Morse-Smale systems

Post-publication activity

Curator: Michael Shub

More-Smale systems are the simplest dynamical systems. They are structurally stable and have intimate connections to the topology of manifolds. We elucidate these concepts below.

Dynamical Systems

By a dynamical system we mean here a $$C^r, r \ge 1\ ,$$ diffeomorphism $$f$$ of a compact differentiable manifold $$M$$ without boundary or a one parameter group $$\phi: \Bbb R \rightarrow$$ $${\rm Diff}^r(M)$$ such that $\phi(t + s) = \phi (t) \phi (s)\ ,$ and the vector field $$X_\phi (m) = \frac{d}{dt} \phi_t (m) |_{t = 0}$$ defined by the tangents to the orbits is well defined and $$C^r\ .$$

Periodic orbits

A periodic orbit of $$f$$ is a set of points $$x, f(x), \dots, f^m (x) = x\ .$$ The minimum such $$m$$ is called period of $$x\ .$$ If the period of $$x$$ is 1, $$x$$ is called a fixed point.

A periodic orbit for $$\phi (t)$$ is a set $$\phi (t) m$$ with $$\phi (t_0) (m) = m$$ for some $$t_0 \in \Bbb R\ .$$ The minimum non-negative such $$t_0$$ is called the period of $$m\ .$$ If the period of $$m$$ is $$0\ ,$$ $$m$$ is called an equilibrium (or fixed or singular) point of the vector field $$X_\phi (m)\ .$$ A periodic orbit of $$\phi (t)$$ which is not a fixed point is diffeomorphic to a circle.

Hyperbolicity

A subset $$\wedge \subset M$$ is invariant for $$f$$ if $$f (\wedge) = \wedge$$ and for $$\phi (t)$$ if $$\phi (t) \wedge = \wedge$$ for all $$t\ .$$ Periodic orbits are examples of invariant sets.

An invariant set $$\wedge$$ for $$f$$ is hyperbolic if there are constants $$C >, 0 < \lambda < 1$$ and a decomposition of the tangent bundle $$T M$$ of $$M$$ left invariant by the derivative $$T f$$ of $$f\ .$$ $T M / \wedge = E^S \oplus E^U \quad \quad \quad T f (E^S) = E^S$ $T f (E^U) = E^U$ and $\| T f^n (x) v \| \le C \lambda^n \| v \| \quad \rm for \ v \in E^S \quad n > 0$ $\|T f^{-n} (x) v \| \le C \lambda^n \| v \| \quad \rm for \ v \in E^U \quad n \ge 0\ .$ An invariant set $$\wedge$$ for $$\phi (t)$$ is hyperbolic if there are constants $$C > 0, 0 < \lambda < 1$$ and a decomposition of the tangent bundle $$T M of M$$ left invariant by the derivatives $$T \phi (t)$$ for all $$t$$ $T M | \wedge = E^S \oplus E^C \oplus E^U$ $T \phi (t) (E^U) = E^U \quad T \phi (t) (E^S) = E^S \quad T \phi (t) (E^C) = E^C$ $$E^C$$ is the bundle tangent to the orbits of $$\phi (t)$$ i.e. the bundle defined by the vector field $$X_\phi (m)$$ and $\|T \phi (t) (x) v \| \le C \lambda^t \| v \| \quad \rm for \quad v \in E^s, \quad t \ge 0$ $\| T \phi (-t) (x) v \| \le C \lambda^t \| v \| \quad \rm for \quad v \in E^U, \quad t \ge 0$ When $$x$$ is an equilibrium of $$\phi (t)\ ,$$ then the bundle $$E^C$$ is just the zero vector and $$E^C$$ need not appear in the decomposition of $$T_x M\ .$$

Stable and unstable manifolds

If $$x, f(x), \dots, f^{m-1} (x)$$ is a hyperbolic periodic orbit of x of period m then the stable and unstable manifolds of $$x$$ $W^S (x) = \{y \in M | f^{mk} (y) \rightarrow x \quad \rm as \ k \rightarrow \infty\}$ $W^U (x) = \{y \in M | f^{mk} (y) \rightarrow x \quad \rm as \ k \rightarrow - \infty \}$ respectively, are 1-1 immersed Euclidean spaces of dimension equal to the dimension of $$E^S$$ and $$E^U$$ respectively and are tangent to $$E^S_x$$ and $$E^U_x$$ at $$X\ .$$

In the case a hyperbolic periodic orbit, C, of a one parameter group of diffeomorphisms, $$\phi (t)$$ the stable and unstable manifolds of $$C$$ are $W^S (C) = \{ y \in M | \phi_t (y) \rightarrow C \quad \rm as \ t \rightarrow \infty \}$ $W^U (C) = \{ y \in M | \phi_t (y) \rightarrow C \quad \rm as \ t \rightarrow - \infty \}$ When $$C$$ is a fixed point they are 1-1 immersed discs tangent to $$E^S_C$$ and $$E^U_C$$ at $$C$$ as for diffeomorphisms. When $$C$$ is diffeomorphic to a circle they are 1-1 immersed cylinders (i.e. vector bundles over $$C$$ oriented or not) tangent to $$E^s \oplus T C$$ and $$T C \oplus E^U$$ at $$C\ .$$

Morse-Smale Dynamical systems: Definition

The diffeomorphism $$f$$ or one parameter group $$\phi (t)$$ is called Morse-Smale iff there are a finite collection of periodic orbits $$P_1, \dots, P_l$$ such that

1. $$P_i$$ is hyperbolic $$i = 1, \dots, l$$
2. $$\bigcup_{i=1}^l W^S (P_i) = M$$
3. $$\bigcup_{i=1}^l W^U (P_i) = M$$
4. $$W^U (P_i)$$ and $$W^S (P_j)$$ are transversal for all $$1 \le i, j \le l\ .$$

The definition implies that for a diffeomorphism $$f$$ and any $$m \in M$$ $$f^n (m) \rightarrow P_i$$ for some $$i$$ as $$n \rightarrow \infty$$ and $$f^{-n} (m) \rightarrow P_j$$ for a different $$P_j$$ as $$n \rightarrow \infty$$ unless $$m$$ is itself periodic.

For a one parameter group the definition implies that $\phi (t) m \rightarrow P_i$ some $$i$$ as $$t \rightarrow \infty$$ $\phi (t) m \rightarrow P_j$ for a different $$P_j$$ as $$t \rightarrow - \infty$$ unless $$m$$ itself is periodic.

Thus the limiting future or past behaviour of orbits of Morse-Smale dynamical systems is simple.

Topological Conjugacy

Two diffeomorphisms $$f : M \rightarrow M, \ g : N \rightarrow N$$ are topologically conjugate if there is a homeomorphism $$h : M \rightarrow N$$ such that $$h$$ maps orbits of $$f$$ to orbits of $$g$$ preserving their "orientation." More precisely $$g(h(x)) = h(f(x))$$ for each $$x \in M\ .$$

Two one-parameter group $\phi (t) : M \rightarrow M$ $\varphi (t) : N \rightarrow N$ are topologically conjugate if there is a homeomorphism $$h : M \rightarrow N$$ taking orbits to orbits i.e. $h \{ \phi (t) (m) \}_{t \in \Bbb R} = \{ \varphi (t) h (m) \}_{t \in \Bbb R}$ for each $$m \in M$$ The parametrization of the orbits is not necessarily preserved, but the natural orientation given by the parametrization is.

Structural Stability

The diffeomorphism $$f$$ is structurally stable if there is a neighborhood $$U$$ of $$f$$ in $$\rm Diff^r (M)$$ such that $$f$$ and $$g$$ are topologically conjugate for all $$g \in U\ .$$

The one parameter group $$\phi (t)$$ is structurally stable if there is a neighborhood $$U$$ of $$X_\phi$$ in the $$C^r$$ vector fields $$\Chi^r(M)$$ such that if $$\varphi (t)$$ is a one parameter group of diffeomorphisms with $$X_\varphi$$ in $$U$$ then $$\phi (t$$) and $$\varphi (t)$$ are topologically conjugate.

Theorem: Morse-Smale dynamical systems are structurally stable.
This theorem of Palis and Smale has generalizations to more complex dynamical systems in which the non-wandering set is assumed to be hyperbolic.

Morse-Smale Gradient Fields and Relations to Topology

The first examples of Morse-Smale dynamical systems are gradient flows. These flows have their origin in optimization theory and the calculus of variations on finite and even infinite dimensional spaces. Here we will restrict our attention to compact differentiable manifolds, $$M$$ without boundary. We equip $$M$$ with a smooth Riemannian metric and consider $$f : M \rightarrow \Bbb R$$ a smooth function. Then $$V (x) = - {\rm grad} f (x)$$ defines a vector field on $$M\ .$$ Any smooth vector field $$V$$ on a compact $$M$$ defines a flow $$\phi_t$$ on $$M\ .$$ $$\phi_t : M \rightarrow M$$ is a one parameter group of diffeomorphisms of $$M$$ which are defined by $$\frac{d}{d t} \phi_t (x) |_{t = 0} = V (x)\ .$$ The $$\phi_t$$ are globally defined for all $$t \in \Bbb R$$ and satisfy $$\phi_{t +s} = \phi_t \phi_s$$ for $$t, s \in \Bbb R\ ,$$ so are a one parameter group of diffeomorphisms. When $$V (x) = - {\rm grad} f (x)$$ the minus sign causes the flow to flow downhill, $$\frac{d}{d t} f (\phi_t (x)) = - \| {\rm grad} f (x) \|^2 \le 0\ .$$

Morse theory proves that for an open and dense (in the $$C^r$$ topology) set of functions f (called Morse functions), the Hessian of f is non-singular at the critical point of f. The vector field $$V = - {\rm grad} f$$ then has only finitely many singularities, say $$p_1, \dots, p_m\ ,$$ where $$V(p_i) = 0\ ,$$ and moreover, near any of the critical points $$p_i$$ there is a local chart so that $$f$$ has the form $f(x) = f (p_i) -x^2_1 -x^2_2 - \dots -x^2_j + x^2_{j+1} + x^2_{j+2} + \dots + x^2_n\ ,$ where $$x = (x_1, \dots, x_n)\ .$$

Thus for any $$x \in M\ ,$$ $$\phi_t (x)$$ converges to some $$p_i$$ as $$t \rightarrow + \infty\ .$$ We may adapt the metric so that near $$p_i, - {\rm grad} f$$ takes the form $(+2x_1, + 2x_2, \dots, +2x_j, -2x_{j+1}, -2x_{j+2}, \dots, -2x_n)$ and $\phi_t (x_1, \dots, x_n) = (e^{+2t} x_1, e^{+2t} x_2, \dots, e^{+2t} x_j, e^{-2t} x_{j+1}, \dots, e^{-2t} x_n)\ .$

This gives the standard picture:

Figure 1:

The points in the $$(x_1, \dots, x_j)$$ space tend to $$p_i$$ as $$t$$ approaches $$- \infty\ .$$ Locally these are discs of dimension $$j$$ called the index of the point $$p\ ,$$ and $$n - j\ .$$ These discs are denoted by $$W^u_{\rm loc} (p_i)$$ and $$W^s_{\rm loc} (p_i)$$ respectively, the local unstable and local stable manifolds of $$p_i\ .$$ The set of $$x \in M$$ such that $$\phi_t (x) \rightarrow p_i$$ as $$t \rightarrow \pm \infty$$ is denoted by $$W^{u, s} (p_i)\ ,$$ the (global) unstable and stable manifolds of $$p_i\ .$$ $$W^u (P_i)$$ and $$W^s (P_i)$$ are 1-1 immersed discs in $$M$$ of dimension $$j$$ and $$n-j$$ respectively. The manifolds $$M$$ is the disjoint union of these stable manifolds, $$M = \bigcup_{i=1, \dots, m} W^s (P_i)\ .$$ $$M$$ is also the union of the unstable manifolds $$M = \bigcup_{i=1, \dots, m} W^u (P_i)\ .$$ So for any $$x \epsilon M, \lim \phi_t (x), t \rightarrow \pm \infty$$ exists and is one of the $$P_i\ .$$ Now add another condition, which was introduced by Smale, that these manifolds $$W^s (P_i), W^u (P_i)$$ are all transversal wherever they meet. The set of such $$f$$ remains open and dense. The vector fields $$V = - \rm grad \ f$$ are called Morse-Smale gradient fields.

Example

Let $$A$$ be a real symmetric matrix with distinct eigenvalues $$\lambda_1 < \lambda_2 < \dots < \lambda_n$$ and corresponding unit eigenvector $$v_1 \dots v_n\ .$$ Then $$f(x) = \frac{1}{2} \ \frac{<x, A(x)>}{<x, x>}$$ defines a Morse-function on the sphere $$S^{n-1}_1\ ,$$ here < , > is the usual inner product in Euclidean space. The critical points of $$f$$ are precisely $$\pm v_i$$ and the index of $$\pm v_i$$ is $$i-1\ .$$ In fact on $$S^{n-1}_1 \rm grad \ f (x) = A (x) - < x,A(x)>X$$ and one can explicitly solve $$\phi_t (x) = \frac{e^{-tA} (x)}{\|e^{-tA}(x)\|}$$ which is a Morse-Smale gradient flow on $$S^{n-1}_1\ .$$ The union of the unstable manifolds of $$\pm v_1, \pm v_2, \dots, \pm v_i$$ is the vector subspace spanned by $$v_1, \dots, v_i$$ intersect $$S^{n-1}_1\ ,$$ while the union of the stable manifolds is the complement of the space spanned by $$v_{i+1}, v_{i+2}, \dots, v_i$$ intersect $$S^{n-1}_1\ .$$

Figure 2:

The function $$f(x)$$ is invariant under the identification $$x \sim - x$$ on $$S^{n-1}_1$$ and the flow $$\phi_t (x)$$ commutes with this identification. Thus $$f$$ and $$\phi_t$$ induce a Morse function and a Morse-Smale gradient flow on $$S^{n-1}_1 / x \sim - x = \Bbb R P (n-1)$$ real projective $$(n-1)$$ space. There is one critical point for each eigenspace corresponding to $$v_1, \dots, v_n$$ of index $$(i-1)\ .$$ Thus there is one critical point for each dimension from 0 to (n-1). The intersection of the $$W^u (\pm v_{i+1})$$ and $$W^S (\pm v_i)$$ in $$S^{n-1}_1$$ must occur in the plane of $$v_i$$ and $$v_{i+1}$$ intersect $$S^{n-1}_1\ .$$ On this circle, the dynamics are always like Figure 2 after identifying $$x \sim - x$$ on $$\Bbb R P (n-1)$$ we get Figure 3 as the dynamics in the $$v_i, v_{i+1}$$ plane in $$\Bbb R P (n-1)\ .$$

Figure 3:

It is by now a standard result of Morse theory that passing a critical value adds a handle to the manifold. More precisely, let $$f: M \rightarrow R$$ be a Morse function. Let $$M_a = f^{-1} (- \infty, a)\ ,$$ so $$\partial M_a = f^{-1} (a)\ .$$

Theorem 1
Suppose that $$f: M^n \rightarrow R$$ is a Morse function. If $$a < b$$ and $$M^n_b - M^n_a$$ contains exactly one critical point $$p$$ of index $$i\ ,$$ then $$M^n_b$$ is diffeomorphic to $$M^n_a \bigcup_{\phi} D^i x D^{n-i}$$ where $$\phi$$ is a diffeomorphism of $$S^{i-1} x D^{n-i}$$ into the boundary of $$M_a\ .$$

The proof of this theorem is a local argument near the critical point$$p\ .$$ In this form the theorem is due to Smale, see Smale 1961a. For general references on Morse theory see Bott 1982 and Milnor 1963. The gradient flow $$- grad \ f\ ,$$ pushes $$M^m_b$$ down to $$M^n_a$$ except for the stable manifold of $$p\ .$$

Figure 4:

Let $$\epsilon > 0$$ be small. Adding a neighborhood of a disc in the unstable manifold of $$p$$ (which intersects $$\partial M^n_{f (p) - \epsilon}$$ transversally) to $$M^n_{f (p) - \epsilon}$$ produces a manifold diffeomorphic to $$M^n_{f (p) + \epsilon}\ .$$ Now since there are no singularities of f in $$M_b - M_{f (p) + \epsilon}$$ or $$M_{f (p) - \epsilon} - M_a$$ pushing along the solutions curves of $$- grad \ f$$ produces diffeomorphisms between $$M_b$$ and$$M_{f (p) + \epsilon} and M_{f (p) - \epsilon}$$ and $$M_a\ .$$

Smale 1961a, 1962b, 1962 exploits this structure in his work on the Poincare conjecture, h-cobordism theorem and structure of manifolds. A good exposition is given in Milnor 1965 which emphasizes the gradient approach. We turn to some of these results, which we summarize in one theorem.

Figure 5:

Let $$- grad \ f$$ be a Morse-Smale vector field. Choose local charts for all the critical points of f so that $$f(x) = f (p) -x^2_1-\dots - x^2_j + x^2_{j+1} + \dots + x^2_n$$ for $$x$$ near $$p\ .$$ This has the effect of orienting the neighborhood of $$p, W^u (p), W^s (p)$$ as $$E^n, E^u$$ and $$E^s$$ with the usual orientation. if $$p, q$$ are critical points of index $$i+1$$ and $$i$$ respectively then $$W^u (p)$$ has dimension $$i+1$$ while $$W^s (q)$$ has dimension $$n-i\ .$$ The transversality hypothesis thus implies that $$W^u (p) \cap W^w (q)$$ consists of a finite number of orbits of the gradient flow $$\phi_t, \phi_t (m_1), \dots, \phi_t (m_j)\ .$$ For each $$m_i$$ we may orient a basis of complementary space to $$W^s (q)$$ in two ways, one from the $$W^u (q)$$ orientation, and one that comes from adding $$(- grad (f) (m_i)$$ as the first element of a basis and using the $$W^u (p)$$ orientation. If these two orientations agree we assign +1 as the index of the intersection; if not, -1. Let $i( p,q) = \sum\limits_{\phi_t (m_i) \subset W^u (p) \cap W^s (q)} index (\phi_t (m_i))\ .$ If $$p_1, \dots, p_k$$ and $$q_1, \dots, q_r$$ are the set of critical points of index i+1 and i respectively we let $$M_{i+1}$$ be the $$(r x k)$$ matrix whose $$(s, t)$$ entry is $$i (q_t, p_s)\ .$$

Theorem 1 (Smale)

Let $$f: M^n \rightarrow R$$ be a Morse function with $$- grad \ f$$ Morse-Smale, then:
A (Morse inequalities) There is a finitely generated chain complex of free abelian groups $$0 \rightarrow C_n \rightarrow C_{n-1} \rightarrow \dots \rightarrow C_0 \rightarrow 0$$ determined by $$f$$with rank $$C_i$$equal to the number of critical points of index $$i$$ and $$\partial_i = M_i$$ in a basis, which gives the homology of $$M^n \ .$$
B (Structure of Manifolds) Conversely, if $$\Pi_1 (M^n) = 0, n \ge 6$$ and $$0 \rightarrow C_n \rightarrow C_{n-1} \rightarrow \dots \rightarrow C_0 \rightarrow 0$$

is a finitely generated chain complex of free abelian groups which has as homology the homology of $$M\ ,$$ then this complex arises from a Morse function on $$M^n$$ as in part A.

Remarks: This is a beautiful theorem with powerful applications in topology including the higher dimensional Poincare conjecture. It also serves as a prototype for theorems relating dynamics and topology. A non-simply connected version of this theorem is proven in Maller 1980. Witten 1982 goes further in this direction and there are tie ins with Floer homology. Part A is by far the simpler part of this theorem. Without explicit computation of the boundary it is even more classical, and does not depend on the transversality condition. We have called Part A the Morse inequalities because they follow from the theorem with a little algebra.

Corollary 1
Let $$f: M^n \rightarrow R$$ be a Morse function. Let $$C_i$$ be the number of critical points of $$f$$ with index $$i$$ and let $$B_i$$ be the $$i^{th}$$ Betti number with coefficients in a field $$F\ .$$ Then one has the following inequalities:

$\begin{array}{ccc} C_0 & \ge & B_0 \\ C_1-C_0 & \ge & B_1-B_0 \\ \vdots & \ge & \vdots \\ \sum\limits_{k=0}^n (-1)^k {C_k}&=&\sum\limits_{k=0}^n (-1)^ k {B_k}\\ \end{array}$ Proof: We can perturb $$f$$ a little if necessary without changing the critical points or their indices to make the transversality hypothesis valid and thus apply Part A of the theorem. Since $$F$$ is a field $$C_i \otimes F$$ is a vector space and we can write $$C_i \otimes F = B_i \oplus H_i (M, F) \oplus B_{i-1}$$ where $$B_i \subset C_i \otimes F$$ is the image $$\partial_{i+1} (C_{i+1} \otimes F)\ .$$ The inequalities of the corollary are now evident.

The proof of the theorem is harder and beyond the scope of what we hope to do here, but Part A is especially instructive and we'll sketch the argument a bit. By the transversality hypothesis $$W^u (p) \cap W^s (q) = \phi$$ if index $$q \ge$$ index $$p\ .$$ Thus $$M^n$$ can be built first from the 0- handles followed by attachments of 1- handles followed by attachments of 2- handles, etc.

Figure 6:

(An i-handle is $$D^i x D^{n-i}$$ which is attached by a diffeomorphism $$\phi$$ defined on $$\partial D^i x D^{n-i}\ .$$ $$D^i x 0$$ is called the core disc and $$0 x D^{n-i}$$ the transverse disc.) This can be seen from the proof of Theorem 1. More formally, there is a sequence of submanifolds, called a handle decomposition of $$M^n\ .$$ $$\phi \subset M^n_0 \subset M^n_1 \subset \dots \subset M^n_n$$ such that $$M^n_{i+1} = M^n_i \cup P^{i+1}_1 \cup \dots \cup P^{i+1}_s$$where $$P^{i+1}_k$$ is a $$(i+1)$$ handle. Now $$\dots \rightarrow H^{i+1} (M^n_{i+1} M^n_i) \rightarrow H^i (M^n_i, M^n_{i-1}) \rightarrow \dots$$is the complex of Theorem 2 Part A.

Examples

In the example which we considered $$R P (n-1), C_i$$ has rank $$1$$ for $$0 \le i \le n-1$$ and $$M_i = (\pm2)$$ or $$(0)$$ as $$i$$ is even or odd respectively, $$i \not = 0\ .$$ Thus $H_0(\Bbb R P(n-1) = Z$ $H_i (\Bbb R P (n-1) = 0$ for $$i$$ even not $$0$$ $H_i (\Bbb R P(n-1) = Z_2$ for $$i$$ odd not $$(n-1)$$ $H_{n-1} (\Bbb R P (n-1) = Z$ if $$(n-1)$$ is odd.

Morse-Smale Diffeomorphisms: Examples and Relations to Topology

Figure 7:

Morse-Smale vector fields and their corresponding one parameter groups are a generalization of Morse-Smale gradient vector fields. The principle difference is the presence of periodic orbits. Fixed points of diffeomorphisms are analogous to singularities of vector fields in the sense that if $$V(x) = 0$$ then $$\phi_t (x) = x$$ for all $$t$$ where $$\phi_t$$ is the one parameter group of diffeomorphisms of $$M$$ generated by $$V\ .$$ Periodic points are analogous to periodic orbits of flows in the sense that if $$\phi_t (x) = x$$ then $$(\phi_t)^n (x) = x$$ for all $$n\ ,$$ but they differ in that a periodic point of a diffeomorphism does not need to lie on a circle of periodic points of the same period. In fact, this last phenomenon is rare for diffeomorphisms.Thus the time $$t \ne 0$$ map of a Morse-Smale vector field $$V$$ is not a Morse-Smale diffeomorphism unless the only periodic orbits of $$V$$ are fixed points. The time $$t \ne 0$$ map, $$\phi_t$$ of a Morse-Smale gradient vector field is a Morse-Smale diffeomorphism, but not every Morse-Smale diffeomorphism arises this way. Here are some other examples.

Figure 8:

1) Let$$g: M \rightarrow M$$ be a periodic diffeomorphism. Let $$f: M \rightarrow R$$ be an equivariant Morse functions that is $$f (g(m)) = f (m)$$ and $$f$$ is Morse. Let $$\phi_t$$ be the flow of $$- \rm grad \ f\ ,$$ fix $$t_0$$ and consider $$\phi_{t_0}$$ composed with $$g\ .$$ A small perturbation of this diffeomorphism will make the stable and unstable manifolds transversal and give a Morse-Smale diffeomorphism which is isotopic to $$g\ .$$

2) Alter one of the standard gradient pictures on the torus, $$T^2$$ by a Dehn twist in the armband.

The induced map is in the isotopy class of the linear map of $$T^2$$ given by the matrix $\begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix}$

3) Let A$T^n \rightarrow T^n$ be quasi-unipotent i.e. every eigenvalue of $$A$$ is a root of unity then A can be perturbed to be Morse-Smale. Over $$Z$$ put $$A$$ in the form $\begin{pmatrix} A_1 & x & \ldots & \ldots & \ldots & x\\ 0 & A_2 & \ddots & & & \vdots\\ \vdots & \ddots & \ddots & \ddots& & \vdots\\ \vdots & & \ddots & \ddots & \ddots & \vdots\\ \vdots & & & \ddots & \ddots & x\\ 0 & \ldots & \ldots & \ldots & 0 & A_j \end{pmatrix}$ where $$A_i$$ is a periodic square matrix. Now perturb by induction. (See Fried-Shub) for this and a nil-manifold analog. The theorem was first proven by Benjamin Halpern.)

4) Let $$f: \Bbb C^n \rightarrow \Bbb C$$ be a polynomial and $$V = f^{-1} (0)\ .$$ Let $$x \epsilon V$$ be an isolated singularity of $$V\ .$$ Let $$S_\epsilon (x)$$ be the $$\epsilon$$-sphere around $$x$$ and $$k = S_\epsilon (x) \cap V\ .$$ Then the map $$S_\epsilon (x) - K \rightarrow S^1$$ by $$Z \rightarrow \frac{f(z)}{\|f(z)\|}$$ is a locally trivial fibration. A generator of $$\prod_1 (S^1)$$ induces a map of the fiber, the monodromy map, which is defined up to isotopy.

Theorem (D. Fried)
The monodromy map of an isolated singularity of a complex hypersurface is isotopic to a Morse-Smale diffeomorphism.

What can be said about the relations between the dynamics and the topology of Morse-Smale diffeomorphisms, in particular what is the analog of Theorem 1. A good survey is Franks (1982). Here we state some results.

All of our examples show quasi-unipotent behaviour. In fact the unstable manifolds of the periodic points are like the cores of the handles of a Morse-Smale gradient flow which are permuted by $$f$$ with a sign due to orientation.

An integral $$n \times n$$ matrix $$A$$ is a signed permutation matrix if each row and column of $$A$$ has exactly one entry equal to $$\pm 1$$ and the other entries 0. An $$(n \times n)$$ integral matrix $$A$$ is a virtual permutation matrix if $\mathbf{A} = \begin{pmatrix} P_1 & & *\\ & \ddots &\\ 0 & & P_j \end{pmatrix}$ where each $$P_i$$ is a signed permutation matrix or the 0 matrix. An endomorphism $$E: \Bbb Z^n \rightarrow \Bbb Z^n$$ is a virtual permutation endomorphism, v.p. for short, if there is a basis of $$\Bbb Z^n$$ with respect to which $$E$$ is a virtual permutation matrix.

Theorem (Shub-Sullivan)

A) If $$f: M \rightarrow M$$ is a Morse-Smale diffeomorphism then there is a finitely generated chain complex C of finitely generated free abelian groups and an endomorphism A of C given by matrices $$A_i$$

Figure 9:

such that:

1) Rank $$C_i =$$ number of periodic points p of f with dim $$W^u_p = i\ .$$

2) Each $$A_i$$ is a virtual permutation matrix

3) $$H_* ($$C, Z) = $$H_* (M, Z)$$

4) $$H_* ($$A) = $$H_* (f)$$ for all coefficients.

B) Conversely: If $$dim M \ge 6, \Pi_1 (M) = 0,$$ f is a diffeomorphism and there is a chain complex C with endomorphism A such that 2), 3), and 4) hold then f is isotopic to a Morse-Smale diffeomorphism.

This is the analog of Theorem 2. Maller 1980, 1981 deals with the non-simply connected version of this theorem. As an immediate corollary we have:

Corollary 3

Let $$f: M \rightarrow M$$ be Morse-Smale. Then $$f_* : H_* (M, \Bbb R) \rightarrow H_* (M, \Bbb R)$$ is quasi-unipotent.

Proof:

Any eigenvalue of $$x_* \ \rm on \ H_* (M, \Bbb R)$$ comes from a chain map of f on cycles which are contained in $$C_i \otimes \Bbb R\ .$$

There is a difference between maps of complexes which are quasi-unipotent on homology and those which can be represented by virtual permutation. Given an endomorphism $$G {h \atop \longrightarrow} G$$ of a finitely generated abelian group the torsion subgroup $$T$$ of $$G$$ is invariant and thus $$h$$ induces a map on $$G / T, h$$ is called quasi-idempotent or q.i. if every eigenvalue of this map is $$0$$ or a root of unity. Let $$QI$$ be the category of q.i. endomorphisms of finitely generated abelian groups. A morphism $$G {h \atop \longrightarrow} G$$ to $$G'{h' \atop \longrightarrow} G'$$ is a map $$k: G \rightarrow G'$$ such that

commutes.

Let $$P$$ be the full subcategory of elements of Q.I. with V.P. resolutions i.e. maps $$G {h \atop \longrightarrow} G$$ such that there is a commutative diagram

where the columns are exact and the $$A_i$$ are V.P.

The group SSF is then defined to be $$K_0 (QI)/K_0 (P)\ .$$

Theorem 13 (Franks-Shub)

A) If $$f: M \rightarrow M$$ is a Morse-Smale diffeomorphism then $$X (f_*) = \sum (-1)^i f_* i$$ is 0 in SSF, where $$f_* i: H_i (M, Z) \rightarrow H_i (M, Z)\ .$$

B) Conversely: If $$f: M \rightarrow M$$ is a diffeomorphism and $$f_*$$ is quasi-unipotent on homology, $$\Pi_1 (M) = 0\ ,$$ dim $$M \ge 6$$ and $$X (f_*) = 0$$ in SSF then f is isotopic to a Morse-Smale diffeomorphism.

Maller 1981 and Maller and Whitehead make progress on non-simply connected versions of this theorem.

What about this group SSF. Might it be zero for example? The answer by the work of Bass and Lenstra is that it is huge. See Bass 1979, 1981 and Lenstra. For example if p is a prime and $$\Theta$$ is a $$p$$th root of unity, then any non-zero element of the ideal class group of $$Z [\Theta]$$ represents a non-zero element of SSF. It's a good challenge to understand the dynamics of diffeomorphism which are quasi-unipotent on homology but not isotopic to Morse-Smale diffeomorphisms. The problem is an example of open problems on the connection between topology and dynamics. A good general reference for results and problems is again Franks. One of my favorite problems here is the entropy conjecture. Let $$f: M \rightarrow M$$ be a smooth diffeomorphism or even map. Let h(f) be the topological entropy of f and let $$\lambda$$ be any eigenvalue of $$f_*: H_* (M, \Bbb R) \rightarrow H_* (M, \Bbb R)\ .$$ Is $$h (f) \ge ln |\lambda |$$ Yomdin proves the $$C^{\infty}$$ entropy conjecture for maps, but it is open for every $$1<r<\infty\ .$$ See also Gromov's exposition of Yomdin's theorem and Shub 2006.

Bibliography

• H. Bass, 1979, The Grothendieck Group of the Category of Abelian Group Automorphisms of Finite Order, Preprint, Columbia University
• H. Bass, 1981, Lenstra's Calculation of G_0 (R [\pi]) and Applications to Morse-Smale Diffeomorphisms in Integral Representations and Applications (Edited by K. W. Roggenkamp) Springer Lecture Notes in Mathematics Number 882, pp. 287-318
• S. Batterson, 1977, Structurally Stable Grassmann Transformations, Trans. American Mathematical Society, Vol. 231, pp. 385-404.
• R. Bott, 1982, Lectures on Morse Theory Old and New, Bulletin (New Series) of the American Mathematical Society, Vol. 7, pp. 331-358.
• J. Franks, 1982, Homology and Dynamical Systems, Conference Board of Mathematical Sciences Regional Conference Series in Mathematics, Number 49, American Mathematical Society, Providence, Rhode Island.
• J. Franks and M. Shub, 1981, The Existence of Morse-Smale Diffeomorphisms Topology, Vol. 20, pp. 273-290.
• M. Gromov, 1987, Entropy, Homology and Semi-algebraic Geometry. In Seminaire Bourbaki, Vol. 1985/86; Asterisque 145-146(5)
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• M. W. Hirsch, C. C. Pugh, and M. Shub, 1977, Invariant Manifolds, Springer Lecture Notes in Mathematics Nubmer 583, Springer Verlag, Berlin.
• H. W. Lenstra Jr., 1981, Grothendieck Groups of Abelian Group Rings, Journal of Pure and Applied Algebra, Vol. 20, pp. 173-193.
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• S. Smale, 1967, Differentiable Dynamical Systems, Bulletin of the American Mathematics Society, Vol. 73, pp. 747-817, with Notes and References in S. Smale, 1980, The Mathematics of Time, pp. 1-82, Springer, New York.
• E.Witten, Supersymmetry and Morse Theory, Jour. Differential Geometry, Vol. 17, 1982, pp. 661-692
• Y. Yomdin, 1987, Volume Growth and Entropy, Israel J. Math, Vol. 57, pp285-300.

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