Shift Radix Systems - A Survey

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Let d≥ 1 be an integer and r=(r₀,,rd₋₁) ∈ Rᵈ. The shift radix system τᵣ: Zᵈ → Zᵈ is defined by τ r(z)=(z₁,,zd₋₁,- r z)ᵗ (z=(z₀,,zd₋₁)ᵗ). τᵣ has the finiteness property if each z ∈ Zᵈ is eventually mapped to 0 under iterations of τᵣ. In the present survey we summarize results on these nearly linear mappings. We discuss how these mappings are related to well-known numeration systems, to rotations with round-offs, and to a conjecture on periodic expansions w.r.t.Salem numbers. Moreover, we review the behavior of the orbits of points under iterations of τᵣ with special emphasis on ultimately periodic orbits and on the finiteness property. We also describe a geometric theory related to shift radix systems.