Monotone Dynamical Systems with Polyhedral Order Cones and Dense Periodic Points

Let X be a subset of R^n whose interior is connected and dense in X, ordered by a polyhedral cone in R^n with nonempty interior. Let T be a monotone homeomorphism of X whose periodic points are dense. Then T is periodic.

x has period k if k is a positive integer and T k x = x. The set of these points is denoted by P k = P k (T ), and the set of periodic pointsby P = P(T ) = k P k .
T is periodic if X = P k , and pointwise periodic if X = P. Our main concern is the following speculation: Conjecture. If P is dense in X, then T is periodic.
We prove this for K a polyhedron: the intersection of finitely many closed affine halfspaces of R n . Theorem 1 (Main). If K is a polyhedron and P is dense in X, then T is periodic.
The following result of D. Montgomery [3] 1 is crucial to the proof: Theorem 2 (Montgomery). Every pointwise periodic homeomorphism of a connected manifold is periodic.
This implies a sharper version of Theorem 1 for analytic maps: Theorem 3. If K is a polyhedron and T is analytic but not periodic, P is nowhere dense.
Two observations simplify Theorem 1: • The hypothesis holds if P is dense in int (X), because int (X) is dense in X.
• The conclusion holds if int (X) ⊂ P.
For then Montgomery's Theorem implies int (X) ⊂ P k , hence X ⊂ P k because int (X) is dense in X and P k is closed in X, Notation i, j, k, l always denote positive integers, and a, b, p, q, u, v, x, y, z points of R n .
x y is a synonym for y x. If x y and x y we write x ≺ or y ≻ x.
The relations x ≪ y and y ≫

Proof of the Main Theorem
We derive three topological consequences, valid even if K is not polyhedral, from the standing assumptions that T is monotone and P is dense: Proof. It suffices to take k = 1. Evidently T P = P, and and T is continuous.
Proof. An application of Zorn's Lemma yields a maximal set J ⊂ [u, v] ∩ P such that: J is totally ordered by ≪, u = max J, v = min J. Maximality implies J is compact and connected and u, v ∈ J, so J is an arc (Wilder [5], Theorem I.11.23).

Proposition 6. Let M ⊂ X be a topological manifold of dimension n − 1 without boundary, closed in X.
(i) P is dense in M.

(ii) If M is periodically invariant, it has a neighborhood base W of periodically invariant open subsets.
(iii) P is dense in each W ∈ W.
Proof. Lefschetz duality (Spanier [4]) shows that every point of M has arbitrarily small open ball neighborhoods W ⊂ X separated by M into two disjoint open sets U and V: There is an arc J ⊂ P ∩ Let T (m) stand for the statement of Theorem 1 for the case n = m. Then T (0) is trivial, and we use the following inductive hypothesis: Hypothesis 7 (Induction). n ≥ 1 and T (n − 1) holds.
Let Q ⊂ R n be a compact n-dimensional polyhedral cell, such as [p, q] with p ≪ q. Its boundary ∂Q is the union of finitely many convex compact (n − 1)cells, the faces of Q. Each face F is the intersection of ∂[p, q] with a unique affine hyperplane E n− 1  The maps T |W λ are periodic by the Induction Hypothesis, so F • ⊂ P. Montgomery's Theorem implies T |F • is periodic, so T |F is also periodic. Since ∂[p, q] is the union of the finitely many faces F, it follows that T |∂[p, q] is periodic.
To complete the inductive proof of the Main Theorem, it suffices by Montgomery's theorem to prove that an arbitrary x ∈ X is periodic. As X is open in R n and P is dense in X, there is an order interval [a, b] ⊂ X such that By Proposition 5, a and b are the endpoints of a compact arc J ⊂ P k ∩ [a, b], totally ordered by ≪. Define p, q ∈ J: p := sup {y ∈ J : y x}, q := inf {y ∈ J : y x}.
If p = q = x then x ∈ P k . Otherwise p ≪ q and x ∈ ∂[p, q], whence x ∈ P by Proposition 8.