We prove that for any dynamical system (X,Σ,m, T ), the maximal operator defined by N∗f(x) = sup n 1 n # { 1 ≤ i : f(T ix) i ≥ 1 n } is almost everywhere finite for f in the Orlicz class L log logL(X), extending a result of Assani [2]. As an application, a weighted return times theorem is also proved.

There is a natural action of SL(2, R) on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup U = 1 * 0 1. We classify the U-invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL(2, R)-orbit of the set of branched covers of a fixed Veech surface.) For the U-action on these submanifolds, this is an ana...

Kummer’s cardinality theorem states that a language is recursive if a Turing machine can exclude for any n words one of the n+ 1 possibilities for the number of words in the language. It is known that this theorem does not hold for polynomial-time computations, but there is evidence that it holds for finite automata: at least weak cardinality theorems hold for finite automata. This paper shows ...

In this paper, we establish some new critical fixed point theorems for the sum $AB+C$ in a Banach algebra relative to the weak topology, where $frac{I-C}{A}$ allows to be noninvertible. In addition, a special class of Banach algebras will be considered.