Groups and dynamical systems. Discrete time dynamics on the E(2) group
It is shown that discrete dynamical systems can be derived from maps defined on a Lie group. Let \(G\) be a Lie group, \(h\in G\) and \(f_ h: G\to G\) defined by \(f_ h(g)=ghg^{-1}\). To a fixed point \(g\in G\) one associates the sequence \((g_ m)_{m\in N}\), \(g_ m=f_ h(g_{m-1})\), called SL-sequence. \(g_ m\) contains \(2^ m-1\) factors of the form \(h\) or \(h^{-1}\), i.e. \[ g_ m=gh_ 1g^{-1}h_ 2gh_ 3g^{-1}\dots h_{2^{m-1}}\dots gh_{2^ m-1}g^{-1},\text{ where }h_ k=h\text{ or }h^{-1}. \] So the form of \(g_ m\) is determined by the binary sequence \(B_ m=b_ 1b_ 2\dots b_{2^ m-1}\), \(b_ k=1\) if \(h_ k=h\), \(b_ k=0\) if \(h_ k=h^{-1}\). \(B_ m\) is called PF-sequence, as being identical with the sequence introduced by \textit{Ch. Davis} and \textit{D. Knuth} [J. Recreat. Math. 3, 61-81 and 133-149 (1970)] in connection with the paper- folding problem. The microdynamics of the SL- and PF-sequence is defined. An explicit formula for \(B_ m\) is obtained. The general results are then used in the case of the group \(\text{E}(2)\), the group of motions of \(R^ 2\), to describe the dynamics of PF- and SL-sequence for \(h=(e^{i\beta},0)\), \(g_ 0=(1,d)\). The fractals associated to microdynamics on the group \(\text{E}(2)\) are described and finally microdynamics on \(\text{E}(2)\) is viewed as a random walk.
scientific article; zbMATH DE number 37361
scientific article