Consider the expanding map given by multiplication by a non-integer
modulo 1.
The typical behavior for a point under this map is that the orbit is
dense and that ergodic averages converge. However, the set of exceptional
points, not having such properties, has full Hausdorff dimension and even
has large intersection properties.
I will show how this can be proven. Central for the argument is the
β-shift, the symbolic space generated by the map. The case when this
shift is not of finite type is tricky and is handled by an approximation
argument.