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= [0; 20, 21, 21, 22, 23, 25, 28, 213, . . .] (α = (1 + √ 5)/2), and that the number ψ is transcendental. Here, bxc denotes the integral part of a real number x, the right-hand side indicates a simple continued fraction, where the powers of 2 appearing in the partial denominators are Fibonacci numbers (1). The binary expansion of ψ can be described by the fixed point of a substitution. For this purpose, we introduce some definitions. K∗ denotes the set of all finite words over an alphabetK = {a, b, c, . . . , d}, i.e., K∗ is the free monoid generated by K with the operation of concatenation and the empty word λ as its unit. K∞ denotes the set of all ω-words w1w2w3 . . . (wn ∈ K). A substitution σ (over K) is a monoid endomorphism σ on K∗ extended to K∞, defined by σ(w) = σ(w1)σ(w2)σ(w3) . . . for w = w1w2w3 . . . ∈ K∞. A fixed point of σ is a ω-word w ∈ K∞ such that σ(w) = w. Any substitution σ over K of the form σ(a) = au (u 6= λ), σ(x) 6= λ (∀x ∈ K) has the unique fixed point w prefixed by a, namely, w = auσ(u)σ2(u) . . . Here, the product τσ denotes the composition of τ and σ, and σ indicates the n-fold iteration σn−1 ◦ σ (n ≥ 1) with σ0(u) = u (u ∈ K∗ ∪K∞).

A class of transcendental numbers having explicit g-adic and Jacobi-Perron expansions of arbitrary dimension