Generalized radix representations and dynamical systems. IV

Abstract

For r = ( r 1,…, r d ) ∈ ℝ d the mapping τ r :ℤ d →ℤ d given by τ r ( a 1,…, a d ) = ( a 2, …, a d ,−⌊ r 1 a 1+…+ r d a d ⌋) where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ℤ d there exists an integer k > 0 with τ r k ( a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).