We prove that, if f is a homeomorphism of the 2-torus isotopic to the identity whose rotation set is a non-degenerate segment and f has a periodic point, then it has uniformly bounded deviations in the direction perpendicular to the segment.

São Paulo, São Paulo, 2018. No presente trabalho iremos provar, usando a folheação de Brouwer-Le Calvez e a teoria de forcing dela derivada, que dado um homeomorsmo do toro f :ou em outras palavras, f não possui difusão sublinear na direção perpendicular à direção do conjunto de rotação Palavras-chave: homeomorsmos do toro, dinâmica topológica, conjunto de rotação, difusão sublinear. iii iv Abstract SALOMÃO, G. S. Inexistence of sublinear diusion for a class of torus homeomorphisms. 2018. 120 f. Tese (Doutorado) -In the present work we will prove, using the Brouwer-Le Calvez foliation and the forcing theory derived from it, that given a torus homeomorphism f : T 2 → T 2 isotopopic to the identity such that its rotation set ρ(f ) = {t ρ 0 | 0 ≤ t ≤ 1} is a line segment with irrational slope and 0 is an extreme point, then there exists M > 0 such that | f n (x) −x, ρ ⊥ 0 | < M, for everyx ∈ R 2 and n ∈ Z, or in other words, f does not have sublinear diusion in the direction perpendicular to the direction of the rotation set.