  The topological snail is a geometric universal plane object, described by
matrices in the special linear group with integer coefficients. It has many
nice properties. From a dynamical point of view, it gives a description of the
set of fixed points and periodic obits of turbulent homeomorphims in the plane.
Those homeomorphisms are the truly complicated one. They have many fixed points
and periodic orbits, as much as the trace of the turbulence matrix that
caracterizes their action on a finite invariant set. The topological snail
generally gives a map of the fixed points and periodic orbits of a such
homeomorphims. One can compute their indexes and turbulent topological types,
and different Nielsen's classes of fixed points naturally give a minoration of
the topological entropy of the considered homeomorphisms by the logarithm of
the spectral radius of the turbulence matrix. One even get a dynamical
interpretation and proof of the extension of the Fermat' theorem to matrices. I
first described the topological snail and its properties at my conference the
31st of may 2024 in Paris, in the case of a purely turbulent homeomorphism,
with an invariant set of three points.
