Beyond substitutive dynamical systems: S-adic expansions

Abstract

An S-adic expansion of an infinite word is a way of writing it as the limit of an infinite product of substitutions (i.e., morphisms of a free monoid). Such a description is related to continued fraction expansions of numbers and vectors. A fundamental example of this relation is between Sturmian sequences and regular continued fractions. We study S-adic words from different perspectives, namely word combinatorics, ergodic theory, and Diophantine approximation, by stressing the parallel with continued fraction expansions.

30 pages

References in corpus (3)

• Multidimensional effective S-adic systems are sofic
• An $S$-adic characterization of minimal subshifts with first difference of complexity $1\le p(n+1)-p(n)\le 2$
• Limit theorems for translation flows (published version)

Cited by in corpus (23)

• Recognizability for sequences of morphisms
• Geometry, dynamics, and arithmetic of $S$-adic shifts
• Dynamical systems arising from random substitutions
• Factor Complexity of S-adic sequences generated by the Arnoux-Rauzy-Poincaré Algorithm
• A combinatorial approach to products of Pisot substitutions
• Almost everywhere balanced sequences of complexity $2n+1$
• Substitution-based structures with absolutely continuous spectrum
• Tower power for $S$-adics
• Rauzy induction of polygon partitions and toral ${\mathbb{Z}}^2$-rotations
• Invariant measures on finite rank subshifts
• Tilings and traces
• Symbolic coding of linear complexity for generic translations of the torus, using continued fractions
• The Jacobs--Keane theorem from the $\cS$-adic viewpoint
• $\mathbf{S}$-adic sequences. A bridge between dynamics, arithmetic, and geometry
• Graph towers, laminations and their invariant measures
• Regularity of aperiodic minimal subshifts
• Abelian bordered factors and periodicity
• Measure transfer and $S$-adic developments for subshifts
• Three characterizations of a self-similar aperiodic 2-dimensional subshift
• Balancedness and coboundaries in symbolic systems
• Absence of absolutely continuous diffraction spectrum for certain S-adic tilings
• A Characterization of Infinite LSP Words
• Characterization of infinite LSP words and endomorphisms preserving the LSP property