**Question 1.** Assume that an infinite word $u\in\{0,1\}^{\mathbb Z}$ is not balanced. Is it true that there exists a finite 0-1 word $w$ such that $0w01w1$ or $1w10w0$ is a factor of $u$? Is it true that both are? (Perhaps, with different $w$.) - **answered in the negative**


**Question 1 (modified).** Assume that an infinite word $u\in\{0,1\}^{\mathbb Z}$ is not balanced. Is it true that there exists a finite (possibly, empty) 0-1 word $w=w\_1\dots w\_k$ such that $0w\_1\dots w\_k01w\_k\dots w\_11$ or $1w\_1\dots w\_k10w\_k\dots w\_10$ is a factor of $u$? Is it true that both are? (Perhaps, with different $w$.) - **ANSWERED by Wolfgang**


**Question 2.** Assume now that $u\in\{0,1\}^{\mathbb Z}$ is balanced and aperiodic (i.e., Sturmian). Is it true that for any $n\ge1$ there exists a finite 0-1 word $w=w\_1\dots w\_k$ with $k\ge n$ such that $w\_1\dots w\_k01w\_k\dots w\_1$ or $w\_1\dots w\_k10w\_k\dots w\_1$ is a factor of $u$? Again, is it true that both are (with different $w$)? - **ANSWERED by domotorp**


I am familiar with Lothaire's famous monograph "Algebraic Combinatorics on Words" but couldn't find these results there. 


**EDIT.** Here's what is meant by `balanced'. For a finite 0-1 word $w$ let $\delta(w)$ denote the number of 1s in $w$. A 0-1 word $w$ (finite or infinite) is called *balanced* if for any $n\ge1$ and any two factors $u$ and $v$ of $w$ of length $n$ we have $|\delta(u)-\delta(v)|\le 1$. 


For instance, $0100101$ is balanced whilst $101000$ is not, since $\delta(101)=2$ and $\delta(000)=0$. 



Two questions from combinatorics on words
I think Question 2 is true and the proof is as follows. Every Sturmian word is equivalent to the [cutting sequence](http://en.wikipedia.org/wiki/Sturmian_word#Cutting_Sequence_of_Irrational) of an irrational number (except, I think, those that have only one extra digit compared to a rational number, for which your statement is true). If you have a line $ax$ that goes through the origin, then its cutting sequence in both directions is the same, i.e., they are each other's reverse. Similarly, whenever the line goes "very close" to a grid point, the part before it will be the reverse of the part after it for a long time. Close to the grid point, you have the 01 or the 10. From [Diophantine approximation](http://en.wikipedia.org/wiki/Diophantine_approximation), it follows that we can indeed achieve "very close" in both situations.


