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Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits

We show how to obtain information about the dynamics of a two-dimensional discrete-time system from its homoclinic and heteroclinic orbits. The results obtained are based on the theory of ‘trellises’, which comprise finite-length subsets of the stable and unstable manifolds of a collection of saddle periodic orbits. For any collection of homoclinic or heteroclinic orbits, we show how to associate a canonical ‘trellis type’ which describes the orbits. Given a trellis type, we show how to compute a ‘graph representative’ which gives a combinatorial invariant of the trellis type. The orbits of the graph give the dynamics forced by the homoclinic/heteroclinic orbits in the sense that every orbit of the graph representative is ‘globally shadowed’ by some orbit of the system, and periodic, homoclinic/heteroclinic orbits of the graph representative are shadowed by similar orbits.