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Interior fixed points of unit-sphere-preserving Euclidean maps

  • 1Email author,
  • 2,
  • 3 and
  • 4
Fixed Point Theory and Applications20122012:183

https://doi.org/10.1186/1687-1812-2012-183

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

Abstract

Schirmer proved that there is a class of smooth self-maps of the unit sphere in Euclidean n-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 spaces X, Y and subsets V X , W Y , a map f : X Y is an extension of a map ϕ : V W if f ( x ) = ϕ ( x ) for all x V . We denote by B n the unit ball in R n , by S n 1 its boundary and by int ( B n ) its interior.

If f : B 1 B 1 is an extension of ϕ : S 0 S 0 = { 1 , 1 } and ϕ has no fixed points, then f must have an interior fixed point, that is, a fixed point in int ( B 1 ) . However, if ϕ has a fixed point, then there need not be any interior fixed points.

If n = 2 , the situation is more complicated. Of course the Brouwer fixed point theorem implies that a map f : B 2 B 2 must have at least one interior fixed point if it is an extension of a map ϕ : S 1 S 1 that has no fixed points. But it was proved in [1] (see also [2]) that if the extension f is smooth, it may still be required to have interior fixed points for certain maps ϕ that have many fixed points. Representing the points of S 1 by complex numbers, let ϕ = ϕ d : S 1 S 1 , for an integer d, be the power map defined by ϕ d ( z ) = z d . If d 2 and f : B 2 B 2 is a smooth extension of ϕ d , then f has at least one interior fixed point. It is also demonstrated in [1] that interior fixed points of extensions need not exist if d 1 or if f is not smooth. Schirmer generalized this interior fixed point result to smooth extensions f : B n B n for n 2 to show in Example 4.7 of [3] that if f is a smooth extension of a ‘sparse’ map ϕ : S n 1 S n 1 , a generalization of ϕ d that is defined below, of degree d such that ( 1 ) n d 2 , then f must have at least one interior fixed point.

Returning to the case n = 1 , if we extend the map ϕ : S 0 S 0 without fixed points to a map f : R 1 R 1 , there still must be a fixed point of f in int ( B 1 ) . The reason for the interior fixed points of the extension f : B 1 B 1 of the map of S 0 without fixed points, namely that ( 1 , 1 ) and ( 1 , 1 ) , lie in different components of B 1 × B 1 Δ , where Δ = { ( x , x ) : x B 1 } , applies also to the extension f : R 1 R 1 since those points are also in different components of B 1 × R 1 Δ .

On the other hand, the reason for the presence of fixed points in int ( B n ) for smooth extensions of certain maps of S n 1 demonstrated in [3] is considerably more subtle. Therefore, it is reasonable to ask whether such fixed points would persist if, instead of smooth extensions f : B n B n of ϕ : S n 1 S n 1 , we consider extensions that are smooth Euclidean maps, that is, maps f : R n R n . Thus, we ask whether there still must be fixed points of f in int ( B n ) if we allow f to map points of int ( B n ) outside of B n .

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 of ϕ : S n 1 S n 1 to its index as a fixed point of an extension. We will therefore devote Section 2 to a discussion of the properties of the fixed point index that we will use. In Section 3, we prove that a smooth extension f : R 2 R 2 of a power map ϕ d : S 1 S 1 for d 2 must have at least one interior fixed point. Section 4 then contains the proof that Schirmer’s result generalizes to smooth extensions f : R n R n of sparse maps ϕ : S n 1 S n 1 that satisfy the same degree restrictions.

2 The fixed point indices

Before extending the results of [1, 2] and [3] to the case of a smooth Euclidean map f : ( R n , S n 1 ) ( R n , S n 1 ) extending ϕ : S n 1 S n 1 , we need to define the relevant fixed point indices. We will consider the restriction f : ( B n , S n 1 ) ( R n , S n 1 ) . Since our goal is to establish conditions for the existence of fixed points on the interior of B n , the behavior of the function outside of B n is not relevant. Therefore, we will make use of the indices i ( B n , f , p ) and i ( S n 1 , ϕ , p ) of an isolated fixed point p S n 1 . We do so by generalizing the approach used in [1] (see also [4]).

For an isolated fixed point p S n 1 , we can choose a small enough neighborhood U so that it contains only this fixed point and no other. We then may write f in this neighborhood of p in terms of a local coordinate system in which U B n is contained in the upper half-space
R + n = { ( x 1 , x 2 , , x n ) R n | x n 0 }
in such a way that p is the origin 0 in this setting and U S n 1 is contained in the subspace
R n 1 = { ( x 1 , x 2 , , x n 1 , 0 ) R n } .
In order to calculate the index of p in each space, we consider the map F : U R n defined by
F ( x 1 , x 2 , , x n ) = { ( x 1 , x 2 , , x n ) f ( x 1 , x 2 , , x n ) , if  x n 0 , ( x 1 , x 2 , , x n ) f ( x 1 , x 2 , , x n 1 , 0 ) , if  x n < 0 .

Note that F sends the origin, lower half-plane and R n 1 to itself respectively. Also, F ( z ) z for z 0 . The index i ( S n 1 , ϕ , p ) is equal to i ( R n 1 , F , 0 ) in this setting as in the traditional definition of the index. Moreover, i ( B n , f , p ) is identified with i ( R n , F , 0 ) , which can be computed as the degree of the map ρ F : S n 1 S n 1 where ρ : R n 0 S n 1 is the retraction defined by ρ ( z ) = z / | z | .

3 Unit-circle-preserving maps of the plane

Brown, Greene and Schirmer proved

Theorem 1 [1, 2]

Let f : B 2 B 2 be a smooth map with a finite number of fixed points such that f ( ζ ) = ζ k for all ζ S 1 for some k 2 , where B 2 is the closed two-dimensional ball with the boundary S 1 . If π is a fixed point of f that lies in S 1 , then either i ( B 2 , f , π ) = 0 or i ( B 2 , f , π ) = 1 .

The contractibility of B 2 implies the following corollary.

Corollary 2 [2]

Suppose f : B 2 B 2 is a smooth map such that f ( ζ ) = ζ k for all ζ S 1 , for some k 2 . Then there exists z int ( B 2 ) such that f ( z ) = z .

We will extend Theorem 1 to maps f : ( B 2 , S 1 ) ( R 2 , S 1 ) by modifying the proof of Theorem 1 in [2]. Corollary 2 will then extend to maps f : ( R 2 , S 1 ) ( R 2 , S 1 ) .

Theorem 3 Let f : ( B 2 , S 1 ) ( R 2 , S 1 ) be a smooth map with a finite number of fixed points such that f ( ζ ) = ζ k for all ζ S 1 for some k 2 . If π is a fixed point of f that lies in S 1 , then either i ( B 2 , f , π ) = 0 or i ( B 2 , f , π ) = 1 .

Proof Let π be a fixed point of f in S 1 . We can write this fixed point in the polar coordinates ( r , θ ) as ( 1 , θ 0 ) . We will introduce new coordinates on a neighborhood U of π as follows:
x 1 = θ θ 0 , x 2 = 1 r .
In the new coordinate setting, the fixed point π is the origin and U S 1 corresponds to the x 1 -axis near 0, and the portion of the interior of the unit ball in U is contained in the upper half-plane. Consider the following map (as described in Section 2) in the new coordinate setting:
F ( x 1 , x 2 ) = { ( x 1 , x 2 ) f ( x 1 , x 2 ) , if  x 2 0 , ( x 1 , x 2 ) f ( x 1 , 0 ) , if  x 2 < 0 .
Now write F = ( F 1 , F 2 ) and define g ( x 1 ) = F 1 ( x 1 , 0 ) . Since f ( ζ ) = ζ k for ζ S 1 , we have
g ( x 1 ) = F 1 ( x 1 , 0 ) = x 1 k x 1 = ( 1 k ) x 1 .
Since the map f is defined to be smooth on B 2 , the map F is smooth on the upper half-plane. Let F + denote the restriction of F to the upper half-plane. We will see that smoothness is only required in a neighborhood of the fixed point at 0. Since we are assuming that k 2 , then
d d x F 1 ( x 1 , 0 ) = g ( x 1 ) = 1 k < 0
and the smoothness of F + implies that
F 1 + ( x 1 , x 2 ) x 1 < 0

for ( x 1 , x 2 ) in an ϵ-neighborhood of the origin, for ϵ > 0 sufficiently small and for x 2 0 . Let Γ be a circle of radius ϵ / 2 about the origin.

Let Γ + and Γ denote the half-circles above and below the x 1 -axis respectively. Since F takes the lower half-plane to itself, we know that F maps Γ to the lower half-plane. Calculating the fixed point index of f at the origin in R 2 is equivalent to finding the winding number of F ( Γ ) around 0. Thus, we need to understand F ( Γ + ) . Since Γ + lies in the upper half-plane, we only consider F + . Assuming F has only a finite number of fixed points, we can choose ϵ small enough so that only one point on Γ or its interior that F maps to the origin is the origin itself. Therefore, we can homotope the restriction of F + to Γ + in R 2 0 to the restriction of F + to the curve Γ δ + for δ > 0 given by
Γ δ + ( t ) = ( ϵ 2 ( 2 t 1 ) , δ ( 1 ( 1 2 t ) 2 ) ) ,

where 0 t 1 .

We write the restriction of F + to Γ δ + in coordinates as
F + ( Γ δ + ( t ) ) = ( F 1 + ( Γ δ + ( t ) ) , F 2 + ( Γ δ + ( t ) ) ) = ( ϕ δ ( t ) , ψ δ ( t ) ) .
The key idea of the proof is that for δ sufficiently small, the smoothness of F + and the fact that
F 1 + ( x 1 , x 2 ) x 1 < 0 implies that d d t ϕ δ ( t ) < 0

for all t. This tells us that the x 1 -coordinate of the curve F + ( Γ δ + ( t ) ) is a strictly monotone function of t. In particular, the curve F + ( Γ δ + ( t ) ) only crosses the x 2 -axis once. This implies the desired result since the winding number of F ( Γ ) can then only be either 0 or −1.

Notice that it is never specified that f maps B 2 into itself. In considering the map F ( z ) = z f ( z ) , although it is assumed that F maps the exterior of the disc to the exterior of the disc, the proof allows the image of the interior of the disc under F to lie anywhere in R 2 . □

Let f ¯ : B 2 R 2 be the restriction of f : ( R 2 , S 1 ) ( R 2 , S 1 ) to B 2 . Since B 2 is contractible, the sum of the indices of ρ f ¯ : B 2 B 2 equals one, and therefore ρ f ¯ ( x ) = x for some x int ( B 2 ) . But then f ( x ) = x as well, so f : ( B 2 , S 1 ) ( R 2 , S 1 ) has a fixed point in the interior of B 2 . Therefore, we can extend Corollary 2 as the following result.

Corollary 4 Let f : ( R 2 , S 1 ) ( R 2 , S 1 ) be a smooth map such that f ( ζ ) = ζ k for all ζ S 1 for some k 2 . Then there exist z int ( B 2 ) such that f ( z ) = z .

4 Interior fixed points of a map f : ( R n , S n 1 ) ( R n , S n 1 )

Definition 1 ([3], p.34)

A smooth map ϕ : S n 1 S n 1 with finitely many fixed points is transversely fixed if d ϕ p I : T p ( S n 1 ) T p ( S n 1 ) is a nonsingular linear map for each fixed point p. For F = { p 1 , , p r } a fixed point class of ϕ, let
i ( F ) = j = 1 r i ( S n 1 , ϕ , p j ) .
The transverse Nielsen number N ( ϕ ) is defined by
N ( ϕ ) = F F | i ( F ) | ,

where F is the set of fixed point classes of ϕ.

A smooth map ϕ : S n 1 S n 1 is sparse if it is transversely fixed and it has N ( ϕ ) fixed points.

In [3], p.45 Schirmer obtained the following result.

Theorem 5 Let ϕ : S n 1 S n 1 be a sparse map of degree d and suppose f : ( B n , S n 1 ) ( B n , S n 1 ) is a smooth map extending ϕ. If ( 1 ) n d 2 , then f must have a fixed point in int ( B n ) .

We will extend this result as a consequence of the following

Theorem 6 Given a smooth map ϕ : S n 1 S n 1 and a smooth map f : ( B n , S n 1 ) ( R n , S n 1 ) extending ϕ, suppose that p S n 1 is an isolated fixed point of f and that d ϕ p I : T p ( S n 1 ) T p ( S n 1 ) is a nonsingular linear transformation. Then either i ( B n , f , p ) = 0 or i ( B n , f , p ) = i ( S n 1 , ϕ , p ) .

Proof The following proof is a modified version of Theorem 5.1 in [1]. We again write f in a small ball that contains p S n 1 as described in Section 2 and the map F is also as defined there. Moreover, for ε > 0 small enough, let
D ε ( x 1 , , x n ) = { ( ε x 1 , ε x 2 , , ε x n ) f ( ε x 1 , ε x 2 , , ε x n ) , if  x n 0 , ( ε x 1 , ε x 2 , , ε x n ) f ( ε x 1 , ε x 2 , , ε x n 1 , 0 ) , if  x n < 0 .

This then means that the index i ( B n , f , p ) = i ( B n , D ε , 0 ) is the degree of ρ D ε : S n 1 S n 1 , where ρ ( x ) = x / | x | for x R n 0 .

Note that
| F ( ε x 1 , ε x 2 , , ε x n ) d F p ( ε x 1 , ε x 2 , , ε x n ) | o ( ε )
because F is a C 1 function. We also have
| d F p ( ε x 1 , ε x 2 , , ε x n ) | C ε

for some C > 0 independent of ε and ( x 1 , x 2 , , x n ) due to the fact that d F p = ( d f p I ) is nonsingular by hypothesis.

Since d F p is a nonsingular linear map, this last degree is easily seen to be 0, or ±1. But the images of D ε of the upper and lower hemisphere are each contained entirely in either the lower or upper half-space. This means that D ε is of degree 0 or D ε is homotopic in R n 0 to the suspension of D ε | S n 2 . □

We can use Theorem 6 to extend Theorem 5 to the case f : ( R n , S n 1 ) ( R n , S n 1 ) . The following is a modified version of part of [3]. Note that despite the fact that the case n = 2 is solved in the previous section, the new material presented below extends the solution to all the cases.

Suppose we have ϕ : S n 1 S n 1 and a smooth map f : ( R n , S n 1 ) ( R n , S n 1 ) extending ϕ. A fixed point class F of f is called a common fixed point class of f and ϕ if there exists an essential fixed point class F ¯ of ϕ which is contained in F.

We will again consider the restriction f : ( B n , S n 1 ) ( R n , S n 1 ) . In the notation of [3], p.39, let
u ( F ) = max { 0 , ( i ( S n 1 , ϕ , F ¯ ) | F ¯ F  and  i ( S n 1 , ϕ , F ¯ ) > 0 ) } , l ( F ) = min { 0 , ( i ( S n 1 , ϕ , F ¯ ) | F ¯ F  and  i ( S n 1 , ϕ , F ¯ ) < 0 ) } .
Then F is a transversally common fixed point class of f and ϕ if
l ( F ) i ( F ) u ( F ) .
Definition 2 The boundary transversal Nielsen number of f : ( B n , S n 1 ) ( R n , S n 1 ) is
N ( f ; B n , S n 1 ) = N ( ϕ ) + N ( f ) N ( f , ϕ ) ,

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

Suppose that ϕ : S n 1 S n 1 has degree d. Then f : B n R n has one fixed point class F with i ( F ) = 1 , and so l ( F ) 0 < i ( F ) . If n = 2 , then ϕ has | 1 d | essential fixed point classes, each of the same index, and i ( S 1 , ϕ , F ¯ ) = L ( ϕ ) = 1 d . Hence, i ( F ) u ( F ) if and only if d 0 , and
N ( f , ϕ ) = { 1 if  d 0 , 0 if  d > 0 .
If n 3 , then ϕ has one fixed point class F ¯ with i ( S n 1 , ϕ , F ¯ ) = 1 + ( 1 ) n 1 d and so
N ( f , ϕ ) = { 1 if  ( 1 ) n 1 d 0 , 0 if  ( 1 ) n 1 d < 0 .

Note that this formula is still true for the case n = 2 .

If ϕ : S n 1 S n 1 is a sparse map of degree d then N ( ϕ ) = | 1 ( 1 ) n d | for all d and all n 2 .

Since N ( f ) = 1 , for the case that ( 1 ) n d 0 , the boundary transversal Nielsen number is
N ( f ; B n , S n 1 ) = N ( ϕ ) + N ( f ) N ( f , ϕ ) = | 1 ( 1 ) n d | + 1 1 = | 1 ( 1 ) n d | .
As for the case that ( 1 ) n d > 0 , the boundary transversal Nielsen number is
N ( f ; B n , S n 1 ) = N ( ϕ ) + N ( f ) N ( f , ϕ ) = | 1 ( 1 ) n d | + 1 0 = | 1 ( 1 ) n d | + 1 .

Hence, we have just proven the following

Proposition 7 If f : ( B n , S n 1 ) ( R n , S n 1 ) is a smooth extension of a sparse map ϕ : S n 1 S n 1 of degree d, with n 2 , then the boundary transversal Nielsen number is
N ( f ; B n , S n 1 ) = { | 1 ( 1 ) n d | if  ( 1 ) n d 0 , | 1 ( 1 ) n d | + 1 if  ( 1 ) n d > 0 .

As defined in [1], p.2, the extension Nielsen number N ( f | ϕ ) is a lower bound for the number of fixed points on S n 1 of continuous extensions of a continuous map ϕ. It is equal to the number of essential (classical) fixed point classes F of f with F S n 1 = . A fixed point class F is representable on S n 1 if there exists a subset F F S n 1 with i ( B n , f , F ) = i ( S n 1 , ϕ , F ) . The smooth extension number N 1 ( f | ϕ ) is the number of essential (classical) fixed point classes F of f which are not representable on S n 1 . It is a lower bound for the number of fixed points in S n 1 of a smooth extension of a smooth and transversally fixed map ϕ.

Proposition 8 If ϕ : S n 1 S n 1 is sparse, then
N 1 ( f | ϕ ) = N ( f ; B n , S n 1 ) N ( ϕ ) , N ( f | ϕ ) = N ( f ; B n , S n 1 ) N ( ϕ ) .

Proof Our proof is modeled on the proofs of Proposition 4.3 and Corollary 4.4 in [3]. For any essential fixed point class F of f, since ϕ is sparse, F S n 1 contains u ( F ) fixed points p such that i ( S n 1 , ϕ , p ) = 1 and l ( F ) fixed points p such that i ( S n 1 , ϕ , p ) = 1 . This means that F is representable on S n 1 if and only if l ( F ) i ( F ) u ( F ) . By the definitions of all the Nielsen numbers involved, we have the result stated above for N 1 ( f | ϕ ) .

The result for N ( f | ϕ ) can be obtained in a similar manner by using Corollary 2.6 from [1] along with the fact that all fixed point classes of a sparse map are essential. □

By the definitions of N ( f , ϕ ) in [5] and the definition of N 1 ( f | ϕ ) defined early, we obtain

Proposition 9 If ϕ : S n 1 S n 1 is sparse, then the number of essential fixed point classes of f which are common but not transversally common is
N 1 ( f | ϕ ) N ( f | ϕ ) = N ( f , ϕ ) N ( f , ϕ ) .

We are now ready to prove the following Theorem.

Theorem 10 Let n 2 , and let ϕ : S n 1 S n 1 be a sparse map of degree d and suppose f : ( B n , S n 1 ) ( R n , S n 1 ) is a smooth map extending ϕ. If ( 1 ) n d 2 , then f must have a fixed point in int ( B n ) .

Proof Since ϕ has degree d and it is sparse, by definition
N ( f | ϕ ) = { 1 if  d ( 1 ) n , 0 if  d = ( 1 ) n
and
N 1 ( f | ϕ ) N ( f | ϕ ) = N ( f ; B n , S n 1 ) N ( ϕ ) N ( f | ϕ ) ( from Proposition 8 ) = N ( f ; B n , S n 1 ) | 1 ( 1 ) n d | N ( f | ϕ ) .
Applying Proposition 7, we have
N 1 ( f | ϕ ) N ( f | ϕ ) = { 0 if  ( 1 ) n d 1 , 1 if  ( 1 ) n d 2 .

Thus, every smooth extension over B n of a sparse map of S n 1 of degree d with ( 1 ) n d 2 has a fixed point on the interior of B n . □

Declarations

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.

Authors’ Affiliations

(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

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