Interior fixed points of unit-sphere-preserving Euclidean maps
Fixed Point Theory and Applications volume 2012, Article number: 183 (2012)
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.
For spaces X, Y and subsets , , a map is an extension of a map if for all . We denote by the unit ball in , by its boundary and by its interior.
If is an extension of and ϕ has no fixed points, then f must have an interior fixed point, that is, a fixed point in . However, if ϕ has a fixed point, then there need not be any interior fixed points.
If , the situation is more complicated. Of course the Brouwer fixed point theorem implies that a map must have at least one interior fixed point if it is an extension of a map that has no fixed points. But it was proved in  (see also ) 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 by complex numbers, let , for an integer d, be the power map defined by . If and is a smooth extension of , then f has at least one interior fixed point. It is also demonstrated in  that interior fixed points of extensions need not exist if or if f is not smooth. Schirmer generalized this interior fixed point result to smooth extensions for to show in Example 4.7 of  that if f is a smooth extension of a ‘sparse’ map , a generalization of that is defined below, of degree d such that , then f must have at least one interior fixed point.
Returning to the case , if we extend the map without fixed points to a map , there still must be a fixed point of f in . The reason for the interior fixed points of the extension of the map of without fixed points, namely that and , lie in different components of , where , applies also to the extension since those points are also in different components of .
On the other hand, the reason for the presence of fixed points in for smooth extensions of certain maps of demonstrated in  is considerably more subtle. Therefore, it is reasonable to ask whether such fixed points would persist if, instead of smooth extensions of , we consider extensions that are smooth Euclidean maps, that is, maps . Thus, we ask whether there still must be fixed points of f in if we allow f to map points of outside of .
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 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 of a power map for must have at least one interior fixed point. Section 4 then contains the proof that Schirmer’s result generalizes to smooth extensions of sparse maps that satisfy the same degree restrictions.
2 The fixed point indices
Before extending the results of [1, 2] and  to the case of a smooth Euclidean map extending , we need to define the relevant fixed point indices. We will consider the restriction . Since our goal is to establish conditions for the existence of fixed points on the interior of , the behavior of the function outside of is not relevant. Therefore, we will make use of the indices and of an isolated fixed point . We do so by generalizing the approach used in  (see also ).
For an isolated fixed point , 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 is contained in the upper half-space
in such a way that p is the origin 0 in this setting and is contained in the subspace
In order to calculate the index of p in each space, we consider the map defined by
Note that F sends the origin, lower half-plane and to itself respectively. Also, for . The index is equal to in this setting as in the traditional definition of the index. Moreover, is identified with , which can be computed as the degree of the map where is the retraction defined by .
3 Unit-circle-preserving maps of the plane
Brown, Greene and Schirmer proved
Let be a smooth map with a finite number of fixed points such that for all for some , where is the closed two-dimensional ball with the boundary . If π is a fixed point of f that lies in , then either or .
The contractibility of implies the following corollary.
Corollary 2 
Suppose is a smooth map such that for all , for some . Then there exists such that .
We will extend Theorem 1 to maps by modifying the proof of Theorem 1 in . Corollary 2 will then extend to maps .
Theorem 3 Let be a smooth map with a finite number of fixed points such that for all for some . If π is a fixed point of f that lies in , then either or .
Proof Let π be a fixed point of f in . We can write this fixed point in the polar coordinates as . We will introduce new coordinates on a neighborhood U of π as follows:
In the new coordinate setting, the fixed point π is the origin and corresponds to the -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:
Now write and define . Since for , we have
Since the map f is defined to be smooth on , the map F is smooth on the upper half-plane. Let 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 , then
and the smoothness of implies that
for in an ϵ-neighborhood of the origin, for sufficiently small and for . Let Γ be a circle of radius about the origin.
Let and denote the half-circles above and below the -axis respectively. Since F takes the lower half-plane to itself, we know that F maps to the lower half-plane. Calculating the fixed point index of f at the origin in is equivalent to finding the winding number of around 0. Thus, we need to understand . Since lies in the upper half-plane, we only consider . 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 to in to the restriction of to the curve for given by
We write the restriction of to in coordinates as
The key idea of the proof is that for δ sufficiently small, the smoothness of and the fact that
for all t. This tells us that the -coordinate of the curve is a strictly monotone function of t. In particular, the curve only crosses the -axis once. This implies the desired result since the winding number of can then only be either 0 or −1.
Notice that it is never specified that f maps into itself. In considering the map , 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 . □
Let be the restriction of to . Since is contractible, the sum of the indices of equals one, and therefore for some . But then as well, so has a fixed point in the interior of . Therefore, we can extend Corollary 2 as the following result.
Corollary 4 Let be a smooth map such that for all for some . Then there exist such that .
4 Interior fixed points of a map
Definition 1 (, p.34)
A smooth map with finitely many fixed points is transversely fixed if is a nonsingular linear map for each fixed point p. For a fixed point class of ϕ, let
The transverse Nielsen number is defined by
where is the set of fixed point classes of ϕ.
A smooth map is sparse if it is transversely fixed and it has fixed points.
In , p.45 Schirmer obtained the following result.
Theorem 5 Let be a sparse map of degree d and suppose is a smooth map extending ϕ. If , then f must have a fixed point in .
We will extend this result as a consequence of the following
Theorem 6 Given a smooth map and a smooth map extending ϕ, suppose that is an isolated fixed point of f and that is a nonsingular linear transformation. Then either or .
Proof The following proof is a modified version of Theorem 5.1 in . We again write f in a small ball that contains as described in Section 2 and the map F is also as defined there. Moreover, for small enough, let
This then means that the index is the degree of , where for .
because F is a function. We also have
for some independent of ε and due to the fact that is nonsingular by hypothesis.
Since is a nonsingular linear map, this last degree is easily seen to be 0, or ±1. But the images of of the upper and lower hemisphere are each contained entirely in either the lower or upper half-space. This means that is of degree 0 or is homotopic in to the suspension of . □
We can use Theorem 6 to extend Theorem 5 to the case . The following is a modified version of part of . Note that despite the fact that the case is solved in the previous section, the new material presented below extends the solution to all the cases.
Suppose we have and a smooth map 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 of ϕ which is contained in F.
We will again consider the restriction . In the notation of , p.39, let
Then F is a transversally common fixed point class of f and ϕ if
Definition 2 The boundary transversal Nielsen number of is
where is the number of essential and transversally common fixed point classes of f and ϕ.
Suppose that has degree d. Then has one fixed point class F with , and so . If , then ϕ has essential fixed point classes, each of the same index, and . Hence, if and only if , and
If , then ϕ has one fixed point class with and so
Note that this formula is still true for the case .
If is a sparse map of degree d then for all d and all .
Since , for the case that , the boundary transversal Nielsen number is
As for the case that , the boundary transversal Nielsen number is
Hence, we have just proven the following
Proposition 7 If is a smooth extension of a sparse map of degree d, with , then the boundary transversal Nielsen number is
As defined in , p.2, the extension Nielsen number is a lower bound for the number of fixed points on of continuous extensions of a continuous map ϕ. It is equal to the number of essential (classical) fixed point classes F of f with . A fixed point class F is representable on if there exists a subset with . The smooth extension number is the number of essential (classical) fixed point classes F of f which are not representable on . It is a lower bound for the number of fixed points in of a smooth extension of a smooth and transversally fixed map ϕ.
Proposition 8 If is sparse, then
Proof Our proof is modeled on the proofs of Proposition 4.3 and Corollary 4.4 in . For any essential fixed point class F of f, since ϕ is sparse, contains fixed points p such that and fixed points p such that . This means that F is representable on if and only if . By the definitions of all the Nielsen numbers involved, we have the result stated above for .
The result for can be obtained in a similar manner by using Corollary 2.6 from  along with the fact that all fixed point classes of a sparse map are essential. □
By the definitions of in  and the definition of defined early, we obtain
Proposition 9 If is sparse, then the number of essential fixed point classes of f which are common but not transversally common is
We are now ready to prove the following Theorem.
Theorem 10 Let , and let be a sparse map of degree d and suppose is a smooth map extending ϕ. If , then f must have a fixed point in .
Proof Since ϕ has degree d and it is sparse, by definition
Applying Proposition 7, we have
Thus, every smooth extension over of a sparse map of of degree d with has a fixed point on the interior of . □
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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.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Khamsemanan, N., Brown, R.F., Lee, C. et al. Interior fixed points of unit-sphere-preserving Euclidean maps. Fixed Point Theory Appl 2012, 183 (2012). https://doi.org/10.1186/1687-1812-2012-183