A Note on a Classical Ergodic Theorem for IFS

The properties of the set of points where Elton's theorem does not hold are given. By constructing cylinder sets in symbolic space, it is proved that for a contractive IFS with probabilities and a continuous function g sa-tisfying that \int_X g \mathrm~d \nu_1 \neq \int_X g \mathrm~d \nu_2, where ν₁ and ν₂ are any two different invariant measures of the IFS, the zero measure invariant set composed of points where Elton's theorem does not hold is either empty or carries full Hausdorff dimension and topological entropy.