Exchange of three intervals: Substitutions and palindromicity

Abstract

Given a symmetric exchange of three intervals, we provide a detailed description of the return times to a subinterval and the corresponding itineraries. We apply our results to morphisms fixing words coding non-degenerate three interval exchange transformation. This allows us to prove that the conjecture stated by Hof, Knill and Simon is valid for such infinite words.

On Words with the Zero Palindromic Defect

“…In [25], we confirm the conjecture for morphisms fixing a codings a non-degenerate exchange of 3 intervals. In [22], the authors prove the validity of the conjecture for marked morphisms.…”

Itineraries Induced by Exchange of Three Intervals

“…Both of these were proven in [12] for general interval exchange transformations with symmetric permutation and thus hold also for 3iets with permutation (321).…”

“…We also focus on substitutions fixing words coding interval exchange transformations. The latter has implications [12] for the question of Hof, Knill and Simon [10] about palindromic substitution invariant words with application to aperiodic Schrödinger operators.…”

On the Zero Defect Conjecture

“…As mentioned in [18], the fixed point u = ψ(u) is again an image of a Sturmian word v under a morphism π : {0, 1} → {a, b, c} and the Sturmian word v itself is a fixed point of a morphism over binary alphabet {0, 1}. Since v is Sturmian, its defect is zero.…”