Equation $x^i{y}^j{x}^k=u^i{v}^j{u}^k$ in words

Abstract

We will prove that the word $a^i{b}^j{a}^k$ is periodicity forcing if $j\ge 3$ and $i+k\ge 3$, where $i$ and $k$ are positive integers. Also we will give examples showing that both bounds are optimal.