Forcing relation on minimal interval patterns

Jozef Bobok¹

¹ KM FSv. ČVUT Thákurova 7 166 29 Praha 6, Czech Republic

Fundamenta Mathematicae, Tome 169 (2001) no. 2, pp. 161-173.

Résumé

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.

Détail

DOI : 10.4064/fm169-2-5

Keywords: cal set pairs subset mathbb compact continuous minimal has piecewise monotone extension mathop conv nolimits pairs cal equivalent map mathop orb nolimits mathop min mathop orb nolimits mathop min defined each mathbb mathop min mathop min increasing mathop orb nolimits mathop min equivalence class relation minimal oriented pattern exhibited continuous interval map there set subset define forcing relation minimal patterns forces continuous interval maps exhibiting exhibit theorem each minimal pattern there maps exhibiting only patterns forced using result prove forcing relation minimal patterns partial ordering theorem extends result where pairs finite considered