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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, in terms of its S-adic representation. This formula is based on Ostrowski’s numeration… (More)

- Charles Holton
- 1998

- Charles Holton, Luca Q. Zamboni
- Theory of Computing Systems
- 1999

Let s = (A, τ) be a primitive substitution. To each decomposition of the form τ (h) = uhv we associate a primitive substitution D [(h,u)] (s) defined on the set of return words to h . The substitution D [(h,u)] (s) is called a descendant of s and its associated dynamical system is the induced system (X h , T h ) on the cylinder determined by h . We show… (More)

In this paper we present a detailed study of the spectral/ergodic properties of threeinterval exchange transformations. Our approach is mostly combinatorial, and relies on the diophantine 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)

- Marcy Barge, Beverly Diamond, Charles Holton
- Theor. Comput. Sci.
- 2003

A primitive, aperiodic substitution on d letters has at most d 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

- Yuval Peres, Elchanan Mossel, +7 authors Pertti Mattila
- 2001

- Charles Holton, Brian F. Martensen Mankato, MN, B. F. Martensen
- 2008

We show that an aperiodic minimal tiling space with only finitely many asymptotic composants embeds in a surface if and only if it is the suspension of a symbolic interval exchange transformation (possibly with reversals). We give two necessary conditions for an aperiodic primitive substitution tiling space to embed in a surface. In the case of… (More)

- Charles Holton, Luca Q. Zamboni
- Theory of Computing Systems
- 2001

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.

- Charles Holton
- 2000

Let A and F be finite sets and suppose for each a ∈ A we have a map φa : F → F. Suppose ω = ω1ω2ω3 . . . 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 (φωn ◦φωn−1 ◦· · ·◦φω1) ∞ n=1 is… (More)