Let $W$ be an irreducible subvariety of codimension $r$ in a smooth affine variety $X$ of dimension $n$ defined over the complex field $C$. Suppose that $W$ is left pointwise fixed by an automorphism of $X$ of infinite order or by a one-dimensional algebraic torus action on $X$. In the present article, we consider whether or not $X$ is then an affine space bundle over $W$ of fiber dimension $n-r$. Our results concern the case $r=1$ or the case $r=2$ and $n\leq3$. As by-products, we obtain algebro-topological characterizations of the affine 3-space.