Generic ergodic optimisation for invertible dynamical systems

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Ergodic optimisation is a subfield of dynamical systems which considers the problem of choosing the initial condition of a dynamical system in order to maximise the long-term average output of a sequence of observations along the trajectory. In 1999, Guocheng Yuan and Brian R. Hunt asked whether for expanding or hyperbolic dynamical systems on manifolds, it is generically true (in the sense of Baire category) that for a C^1 observable this average is maximised by taking the initial condition to be an appropriate periodic point. This question was answered positively by Gonzalo Contreras in the expanding case in the 2016 article "Ground States are generically a Periodic Orbit" (published in Inventiones Mathematicae). The overall objective of the present project is to extend Contreras' result to invertible, hyperbolic dynamical systems using the methodology of ergodic optimisation for amphidynamical systems introduced by Thierry Bousch, answering the question of Yuan and Hunt in this broader context. Initially the project will focus on invertible symbolic dynamical systems, with a view to extending the results to invertible dynamical systems such as Anosov diffeomorphisms if time permits. Extensions of these results to dynamical systems in continuous time may also be explored if time permits.