1. Introduction

The following postulates and notation are used throughout:

• $K\subset {{\mathbb{R}}^{n}}$ (Euclidean $n$-space) is a solid order cone: a closed convex cone that has nonempty interior Int$K$ and contains no affine line.

• ${\mathbb{R}}^{n}$ has the (partial) order $\succeq$ determined by $K$:

$y \succeq x\Leftrightarrow y-x\in K,$

referred to as the $K$-order.

• $X\subset {\mathbb{R}}^{n}$ is a nonempty set whose Int $X$ is connected and dense in $X$.

• T: $X\approx X$ is homeomorphism that is monotone for the $K$-order:

$x \succeq y\Rightarrow Tx \succeq Ty.$

A point $x\in X$ has period $k$ provided $k$ is a positive integer and $T^kx=x$. The set of such points is $P_k =P_k (T)$, and the set of periodic points is $P=P(T)=\bigcup_kP_k$. $T$ 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.

The assumptions on $X$ show that $T$ is periodic iff $T|Int X$ is periodic. Therefore we assume henceforth:

•  $X$ is connected and open ${\mathbb{R}}^{n}$.

We prove the conjecture under the additional assumption that $K$ is a polyhedron, the intersection of finitely many closed affine halfspaces of ${\mathbb{R}}^{n}$:

Theorem 1 (Main). Assume $K$ is a polyhedron, T: $X\approx X$ is monotone for the $K$-order, and $P$ is dense in $X$. Then $T$ is periodic.

For analytic maps there is an interesting contrapositive:

Theorem 2. Assume $K$ is a polyhedron and T: $X\approx X$ is monotone for the $K$-order. If $T$ is analytic but not periodic, $P$ is nowhere dense.

Proof. As $X$ is open and connected but not contained in any of the closed sets $P_k$, analyticity implies each $P_k$ is nowhere dense. Since $P=\bigcup_{k=1}^\infty P_k$, a well known theorem of Baire ^[1] implies $P$ is nowhere dense.

The following result of D. Montgomery ^[4]*is crucial for the proof of the Main Theorem:

^*See also S. Kaul ^[3].

Theorem 3 (Montgomery). Every pointwise periodic homeomorphism of a connected manifold is periodic.

Notation

$i, j, k, l$ denote positive integers. Points of ${\mathbb{R}}^{n}$ are denoted by $a, b, p, q, u, v, w, x, y, z$.

$x\preceq y$ is a synonym for $y \succeq x$. If $x\preceq y$ and $x\ne y$ we write $x\prec$ or $y\succ x$.

The relations $x\ll y$ and $y\gg x$ mean $y-x \in Int K$.

A set $S$ is totally ordered if $x, y \in S \Rightarrow x \preceq y$ or $x\succeq y$.

If $x\preceq y$, the order interval $[x, y]$ is $\{z: x\preceq z \preceq y\} = K_x \cap{-K_y}$.

The translation of $K$ by $x\in {\mathbb{R}}^{n}$ is $K_x:=\{w+x, w\in K.\}$

The image of a set or point $\xi$ under a map $H$ is denoted by $H\xi$ or $H(\xi)$. A set $S$ is positively invariant under $H$ if $HS \subset S$, invariant if $H\xi=\xi$, and periodically invariant if $H^k\xi=\xi$.

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2. Proof of the Main Theorem

The following four topological consequences of the standing assumptions are valid even if $K$ is not polyhedral.

Proposition 4. Assune $p, q \in P_k$ are such that

$p\ll q, \qquad p, q\in P_k. \qquad [p, q]\subset X.$

Then $T^k([p, q]=[p, q]$.

proof. It suffices to take $k=1$. Evidently $TP=P$, and $T[p, q]\subset[p, q]$ because $T$ is monotone, whence $Int{ [p, q]}\cap P$ is positively invariant under $T$. The conclusion follows because $Int{[p, q]}\cap P$ is dense in $[p, q]$ and $T$ is continuous.

Proposition 5. Assume $a, b\in P_k, a\ll b$, and $[a, b]\subset X$. There is a compact arc $J\subset P_k\cap [a, b]$ that joins $a$ to $b$, and is totally ordered by $\ll$.

proof. An application of Zorn's Lemma yields a maximal set $J\subset [a, b]\cap P$ such that: $J$ is totally ordered by $\ll, \ a=\max J, \ b=\min J$. Maximality implies $J$ is compact and connected and $a, b \in J$, so $J$ is an arc (Wilder ^[7], Theorem I.11.23).

Proposition 6. Let $M\subset X$ be a homeomorphically embedded topological manifold of dimension $n-1$, with empty boundary.

(i) $P$ is dense in $M$.

(ii) If $M$ is periodically invariant, it has a neighborhood base $B$ of periodically invariant open sets.

proof. (i) $M$ locally separates $X$, by Lefschetz duality ^[5] (or dimension theory ^[6]. Therefore we can choose a family $V$ of nonempty open sets in $X$ that the family of sets $V_M:=\{V\cap M: V\in V)$ satisfies:

•  $V_M$ is a neighborhood basis of $M$,

•  each set $V\cap M$ separates $V$.

By Proposition 5, for each $V\in V$ there is a compact arc $J_V\cap P\cap V$ whose endpoints $a_V, b_v$ lie in different components of $V\, \verb=\=\, M$. Since $J_V$ is connected, it contains a point in $V\cap M\cap P$. This proves (i).

(ii) With notation as above, let $B_V:=[a_V, b_V]\, \verb=\=\, \partial [a_V, b_V]$. The desired neighborhood basis is $B:=\big\{B_V: V\in V\big\}.$

From Propositions 4 and 6 we infer:

Proposition 7. Suppose $p, q\in P$, $p\ll q$ and $[p, q]\subset X$. Then $P$ is dense in $\partial [p, q]$.

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 (Induction). $n \ge 1$ and $T (n-1)$ holds.

Let $Q\subset {\mathbb{R}}^{n}$ be a compact $n$-dimensional polyhedron. Its boundary $\partial Q$ is the union of finitely many convex compact $(n-1)$-cells, the faces of $Q$. Each face $F$ is the intersection of $\partial [p, q]$ with a unique affine hyperplane $E^{n-1}$. The corresponding open face $F^\circ:=F\, \verb=\=\, \partial F$ is an open $(n-1)$-cell in $E^{n-1}$. Distinct open faces are disjoint, and their union is dense and open in $\partial Q$.

Proposition 8. Assume $p, q\in P_k, \ p\ll q, \ [p, q]\subset X$. Then $T|\partial [p, q]$ is periodic.

^8224; This result is adapted from Hirsch & Smith ^[2],Theorems 5,11 & 5,15.

proof. $[p, q]$ is a compact, convex $n$-dimensional polyhedron, invariant under $T^k$ (Proposition 4). By Proposition 6 applied to $M:=\partial [p, q]$, there is a neighborhood base $B$ for $\partial [p, q]$ composed of periodically invariant open sets. Therefore if $F^\circ\subset \partial [p, q]$ is an open face of $[p, q]$, the family of sets

$B_{F^\circ}:=\{W\in B: W\subset F^\circ\}$

is a neighborhood base for $F^\circ$, and each $W\in B_{F^\circ}$ is a periodically invariant open set in which $P$ is dense.

For every face $F$ of $[p, q]$ the Induction Hypothesis shows that $F^{\circ} \subset P$. Therefore Montgomery's Theorem implies $T|F^{\circ}$ is periodic, so $T|F$ is periodic by continuity. Since $\partial [p, q]$ is the union of the finitely many faces, it follows that $T|\partial [p, q]$ is periodic.

To complete the inductive proof of the Main Theorem, it suffices by Montgomery's theorem to prove that an arbitrary $x\in X$ is periodic. As $X$ is open in ${\mathbb{R}}^{n}$ and $P$ is dense in $X$, there is an order interval $[a, b]\subset X$ such that

$a\ll x \ll b, \qquad a, b \in P_k.$

By Proposition 5, $a$ and $b$ are the endpoints of a compact arc $J\subset P_k\cap [a, b]$, totally ordered by $\ll$. Define $p, q\in J$:

$p:=\sup\, \{y\in J: y\preceq x\}, \qquad q:=\inf\, \{y\in J: y\succeq x\}.$

If $p=q=x$ then $x\in P_k$. Otherwise $p\ll q$, implying $x\in\partial [p, q]$, whence $x\in P$ by Proposition 8

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Conflict of Interest

The author declares no conflicts of interest in this paper.

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