Lefschetz fixed-point theory applies to smooth maps F : X → Xwhere X is a com- pact oriented manifold [Guillemin and Pollack, 2010; Hirsch, 2012] or a compact triangulable space [Armstrong, 2013]1. ThusX= Rn is excluded, but ifXis a com- pact subset ofRnsuch as a simplex, then it is allowed. This also means that if a map F :Rn → Rn is known to have no fixed points for large values of its argument, the theory can often be applied by considering the restriction of F to a compact subset of Rn, such as a ball of large enough radius for which F in Eq. (8.1) needs to be positively invariant.
Lefschetz fixed-point theory involves derivatives. Any smooth map has the pro- perty that at any point x ∈ X, there is a linear derivative mapping, call it dFx, and if Xlooks locally like Rm, then the derivative map can be represented by the m×m
Jacobian matrix in the local coordinate basis.
Interest is centred for the purposes of this chapter on those maps which have a finite number of fixed points (including possibly zero) inX, though of course, maps with an infinite set of fixed points exist, for example F(x) = x, i.e. the identity map. A fixed pointxis called aLefschetz fixed point ofFif the eigenvalues ofdFxare unequal to 1. A fixed point being a Lefschetz fixed point is sufficient but not necessary to ensure that x is an isolated fixed point of F, i.e. there is an open neighbourhood around x in which no other fixed point occurs. Because X is compact, and if it is known that all fixed points of F are isolated, say because they are all Lefschetz fixed points, it easily follows that the number of fixed points is necessarily finite. For completeness, an argument by contradiction is recorded, which seems standard. If there were an infinite number of fixed points, xi,i = 1, 2, . . . , compactness of X implies there is a convergent subsequence xi1,xi2, . . . , with limit point ¯x, and again
by compactness ¯x ∈ X. Now F is continuous so F(xij) → F(x¯) since xij → x¯ as
j→∞. Thenxij−F(xij)→ x¯−F(x¯)as j→∞. Sincexij−F(xij) =0∀j, it is evident
1The notion of orientation of a manifold is described in the references; roughly, a manifold is oriented
if one can attach an infinitesimal set of coordinate axes to an arbitrary point on the manifold, and then move the point with the axes attached knowing that one can never move to reverse the orientation. A
that ¯xis a fixed point of F. However, being a limit point it is not isolated, hence the contradiction.
The Lefschetz property holding at a particular fixed point x also implies that at the point x, the (linear) mapping Im−dFx is an isomorphism of the tangent space
Tx(X) at x. If it preserves orientation, then its determinant is positive, while if it reverses orientation, its determinant is negative. Thelocal Lefschetz number of F at a fixed pointx, written Lx(F), is defined as+1 or−1 according as the determinant of
Im−dFxis positive or negative.2
The mapFis termed aLefschetz mapif and only if all its fixed points are Lefschetz fixed points (and there are then, as noted above, a finite number of fixed points). The
Lefschetz number of F, writtenL(F), is defined as
L(F) =
∑
F(x)=x
Lx(F) (8.3)
There is an alternative definition of the Lefschetz number not provided here which can be shown to be equivalent to that appearing here, based on topologi- cal considerations, and provided in [Hirsch, 2012; Guillemin and Pollack, 2010]. It isnot restricted to maps with a finite number of fixed points. Moreover, using this alternative definition, one sees that L(F) is a homotopy invariant3, and this particu- lar property does not require limitation to those maps with a finite number of fixed points. Further, the alternative approach yields a connection between the Lefschetz number of the identity map (which has an infinite number of fixed points) and anot- her topological invariant of the underlying space X, viz the Euler characteristic4, [Hirsch, 2012; Guillemin and Pollack, 2010; Matsumoto, 2002].
The key result (see e.g. [Hirsch, 2012] for the case of a compact oriented manifold and [Armstrong, 2013] for the case of a compact triangulable space) is as follows: Theorem 8.1. The Lefschetz number of the identity mapId : X→X where X is a compact oriented manifold or a compact triangulable space isχ(X), the Euler characteristic of X.
A key consequence of this theorem is that if a map Fis homotopically equivalent to Id, i.e. if there exists a smooth map H : X×I → X such thatH(x, 0) = F(x)and H(x, 1) =Id(x) =xthen
L(F) =χ(X) (8.4)
Hence one has the following theorem:
2Reference [Guillemin and Pollack, 2010] uses dF
x−Im rather than Im−dFx, which is used by
[Hirsch, 2012]. The latter form is what is required.
3Smooth mapsF :X →XandG :X →Xare said to be homotopic if there exists a smooth map
H : X×I → X×I with H(x, 0) = F(x), and H(x, 1) = G(x). Saying L(F) is a homotopy invariant
meansL(F) =L(G)for anyGwhich is homotopic toF.
4The Euler characteristic is an integer number associated with a topological space, including a space
that in some sense is a limit of a sequence of multidimensional polyhedra, e.g. a sphere, and a key property is that distortion or bending of the space leaves the number invariant. Euler characteristics are known for a great many topological spaces.
Theorem 8.2(Specialisation of Lefschetz–Hopf Theorem). Let X be a compact oriented manifold or a compact triangulable space, and suppose F : X → X is a Lefschetz map, i.e. there are a finite number of fixed points at each of whichIm−dFx is an isomorphism, and is
homotopically equivalent to the identity map. Then there holds
L(F) =
∑
F(x)=x
Lx(F) =χ(X) (8.5)
where Lx(F) is +1 or −1 according as det(Im−dFx) has positive or negative sign, and χ(X)is the Euler characteristic of X.
8.2.2 A General Convergence Result
The general result established in [Anderson and Ye, 2018] will now be presented. It is assumed throughout this subsection that a mapping Fhas at least one fixed point (this perhaps being established by a standard fixed point result, e.g. Brouwer Fixed Point Theorem, [Guillemin and Pollack, 2010; Hirsch, 2012]). The result will establish that certain properties of the mapping F and the associated space X guarantee that F has a unique fixed point and the system Eq. (8.1) is locally exponentially stable about the unique fixed point. The main result, proved using the Lefschetz theory, is as follows.
Theorem 8.3. Consider a smooth map F : X → X where X is a compact, oriented and convex manifold or a convex triangulable space of arbitrary dimension. Suppose that the eigenvalues ofdFx have magnitude less than 1 for all fixed points ofF. ThenF has a unique
fixed point, and in a local neighborhood about the fixed point, Eq. (8.1) converges to the fixed point exponentially fast.
Proof. Observe first that the compactness and convexity properties ofX guarantee it
is homotopy equivalent to the unitm-dimensional diskDmand accordingly then ho- motopy equivalent to a single point. This means thatχ(X) =1, see e.g. [Matsumoto, 2002, pp. 140].
Next, observe that, because X is convex, H = tId+ (1−t)F which maps x to
tx+ (1−t)F(x), is a mapping from X to Xfor every t ∈ [0, 1] and the smoothness properties of H (which come from the smoothness ofF and the specific dependence on t) then guarantee that F and Id are homotopically equivalent. By Theorem 8.2, there holds
L(F) =1 (8.6)
Now for any real matrix A ∈ Rm×m for which the eigenvalues are less than one in magnitude, it is easily seen that the matrix Im−Ahas eigenvalues all with positive real part, from which it follows that the determinant of Im−Ais positive, since the determinant is equal to the product of the eigenvalues. Hence for any fixed point x of F, one observes by identifying AwithdFx that there necessarily holds Lx(F) = 1. By Eq. (8.3) and Eq. (8.6), it follows that 1 = ∑F(x)=x1, or that there is precisely one fixed point.
Convergence of Eq. (8.1) to the unique fixed point from any initial value in its re- gion of attraction is necessarily exponentially fast. In a neighbourhoodDaround the unique fixed point, the eigenvalue property of dFx guarantees exponential conver- gence. The region of attraction for the fixed point is in most instances larger thanD, and one denotes asU ⊂Xan arbitrarycompactspace within the region of attraction, and containingD. For any initialx∈ U, the sequencex,F(x),F(F(x)), . . . converges to the neighbourhood D in a finite number of steps, and because the setU is com- pact, there is a number of steps, ¯N < ∞ say, such that from all initial conditions in
U, the neighbourhood is reached in no more than ¯Nsteps. The finiteness of ¯N then implies that exponentially fast convergence occurs∀x(0)∈ U.
Remark 8.1. The proof of the theorem using Lefschetz ideas will clearly generalise in the following way. Suppose that F is homotopic to the identity and X is not homotopic to the unit ball, while all fixed points are Lefschetz with the property that Im−dFx has positive
determinant. Then the number of fixed points will be χ(X), and Eq. (8.1) will have χ(X)
locally exponentially stable equilibria. If for exampleF mapped S2 to S2(i.e. the unit sphere
embedded in three-dimensional Euclidean space) and never mapped a point to its antipodal point, i.e. there was no xfor whichF(x) =−x, it will be homotopic to the identity map and then there will be two fixed points, sinceχ(S2) =2. To construct the homotopy, observe that,
because of the exclusion thatFcan map any point to an antipodal point, there is a well-defined homotopy provided by
H(x,t) = (1−t)x+tF(x)
k(1−t)x+tF(x)k2