An ergodic family of maps on sequence spaces (Q918009)

Let k be a finite field and let \({\mathcal E}\) denote the set of all infinite sequences \((a_ 0,a_ 1,a_ 2,...)\) where \(a_ i\in k\). The set \({\mathcal E}\) can be identified with the ring R of all power series over k in indeterminate X by the correspondence \[ (a_ 0,a_ 1,a_ 2,...)\quad \leftrightarrow \quad \sum^{\infty}_{i=0}a_ iX^ i. \] The author studies the dynamics of a family of linear maps F: \(R\to R\) which when put in standard form are given by \[ \alpha F(a)=\beta a-a_ 0\nu,\quad a\in R \] where \(\alpha =X\) or I and \(\beta\),\(\nu\) are fixed elements of R. The author proves several results (we list a few below) and illustrates these results using computer graphics. Theorem. If \(\alpha =X\) then (i) F maps all eventually spatially periodic points to eventually spatially periodic points iff \(\beta\) and \(\nu\) are rational power series. (ii) F maps all spatially periodic points to spatially periodic points iff \(\beta\) and \(\nu\) are rational with deg(\(\beta\))\(\leq 1\) and deg(\(\nu\))\(\leq 0.\) Theorem. Let \(\alpha =X\) and assume that \(\beta\) and \(\nu\) are rational with nonpositive degree. Then every temporally periodic point of F is also spatially periodic and the temporal period divides the spatial period.