Remember the Adding Machine

the $n$-ary adding machine and solvable groups

we describe under various conditions abelian subgroups of the automorphism‎ ‎group $mathrm{aut}(t_{n})$ of the regular $n$-ary tree $t_{n}$‎, ‎which are‎ ‎normalized by the $n$-ary adding machine $tau =(e‎, ‎dots‎, ‎e,tau )sigma _{tau‎ ‎}$ where $sigma _{tau }$ is the $n$-cycle $left( 0,1‎, ‎dots‎, ‎n-1right) $‎. ‎as‎ ‎an application‎, ‎for $n=p$ a prime number‎, ‎and for $n=4$‎, ‎we prove that...

Statistical Machine Translation Adding Rule-based Machine Translation

Topological Entropy and Adding Machine Maps

We prove two theorems which extend known results concerning periodic orbits and topological entropy in one-dimensional dynamics. One of these results concerns the adding machine map (also called the odometer map) fα defined on the α-adic adding machine ∆α. We let H(fα) denote the greatest lower bound of the topological entropies of F , taken over all continuous maps F of the interval which cont...

THE n-ARY ADDING MACHINE AND SOLVABLE GROUPS

We describe under various conditions abelian subgroups of the automorphism group Aut(Tn) of the regular n-ary tree Tn, which are normalized by the n-ary adding machine τ = (e, . . . , e, τ)στ where στ is the n-cycle (0, 1, . . . , n− 1). As an application, for n = p a prime number, and for n = 4, we prove that every soluble subgroup of Aut(Tn), containing τ is an extension of a torsion-free met...

An Ergodic Adding Machine on the Cantor Set

We calculate all ergodic measures for a specific function F on the unit interval. The supports of these measures consist of periodic orbits of period 2n and the classical ternary Cantor set. On the Cantor set, F is topologically conjugate to an “adding machine” in base 2. We show that F is representative of the class of functions with zero topological entropy on the unit interval, already analy...