This paper studies homeomorphisms of surfaces isotopic to the identity by means of purely topological methods and Brouwer theory. The main development is a novel theory of orbit forcing using maximal isotopies and transverse foliations. This allows us to derive new proofs for some known results as well as some new applications, among which we note the following: we extend Franks and Handel's classification of zero entropy maps of S 2 for non-wandering homeomorphisms; we show that if f is a Hamiltonian homeomorphism of the annulus, then the rotation set of f is either a singleton or it contains zero in the interior, proving a conjecture posed by Boyland; we show that there exist compact convex sets of the plane that are not the rotation set of some torus homeomorphisms, proving a first case of the Franks-Misiurewicz Conjecture; we extend a bounded deviation result relative to the rotation set to the general case of torus homeomorphisms. F. A. Tal was partially supported by CAPES, FAPESP and CNPq-Brasil. 1 arXiv:1503.09127v3 [math.DS] 8 Nov 2017The study of non-contractible periodic orbits for Hamiltonian maps of sympletic manifolds has been receiving increased attention (see for instance [GG]). A natural question in the area, posed by V. Ginzburg, is to determine if the existence of non-contractible periodic points is generic for smooth Hamiltonians. A consequence of Corollary I is an affirmative answer for the case of the torus:Proposition J. Let Ham ∞ (T 2 ) be the set of Hamiltonian C ∞ diffeomorphisms of T 2 endowed with the Whitney C ∞ -topology. There exists a residual subset A of Ham ∞ (T 2 ) such that every f in A has non-contractible periodic points.Let us explain now the results related to the entropy. For example we can give a short proof of the following improvement of a result due to Handel [H1].Theorem K. Let f : S 2 → S 2 be an orientation preserving homeomorphism such that the complement of the fixed point set is not an annulus. If f is topologically transitive then the number of periodic points of period n for some iterate of f grows exponentially in n. Moreover, the entropy of f is positive.Another entropy result we obtain is related to the existence and continuous variation of rotation numbers for homeomorphisms of the open annulus. A stronger version of this result for diffeomorphisms was already proved in an unpublished paper of Handel [H2]. Given a homeomorphism of T 1 × R and a liftf to R 2 , we say that a point z ∈ T 1 × R has a rotation number rot(z) if the ω-limit of its orbit is not empty, and if for any compact set K ⊂ T 1 × R and every increasing sequence of integers n k such that f n k (z) ∈ K and anyž ∈ π −1 (z), lim k→∞ 1 n k π 1 (f n k (ž) − π 1 (ž) = rot(z),where π is the covering projection from R 2 to T 1 × R and π 1 : R 2 → R is the projection on the first coordinate.