  We study two actions of big mapping class groups. The first is an action by
isometries on a Gromov-hyperbolic graph. The second is an action by
homeomorphisms on a circle in which the vertices of the graph naturally embed.
  The first two parts of the paper are devoted to the definition of objects and
tools needed to introduce these two actions; in particular, we define and prove
the existence of equators for infinite type surfaces, we define the hyperbolic
graph and the circle needed for the actions, and we describe the
Gromov-boundary of the graph using the embedding of its vertices in the circle.
  The third part focuses on some fruitful relations between the dynamics of the
two actions. For example, we prove that loxodromic elements (for the first
action) necessarily have rational rotation number (for the second action). In
addition, we are able to construct non trivial quasimorphisms on many subgroups
of big mapping class groups, even if they are not acylindrically hyperbolic.
