Balanced Word to Balanced (Sturmian?) Sequence

by co.sine   Last Updated December 06, 2018 15:20 PM

Let $E \in \{0,1\}^{n}, n\in \mathbb{N}$, be a balanced finite word: for every two subwords $U,V$ of the same length, the number of $1$'s in $U$ differs from the number of $1$'s in $V$ by at most one.

  • Can $E$ be continued to an infinite balanced sequence?
  • Can $E$ be continued to an infinite Sturmian sequence? (A subcase of the previous question.)

Furthermore, in both cases, what is the cardinality of the set of possible extensions of $E$?

This came up when I tried to solve an exercise, where I have to show that Sturmian sequences are dense in the space of balanced sequences (with regard to the usual topology of symbolic dynamics).

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