On Coupling Lemma and Stochastic Properties with Unbounded Observables for 1-d Expanding Maps

Fatou's Lemma for Multifunctions with Unbounded Values

Fatou’s lemma in finitely many dimensions goes back to Aumann [2] and Schmeidler [32]. It plays an important technical role in the usual proofs of competitive equilibrium existence. Related versions of Fatou’s lemma were given by Artstein and Hildenbrand-Mertens [1, 24], and in [3] a version was given that subsumes the aforementioned ones. In another development, Olech introduced the use of con...

Stochastic-like Behaviour in Nonuniformly Expanding Maps

Statistical properties of intermittent maps with unbounded derivative

Fatou’s Lemma for Unbounded Gelfand Integrable Mappings

It is shown that, in the framework of Gelfand integrable mappings, the Fatou-type lemma for integrably bounded mappings, due to Cornet–Medecin [14] and the Fatou-type lemma for uniformly integrable mappings due to Balder [9], can be generalized to mean norm bounded integrable mappings.

Quantitative recurrence properties of expanding maps

Under a map T , a point x recurs at rate given by a sequence {rn} near a point x0 if d(T(x), x0) < rn infinitely often. Let us fix x0, and consider the set of those x’s. In this paper, we study the size of this set for expanding maps and obtain its measure and sharp lower bounds on its dimension involving the entropy of T , the local dimension near x0 and the upper limit of 1 n log 1 rn . We ap...