Explicit formulae and topological descriptions of action-minimizing sets of a full shift with an uncountable alphabet $[0,1]$

Abstract

We completely solve ergodic optimization of a full shift with an uncountable alphabet $[0,1]$, which is one of the most well-known examples of infinite dimensional dynamical systems with positive mean dimension (and thus with infinite topological entropy), for potentials depending only on the first two coordinates with the twist condition as well as giving explicit formulae of the associated Mather set and the Aubry set. Moreover, we investigate the total disconnectedness of the (quotient) Aubry set, in which case the differentiability of the potential function makes a crucial difference. Although these results imply that the (quotient) Aubry set is small enough, we give a complete characterization of an analogical object of the Aubry set, called the Ma\~{n}\'e set, and show that it is much larger than the Aubry set so that it contains cubes of any finite dimension. In our proofs, estimates for ``connecting orbits" play key roles, and we establish them by combining three different perspectives in our symbolic setting; weak KAM subaction approach for symbolic dynamics, Ma\~{n}\'e's formulation in Lagrangian systems, and Bangert's variational approach for twist maps.