Forcing relation on minimal interval patterns
Volume 169 / 2001
Abstract
Let ${\cal M}$ be the set of pairs $(T,g)$ such that $T\subset {\mathbb R}$ is compact, $g: T\to T$ is continuous, $g$ is minimal on $T$ and has a piecewise monotone extension to $\mathop {\rm conv}\nolimits T$. Two pairs $(T,g),(S,f)$ from ${\cal M}$ are equivalent if the map $h:\mathop {\rm orb}\nolimits (\mathop {\rm min}T,g)\to \mathop {\rm orb}\nolimits (\mathop {\rm min}S,f)$ defined for each $m\in {\mathbb N}_0$ by $h(g^m(\mathop {\rm min}T))=f^m(\mathop {\rm min}S)$ is increasing on $\mathop {\rm orb}\nolimits (\mathop {\rm min}T,g)$. An equivalence class of this relation—a minimal (oriented) pattern $A$—is exhibited by a continuous interval map $f:I\to I$ if there is a set $T\subset I$ such that $(T,f|T)=(T,f)\in A$. We define the forcing relation on minimal patterns: $A$ forces $B$ if all continuous interval maps exhibiting $A$ also exhibit $B$. In Theorem 3.1 we show that for each minimal pattern $A$ there are maps exhibiting only patterns forced by $A$. Using this result we prove that the forcing relation on minimal patterns is a partial ordering. Our Theorem 3.2 extends the result of [B], where pairs $(T,g)$ with $T$ finite are considered.
