  In this work we develop a new criterion for the existence of topological
horseshoes for surface homeomorphisms in the isotopy class of the identity.
Based on our previous work on forcing theory, this new criterion is purely
topological and can be expressed in terms of equivariant Brouwer foliations and
transverse trajectories. We then apply this new tool in the study of the
dynamics of homeomorphisms of surfaces with zero genus and null topological
entropy and we obtain several applications. For homeomorphisms of the open
annulus $\mathbb{A}$ with zero topological entropy, we show that rotation
numbers exists for all points with nonempty omega limits, and that if
$\mathbb{A}$ is a generalized region of instability then it admits a single
rotation vector. We also offer a new proof of a recent result of Passegi,
Potrie and Sambarino, showing that zero entropy dissipative homeomorphisms of
the annulus having as an atractor a circloid have a single rotation number.
  Our work also studies homeomorphisms of the sphere without horseshoes. For
these maps we present a structure theorem in terms of fixed point free
invariant sub-annuli, as well as a very restricted description of all possible
dynamical behavior in the transitive subsets. This description ensures, for
instance, that transitive sets can contain at most $2$ distinct periodic orbits
and that, in many cases, the restriction of the homeomorphism to the transitive
set must be an extension of an odometer. In particular, we show that any
nontrivial and stable transitive subset of a dissipative diffeomorphism of the
plane is always infinitely renormalizable in the sense of
Bonatti-Gambaudo-Lion-Tresser.
