Strongly dissipative surface diffeomorphisms

We introduce a class of volume-contracting surface diffeomorphisms whose dynamics is intermediate between one-dimensional dynamics and general surface dynamics. For that type of systems one can associate to the dynamics a reduced one-dimensional model and it is proved a type of $C^{\mathrm{\infty }}-$closing lemma on the support of every ergodic measure. We also show that this class contains H'enon maps with Jacobian in $(-1\mathrm{/}4,1\mathrm{/}4)$.