist ifd≤ or iff is not smooth. Schirmer generalized this interior fixed point result to smooth extensions f:Bn_→Bn_{forn≥ to show in Example . of [] that iff is a smooth extension of a}

‘sparse’ mapφ:Sn–_→Sn–_{, a generalization ofφ}

dthat is defined below, of degreedsuch

that (–)n_{d≥, thenf must have at least one interior fixed point.}

Returning to the casen= , if we extend the mapφ:S_→S_{without fixed points to a}

mapf:R_→R_{, there still must be a fixed point off inint(B}_{). The reason for the interior}

fixed points of the extensionf:B_→B_{of the map ofS}_{without fixed points, namely that}

(–, ) and (, –), lie in different components ofB_×B_{\, where={(x,x) :x∈B}_},

ap-plies also to the extensionf:R_→R_{since those points are also in different components}

ofB_×R_\.

On the other hand, the reason for the presence of fixed points inint(Bn_{) for smooth}

ex-tensions of certain maps ofSn–_{demonstrated in [] is considerably more subtle.}

There-fore, it is reasonable to ask whether such fixed points would persist if, instead of smooth extensions f: Bn_→Bn _{ofφ: S}n–_→Sn–_{, we consider extensions that are smooth}

Eu-clidean maps, that is, mapsf:Rn_→Rn_{. Thus, we ask whether there still must be fixed}

points off inint(Bn) if we allowf to map points ofint(Bn) outside ofBn.

We will prove that the interior fixed points do persist, even in this more general setting. As in the case of self-maps of balls, the interior fixed points of Euclidean maps are detected by means of a theorem that relates the index of a fixed point ofφ:Sn–_→Sn–_{to its index}

as a fixed point of an extension. We will therefore devote Section  to a discussion of the properties of the fixed point index that we will use. In Section , we prove that a smooth extensionf:R_→R_{of a power mapφ}

d:S→Sford≥ must have at least one

in-terior fixed point. Section  then contains the proof that Schirmer’s result generalizes to smooth extensionsf:Rn→Rnof sparse mapsφ:Sn–→Sn–that satisfy the same degree restrictions.

2 The fixed point indices

Before extending the results of [, ] and [] to the case of a smooth Euclidean map f : (Rn_,Sn–_)→(Rn_,Sn–_{) extendingφ:S}n–_→Sn–_{, we need to define the relevant fixed}

point indices. We will consider the restrictionf: (Bn,Sn–)→(Rn,Sn–). Since our goal is to establish conditions for the existence of fixed points on the interior ofBn, the behavior of the function outside ofBn_{is not relevant. Therefore, we will make use of the indices}

i(Bn_{,f,p) andi(S}n–_{,φ,p) of an isolated fixed pointp∈S}n–_{. We do so by generalizing the}

approach used in [] (see also []).

For an isolated fixed pointp∈Sn–_{, we can choose a small enough neighborhoodUso}

that it contains only this fixed point and no other. We then may writef in this neighbor-hood ofpin terms of a local coordinate system in whichU∩Bnis contained in the upper half-space

Rn +=

(x,x, . . . ,xn)∈Rn|xn≥

in such a way thatpis the originin this setting andU∩Sn–_{is contained in the subspace}

Rn–_=(x

,x, . . . ,xn–, )∈Rn

.

In order to calculate the index ofpin each space, we consider the mapF:U→Rndefined by

F(x,x, . . . ,xn) =

⎧ ⎨ ⎩

(x,x, . . . ,xn) –f(x,x, . . . ,xn), ifxn≥,

(x,x, . . . ,xn) –f(x,x, . . . ,xn–, ), ifxn< .

Note thatFsends the origin, lower half-plane andRn–_{to itself respectively. Also,F(z)=z}

forz= . The indexi(Sn–_{,φ,p) is equal toi(R}n–_{,F,) in this setting as in the traditional}

definition of the index. Moreover,i(Bn,f,p) is identified withi(Rn_{,F,), which can be}

3 Unit-circle-preserving maps of the plane

Brown, Greene and Schirmer proved

Theorem [, ] Let f :B_→B _{be a smooth map with a finite number of fixed points}

such that f(ζ) =ζkfor allζ ∈S _{for some k≥,where B}_{is the closed two-dimensional}

ball with the boundary S_{.Ifπis a fixed point of f that lies in S}_{,then either i(B}_{,f,π) = }

or i(B_{,f,π) = –.}

The contractibility ofB_{implies the following corollary.}

Corollary [] Suppose f :B_→B_{is a smooth map such that f(ζ) =ζ}k_{for allζ∈S}_,for

some k≥.Then there exists z∈int(B_{)such that f(z) =z.}

We will extend Theorem  to mapsf : (B_,S_)→(R_,S_{) by modifying the proof of}

The-orem  in []. Corollary  will then extend to mapsf: (R_,S_)→(R_,S_).

Theorem  Let f : (B_,S_)→(R_,S_{)be a smooth map with a finite number of fixed points}

such that f(ζ) =ζkfor allζ ∈S_{for some k≥.Ifπis a fixed point of f that lies in S}_,then

either i(B_{,f,π) = or i(B}_{,f,π) = –.}

Proof Letπbe a fixed point offinS_{. We can write this fixed point in the polar coordinates}

(r,θ) as (,θ). We will introduce new coordinates on a neighborhoodUofπas follows:

x=θ–θ, x=  –r.

In the new coordinate setting, the fixed pointπis the origin andU∩S_{corresponds to the}

x-axis near , and the portion of the interior of the unit ball inUis contained in the upper

half-plane. Consider the following map (as described in Section ) in the new coordinate setting:

F(x,x) =

⎧ ⎨ ⎩

(x,x) –f(x,x), ifx≥,

(x,x) –f(x, ), ifx< .

Now writeF= (F,F) and defineg(x) =F(x, ). Sincef(ζ) =ζkforζ∈S, we have

g(x) =F(x, ) =x–kx= ( –k)x.

Since the mapf is defined to be smooth onB_{, the mapFis smooth on the upper}

half-plane. LetF+_{denote the restriction ofFto the upper half-plane. We will see that}

smooth-ness is only required in a neighborhood of the fixed point at . Since we are assuming that k≥, then

d

dxF(x, ) =g _(x

) =  –k< 

and the smoothness ofF+_{implies that}

∂F+ (x,x)

∂x

for (x,x) in an -neighborhood of the origin, for >  sufficiently small and forx≥.

Letbe a circle of radius / about the origin.

Let+and–denote the half-circles above and below thex-axis respectively. Since

F takes the lower half-plane to itself, we know thatF maps– to the lower half-plane. Calculating the fixed point index off at the origin inRis equivalent to finding the winding number ofF() around . Thus, we need to understandF(+). Since+lies in the upper half-plane, we only considerF+_{. AssumingFhas only a finite number of fixed points, we}

can choose small enough so that only one point onor its interior thatFmaps to the origin is the origin itself. Therefore, we can homotope the restriction ofF+_to+_inR_\

to the restriction ofF+_{to the curve}+

δ forδ>  given by

+_δ(t) =

(t– ),δ

 – ( – t) ,

where ≤t≤.

We write the restriction ofF+_to+

δ in coordinates as

F+_δ+(t) =F_+_δ+(t),F_++_δ(t) =φδ(t),ψδ(t) .

The key idea of the proof is that forδsufficiently small, the smoothness ofF+_{and the fact}

that

∂F_+(x,x)

∂x

<  implies that d

dtφδ(t) < 

for allt. This tells us that thex-coordinate of the curveF+(+δ(t)) is a strictly monotone

function oft. In particular, the curveF+(+_δ(t)) only crosses thex-axis once. This implies

the desired result since the winding number ofF() can then only be either  or –. Notice that it is never specified thatf mapsB_{into itself. In considering the mapF(z) =}

z–f(z), although it is assumed thatFmaps the exterior of the disc to the exterior of the disc, the proof allows the image of the interior of the disc underFto lie anywhere inR_.

Letf¯:B_→R_{be the restriction off: (R}_,S_)→(R_,S_{) toB}_{. SinceB}_{is contractible,}

the sum of the indices ofρf¯: B→Bequals one, and thereforeρ¯f(x) =xfor somex∈ int(B_{). But thenf(x) =xas well, sof: (B}_,S_)→(R_,S_{) has a fixed point in the interior}

ofB_{. Therefore, we can extend Corollary  as the following result.}

Corollary  Let f : (R_,S_)→(R_,S_{)be a smooth map such that f(ζ) =ζ}k_{for allζ ∈S}

for some k≥.Then there exist z∈int(B_{)such that f(z) =z.}

4 Interior fixed points of a mapf: (Rn_,Sn–1_)→(Rn_,Sn–1₎

Definition ([], p.) A smooth mapφ:Sn–_→Sn–_{with finitely many fixed points is}

transversely fixedifdφp–I:Tp(Sn–)→Tp(Sn–) is a nonsingular linear map for each fixed

pointp. ForF={p, . . . ,pr}a fixed point class ofφ, let

i(F) =

r

j=

Thetransverse Nielsen number N(φ) is defined by

N(φ) =

F∈F

i(F),

whereFis the set of fixed point classes ofφ.

A smooth mapφ:Sn–_→Sn–_{issparseif it is transversely fixed and it hasN}

(φ) fixed points.

In [], p. Schirmer obtained the following result.

Theorem  Letφ:Sn–_→Sn–_{be a sparse map of degree d and suppose f : (B}n_,Sn–_)→

(Bn_,Sn–_{)is a smooth map extendingφ.If(–)}n_{d≥,then f must have a fixed point in}

int(Bn_).

We will extend this result as a consequence of the following

Theorem  Given a smooth map φ:Sn–_→Sn– _{and a smooth map f : (B}n_,Sn–_)→

(Rn_,Sn–_{)extendingφ,suppose that p∈S}n–_{is an isolated fixed point of f and that dφ} p–I:

Tp(Sn–)→Tp(Sn–)is a nonsingular linear transformation.Then either i(Bn,f,p) = or

i(Bn,f,p) =i(Sn–,φ,p).

Proof The following proof is a modified version of Theorem . in []. We again writef in a small ball that containsp∈Sn–_{as described in Section  and the mapF is also as}

defined there. Moreover, forε>  small enough, let

Dε(x, . . . ,xn) =

⎧ ⎨ ⎩

(εx,εx, . . . ,εxn) –f(εx,εx, . . . ,εxn), ifxn≥,

(εx,εx, . . . ,εxn) –f(εx,εx, . . . ,εxn–, ), ifxn< .

This then means that the indexi(Bn_{,f,p) =i(B}n_,D

ε,) is the degree ofρ◦Dε:Sn–→Sn–,

whereρ(x) =x/|x|forx∈Rn_\.

Note that

F(εx,εx, . . . ,εxn) –dFp(εx,εx, . . . ,εxn)≤o(ε)

becauseFis aCfunction. We also have

dFp(εx,εx, . . . ,εxn)≥Cε

for someC>  independent ofεand (x,x, . . . ,xn) due to the fact thatdFp= –(dfp–I) is

nonsingular by hypothesis.

SincedFpis a nonsingular linear map, this last degree is easily seen to be , or±. But

the images ofDεof the upper and lower hemisphere are each contained entirely in either

the lower or upper half-space. This means thatDεis of degree  orDε is homotopic in Rn_{\to the suspension ofD}

ε|Sn–.

We can use Theorem  to extend Theorem  to the casef : (Rn_,Sn–_)→(Rn_,Sn–_{). The}

is solved in the previous section, the new material presented below extends the solution to all the cases.

Suppose we haveφ:Sn–→Sn–and a smooth mapf : (Rn,Sn–)→(Rn,Sn–) extend-ingφ. A fixed point classF off is called acommon fixed point classoff andφif there exists an essential fixed point classFofφwhich is contained inF.

We will again consider the restrictionf : (Bn_,Sn–_)→(Rn_,Sn–_{). In the notation of [],}

p., let

u(F) =max,iSn–,φ,F F⊂FandiSn–,φ,F >  , l(F) =min

,iSn–,φ,F F⊂FandiSn–,φ,F <  .

ThenFis atransversally common fixed point classoff andφif

l(F)≤i(F)≤u(F).

Definition  Theboundary transversal Nielsen numberoff : (Bn_,Sn–_)→(Rn_,Sn–_{) is}

Nf;Bn,Sn– =N(φ) +N(f) –N(f,φ),

whereN(f,φ) is the number of essential and transversally common fixed point classes of f andφ.

Suppose thatφ:Sn–_→Sn–_{has degreed. Thenf:B}n_→Rn_{has one fixed point classF}

withi(F) = , and sol(F)≤ <i(F). Ifn= , thenφhas| –d|essential fixed point classes, each of the same index, andi(S_{,φ,F) =L(φ) =  –d. Hence,i(F)≤u(F) if and only if}

d≤, and

N(f,φ) =

⎧ ⎨ ⎩

 ifd≤,

 ifd> .

Ifn≥, thenφhas one fixed point classFwithi(Sn–_{,φ,F) =  + (–)}n–_{dand so}

N(f,φ) = ⎧ ⎨ ⎩

 if (–)n–_d≥,

 if (–)n–_{d< .}

Note that this formula is still true for the casen= . Ifφ:Sn–_→Sn–_{is a sparse map of degreedthenN}

(φ) =| – (–)n_{d|for alldand all}

n≥.

SinceN(f) = , for the case that (–)n_{d≤, the boundary transversal Nielsen number is}

Nf;Bn,Sn– =N(φ) +N(f) –N(f,φ)

= – (–)nd+  – 

As for the case that (–)n_{d> , the boundary transversal Nielsen number is}

Nf;Bn,Sn– =N(φ) +N(f) –N(f,φ)

= – (–)nd+  – 

= – (–)nd+ .

Hence, we have just proven the following

Proposition  If f: (Bn,Sn–)→(Rn,Sn–) is a smooth extension of a sparse map φ : Sn–_→Sn–_{of degree d,with n≥,then the boundary transversal Nielsen number is}

Nf;Bn,Sn– = ⎧ ⎨ ⎩

| – (–)n_{d| if(–)}n_d≤, | – (–)n_{d|+  if(–)}n_{d> .}

As defined in [], p.,the extension Nielsen number N(f|φ) is a lower bound for the number of fixed points onSn–_{of continuous extensions of a continuous mapφ. It is equal}

to the number of essential (classical) fixed point classesFoff withF∩Sn–=∅. A fixed point classFisrepresentableonSn–_{if there exists a subsetF⊂F∩S}n–_withi(Bn_{,f,F) =}

i(Sn–_{,φ,F). Thesmooth extension number N}_{(f|φ) is the number of essential (classical)}

fixed point classesFoff which are not representable onSn–_{. It is a lower bound for the}

number of fixed points inSn–_{of a smooth extension of a smooth and transversally fixed}

mapφ.

Proposition  Ifφ:Sn–→Sn–is sparse,then

N(f|φ) =Nf;Bn,Sn– –N(φ), N(f|φ) =Nf;Bn,Sn– –N(φ).

Proof Our proof is modeled on the proofs of Proposition . and Corollary . in []. For any essential fixed point classFoff, sinceφis sparse,F∩Sn–containsu(F) fixed points psuch thati(Sn–_{,φ,p) =  andl(F) fixed pointspsuch thati(S}n–_{,φ,p) = –. This means}

thatFis representable onSn–_{if and only ifl(F)≤i(F)≤u(F). By the definitions of all the}

Nielsen numbers involved, we have the result stated above forN_(f|φ).

The result forN(f|φ) can be obtained in a similar manner by using Corollary . from [] along with the fact that all fixed point classes of a sparse map are essential.

By the definitions ofN(f,φ) in [] and the definition ofN(f|φ) defined early, we obtain

Proposition  Ifφ:Sn–_→Sn–_{is sparse,then the number of essential fixed point classes}

of f which are common but not transversally common is

N(f|φ) –N(f|φ) =N(f,φ) –N(f,φ).

Theorem  Let n≥,and letφ:Sn–_→Sn–_{be a sparse map of degree d and suppose}

f : (Bn_,Sn–_)→(Rn_,Sn–_{)is a smooth map extendingφ.If(–)}n_{d≥,then f must have a}

fixed point inint(Bn_).

Proof Sinceφhas degreedand it is sparse, by definition

N(f|φ) = ⎧ ⎨ ⎩

 ifd= (–)n_,

 ifd= (–)n

and

N(f|φ) –N(f|φ) =Nf;Bn,Sn– –N(φ) –N(f|φ) (from Proposition ) =Nf;Bn,Sn– – – (–)nd–N(f|φ).

Applying Proposition , we have

N(f|φ) –N(f|φ) = ⎧ ⎨ ⎩

 if (–)n_d≤,

 if (–)n_d≥.

Thus, every smooth extension overBnof a sparse map ofSn–of degreedwith (–)nd≥

has a fixed point on the interior ofBn.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Author details

1_{School of Information, Computer, and Communication Technology, Sirindhorn International Institute of Technology}

(SIIT), Thammasat University, Prathum Thani, Thailand. 2_{Department of Mathematics, University of California, Los Angeles,} CA, USA.3_{Bard High School Early College, New York, NY, USA.}4_{Department of Mathematics, Chiang Mai University,} Chiang Mai, Thailand.

Acknowledgements

This work originated with Catherine Lee’s PhD dissertation under the supervision of Professor Robert F. Brown. The first author has been supported by the Thailand Research Fund grant number MRG5580232 under the mentorship of Professor Robert F. Brown of the Department of Mathematics, UCLA and Professor Sompong Dhompongsa of the Department of Mathematics, Chiang Mai University. She wishes to thank TRF as well. Last but not least, the authors would like to express their appreciation to both reviewers for their thorough reviews and many useful suggestions and comments.

Received: 29 June 2012 Accepted: 4 October 2012 Published: 17 October 2012

References

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2. Brown, RF, Greene, RE: An interior fixed point property of the disc. Am. Math. Mon.101(1), 39-47 (1994)

3. Schirmer, H: Nielsen theory of transversal fixed point sets. Fundam. Math.141(1), 31-59 (1992). With an appendix by Robert E. Greene

4. Dold, A: Fixed point index and fixed point theorem for Euclidean neighborhood retract. Topology4, 1-9 (1965) 5. Schirmer, H: A relative Nielsen number. Pac. J. Math.122, 459-473 (1986)

doi:10.1186/1687-1812-2012-183