A \(p\)-adic behaviour of dynamical systems
Let \(\mathbb{C}_p\) be the completion of the algebraic closure of the field \(\mathbb{Q}_p\) of \(p\)-adic numbers. For \(n\in \{2,3,\dots\}\) the map \(f:x\to x^n\) is studied in the spirit of the theory of dynamical systems. Thus, one is interested in the orbits \(x_0, x_1,\dots, \) where \(x_{k+1}= f(x_i)=: f^{(k+1)} (x_0)\), fixed points, Siegel disks, cycles, attractors, etc. As the behavior of \(f^{(k)}\) is obvious on \(\{x: |x|> 1\}\) and \(\{x: |x|< 1\}\) the study is concentrated on balls contained in the unit sphere \(\{x: |x|=1\}\) (which is a closed and open set in \(\mathbb{C}_p\)). As the fixed points of \(f\) belong to the group of roots of unity in \(\mathbb{C}_p\), the structure of this group is introduced. Several results and examples are given. Next, the theory is applied to dynamical systems in \(\mathbb{Q}_p\) (which has only \((p-1)\)th roots of unity for \(p\neq 2\)). Fuzzy cycles (iteration generates a cycle of mutually disjoint balls \(B_1,\dots, B_n\)) are studied. Several examples are found by computer calculations.
scientific article; zbMATH DE number 1441657