The S-adic Pisot conjecture on two letters

The S-adic Pisot Conjecture on Two Letters

We prove an extension of the well-known Pisot substitution conjecture to the S-adic symbolic setting on two letters. The proof relies on the use of Rauzy fractals and on the fact that strong coincidences hold in this framework.

On the Pisot substitution conjecture

On alpha-adic expansions in Pisot bases

We study α-adic expansions of numbers, that is to say, left infinite representations of numbers in the positional numeration system with the base α, where α is an algebraic conjugate of a Pisot number β. Based on a result of Bertrand and Schmidt, we prove that a number belongs to Q(α) if and only if it has an eventually periodic α-expansion. Then we consider α-adic expansions of elements of the...

A proof of Pisot ’ s d th root conjecture

Let {b(n) : n ∈ N} be the sequence of coefficients in the Taylor expansion of a rational function R(X) ∈ Q(X) and suppose that b(n) is a perfect d power for all large n. A conjecture of Pisot states that one can choose a d root a(n) of b(n) such that ∑ a(n)Xn is also a rational function. Actually, this is the fundamental case of an analogous statement formulated for fields more general than Q. ...

Towards a statement of the S-adic conjecture through examples

The S-adic conjecture claims that there exists a condition C such that a sequence has a sub-linear complexity if and only if it is an S-adic sequence satisfying Condition C for some finite set S of morphisms. We present an overview of the factor complexity of S-adic sequences and we give some examples that either illustrate some interesting properties or that are counter-examples to what could ...