GENERAL SHIFT RADIX SYSTEMS AND DISCRETE ROTATION

ATTILA PETHŐ FACULTY OF INFORMATICS, UNIVERSITY OF DEBRECEN PETHO.ATTILA@INF.UNIDEB.HU

Abstract

My talk is based on joint research with Carolin Hannusch.

Let F be a bounded fundamental domain of the action of ℤ^{k} on ℝ^{k}. For any v ∈ ℝ^{k} there exists a unique a ∈ ℤ^{k}, such that v − a ∈ F, which is denoted by ⌊v⌋_{F}.

For fixed matrices R_{1}, …, R_{n} ∈ ℝ^{k×k} define the sequence of integer vectors by the initial terms a_{1}, … a_{n} ∈ ℤ^{k} and for m > n by the nearly linear recursive relation

(1)  a_{m} = − ⌊∑_{ℓ=1}^{n} R_{ℓ}a_{m−n+ℓ−1}⌋_{F} .

With the n-tuple R = (R_{1}, …, R_{n}) of matrices define the mapping τ_{R} : ℤ^{k×n} ↦ ℤ^{k×n} such that if A = (a_{1}, …, a_{n}) ∈ ℤ^{k×n} then

(2)  τ_{R}(A) = (a_{2}, …, a_{n}, a_{n+1}),

where a_{n+1} is defined by (1) with m = n + 1. The mapping τ_{R}, which is a discrete dynamical system, is called general srs, or short gsrs.

In the first part we show that gsrs is a generalization not only of srs introduced by Akiyama, Borbély, Brunotte, Pethő and Thuswaldner in 2005, but also of GNS over general orders, studied very recently by Evertse, Győry, Pethő and Thuswaldner.

In the second part we concentrate on the special case n = 1 and F = [0, 1)^{k}, i.e., when a_{0} ∈ ℤ^{k} and

τ_{R}(a_{m}) = ⌊Ra_{m}⌋ , m = 0, 1, …

with a k × k real matrix R and the integer part function is meant coordinate wise. It is easy to see that every orbits of τ_{R} is bounded, i.e, ultimately periodic, if all eigenvalues of R lie in the unit circle. In the other hand there exist unbounded orbits if some eigenvalue lie outside the unit circle. The fascinating question is, like for srs, what happens if all eigenvalues of R lie on the unit circle? This class includes the rotations of the plane, in which case τ_{R} is a digital rotation. We performed lot of experiments with different values of R and a_{0} and we found only ultimately periodic orbits. If the angle of the rotation is 2π/3, π/2 and π/4 we show infinite classes of initial vectors for which the orbits are indeed ultimately periodic.

We show under general conditions on R that τ_{R} is not surjective.