Heaviness - an Extension of a Lemma of Y. Peres

We provide an elementary proof of Y. Peres’ lemma on the existence in certain dynamical systems of what we term heavy points, points whose ergodic averages consistently dominate the expected value of the ergodic averages. We also derive several generalizations of Peres’ lemma by employing techniques from the simplified proof. 1. The Lemma of Peres and its Immediate Generalizations The following lemma is derived by Yuval Peres [7] using the maximal ergodic theorem. While this original proof is quite short and natural, we will see that elementary methods yield more general results. Before proceeding to these broader results, however, the original lemma deserves mention. Lemma 1 (Peres). Let T : X → X be a continuous transformation of a compact space, and let μ be a probability measure preserved by T . For every continuous f : X → R there exists some x ∈ X such that (1) ∀N ∈ N 1 N N−1