Transcendence for Pisot morphic words over an algebraic base

It is known that for a uniform morphic sequence u = ⟨ u n ⟩ n = 0 ∞ u = uₙ ₙ₌₀^∞ bold italic u equals left angle bracket u Subscript n Baseline right angle bracket Subscript n equals 0 Superscript infinity and an algebraic number β β beta such that | β | > 1 |β |>1 StartAbsoluteValue beta EndAbsoluteValue greater than 1 , the number [ [ u ] ] β := ∑ n = 0 ∞ ( u n / β n ) [\![ u ]\!] _β :=∑ ₙ₌₀^∞ (uₙ/β ⁿ) left bracket left bracket bold italic u right bracket right bracket Subscript beta Baseline colon equals sigma summation Underscript n equals 0 Overscript infinity Endscripts left parenthesis u Subscript n Baseline divided by beta Superscript n Baseline right parenthesis either lies in Q ( β ) Q(β ) double struck upper Q left parenthesis beta right parenthesis or is transcendental. In this paper, we show a similar rational–transcendental dichotomy for sequences defined by irreducible Pisot morphisms on binary alphabets. Subject to the Pisot conjecture (an irreducible Pisot morphism has pure discrete spectrum), we generalise the latter result to arbitrary finite alphabets. In certain cases, we are able to show transcendence of [ <mml:m