We investigate powers of prefixes in Sturmian sequences. We obtain an explicit formula for ice(ω), the initial critical exponent of a Sturmian sequence ω, defined as the supremum of all real numbers p > 0 for which there exist arbitrary long prefixes of ω of the form u p , in terms of its S-adic representation. This formula is based on Ostrowski's… (More)
In this paper we present a detailed study of the spectral/ergodic properties of three-interval exchange transformations. Our approach is mostly combinatorial, and relies on the dio-phantine results in Part I and the combinatorial description in Part II. We define a recursive method of generating three sequences of nested Rokhlin stacks which describe the… (More)
A substitution naturally determines a directed graph with an ordering of the edges incident at each vertex. We describe a simple method by which any primitive substitution can be modified (without materially changing the bi-infinite fixed points of the substitution) so that points in the substitution minimal shift are in bijective correspondence with… (More)
We study powers of prefixes of Sturmian words.
A primitive, aperiodic substitution on d letters has at most d 2 asymptotic orbits; this bound is sharp. Since asymptotic arc components in tiling spaces associated with substitutions are in 1–1 correspondence with asymptotic words, this provides a bound for those as well.
Let A and F be finite sets and suppose for each a ∈ A we have a map ϕa : F → F. Suppose ω = ω 1 ω 2 ω 3. .. is a sequence in A which is ultimately primitive substitutive, i.e., a tail of ω is the image (under a letter to letter morphism) of a fixed point of a primitive substitution. We show that the induced sequence of iterates (ϕω n • ϕω n−1 • · · · • ϕω… (More)
In this paper we outline a connection between substitution dynamical systems and systems of partial isometries of IR. Let τ be a primitive substitution and (X, T) the associated minimal subshift. Associated to each real geometric realization G of τ is a system of partial isometries I G. The subshift (X, T) generated by τ provides a symbolic coding of the… (More)