Indices of states in dynamical system of binary vectors associated with palms orientations

A. V. Zharkova

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 4, pp. 475-484

Résumé

Dynamical system of binary vectors associated with palms orientations is considered. A tree is called a palm with $s+c$ edges if it is a union of $c+1$ paths with common end vertex and all of these paths except perhaps one (with $s$ edges) have a length 1. The system splits into finite subsystems according to the dimension of states. States of a finite dynamical system ($B^{s+c}$,$\gamma$) are all possible orientations of a given palm with $s+c$ edges. They are naturally encoded by binary vectors and the evolutionary function $\gamma$ transforms a given palm orientation by reversing all arcs that enter sinks and there is no other difference between the given state and the next one. An algorithm to calculate indices of states in this dynamical system is proposed and it is proved that the depth of the basin of the finite dynamical system ($B^{s+c}$, $\gamma$), $s>0$, $c>1$, is equal to $s$.