  We study some forward invariant sets appearing in the dynamics of the
exponential family. We prove that the Hausdorff dimension of the sets under
consideration is not larger than $1$. This allows us to prove, as a
consequence, a result for some dynamically defined indecomposable continua
which appear in the dynamics of the exponential family. We prove that the
Hausdorff dimension of these continua is equal to one.
