On Sturmian graphs
نویسندگان
چکیده
In this paper we define Sturmian graphs and we prove that all of them have a certain “counting” property. We show deep connections between this counting property and two conjectures, by Moser and by Zaremba, on the continued fraction expansion of real numbers. These graphs turn out to be the underlying graphs of compact directed acyclic word graphs of central Sturmian words. In order to prove this result, we give a characterization of the maximal repeats of central Sturmian words. We show also that, in analogy with the case of Sturmian words, these graphs converge to infinite ones.
منابع مشابه
Usefulness of Directed Acyclic Subword Graphs in Problems Related to Standard Sturmian Words
The class of finite Sturmian words consists of words having particularly simple compressed representation, which is a generalization of the Fibonacci recurrence for Fibonacci words. The subword graphs of these words (especially their compacted versions) have a very special regular structure. In this paper we investigate this structure in more detail than in previous papers and show how several ...
متن کاملOuter Billiards, Arithmetic Graphs and Multigrid Flows
I considered the first construction in my book, Outer Billiards on Kites [S2]. The arithmetic graphs served as the main tool for understanding the dynamics of outer billiards on kites. See §2 for definitions. The second construction, which is quite general, is based on patterns of oriented lines in the Euclidean plane. See §3 for definitions. The concrete instance of the multigrid construction ...
متن کاملOn Lazy Representations and Sturmian Graphs
In this paper we establish a strong relationship between the set of lazy representations and the set of paths in a Sturmian graph associated with a real number α. We prove that for any non-negative integer i the unique path weighted i in the Sturmian graph associated with α represents the lazy representation of i in the Ostrowski numeration system associated with α. Moreover, we provide several...
متن کاملSturmian graphs and integer representations
In this paper we consider a numeration system, originally due to Ostrowski, based on the continued fraction expansion of a real number α. We prove that this system has deep connections with the Sturmian graph associated with α. We provide several properties of the representations of the natural integers in this system. In particular, we prove that the set of lazy representations of the natural ...
متن کاملGeneralized balances in Sturmian words
One of the numerous characterizations of Sturmian words is based on the notion of balance. An innnite word x on the f0;1g alphabet is balanced if, given two factors of x, w and w 0 , having the same length, the diierence between the number of 0 0 s in w (denoted by jwj0) and the number of 0 0 s in w 0 is at most 1, i.e. jjwj0 ? jw 0 j0j 1. It is well known that an aperiodic word is Sturmian if ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete Applied Mathematics
دوره 155 شماره
صفحات -
تاریخ انتشار 2007