  Let $k$ be an algebraically closed field and $Q$ be an acyclic quiver with
$n$ vertices. Consider the category ${\rm rep}(Q)$ of finite dimensional
representations of $Q$ over $k$. The exceptional representations of $Q$, that
is, the indecomposable objects of ${\rm rep}(Q)$ without self-extensions,
correspond to the so-called real Schur roots of the usual root system attached
to $Q$. These roots are special elements of the Grothendieck group
$\mathbb{Z}^n$ of ${\rm rep}(Q)$. When we identify the dimension vectors of the
representations (that is, the non-negative vectors of $\mathbb{Z}^n$) up to
positive multiple, we see that the real Schur roots can accumulate in some
directions of $\mathbb{R}^n \supset \mathbb{Z}^n$. This paper is devoted to the
study of these accumulation points. After giving new properties of the
canonical decomposition of dimension vectors, we show how to use this
decomposition to describe the rational accumulation points. Finally, we study
the irrational accumulation points and we give a complete description of them
in case $Q$ is of weakly hyperbolic type.
