Abstract

A (line) system is a pair $(T,g)$, where $T\subset R$ is compact and a map $g:\; T\to \; T$ is continuous. A system $(T,g)$ is transitive if there is a point $x\in \; T$ such that the set $g(x),g2(x),⩿$ is dense in $T$ - we denote the set of all such systems by $T$. A system $(T,g)\in \; T$ is maximal - we call it a roof - if it satisfies Two systems $(T,g),(S,f)\in T$ are $\sim$-equivalent if there are transitive points $t\in T$ and $s\in \; S$ such that the map $h:\; gi(t):\; i=0,1,...\; \to \; fi(s)\; :\; i=0,1,...$ defined by $h(gi(t))=fi(s)$ is increasing.
In our lecture we use the usual definition of forcing relation (known for periodic patterns) on $\sim$-equivalence classes of roofs - we call them roof patterns. We show that this relation is a partial ordering on a subset of non-fractal roof patterns. In order to compare this approach with the older one used for periodic and minimal patterns, we define a bottom of roof (closed subset with some property of minimality) and say that two bottoms are $owtie$-equivalent if they have $\sim$-equivalent roofs.
We show that each periodic and minimal system is a bottom (but there are also other bottoms). We also show that a forcing relation on bottom patterns ($owtie$-classes) naturally derived from the one on roof patterns is really an extension of the forcing relation known from the literature.