A Note on the Uniform Ergodicity of Dynamical Systems

We study the uniform ergodicity property for non-invertible topological and measure-preserving dynamical systems. It is shown that for topological dynamical systems uniform ergodicity is equivalent to eventually periodicity and that for measure preserving systems it is equivalent to periodicity. To obtain our results, we prove a result on the long-term behavior of lattice homomorphisms that have $1$ isolated in its spectrum.