On the existence of maximizing measures for irreducible countable Markov shifts: a dynamical proof (Q2925258)

For a real function \(f\) on a dynamical system, a shift-invariant Borel probability measure is said to be maximizing if it gives \(f\) the maximal integral among all Borel invariant probability measures. Here, the question of the existence of such measures is addressed for an irreducible Markov shift over a countably infinite alphabet \(\Sigma_A(\mathbb{N})\) and functions of bounded variation under an additional technical condition, namely ``coercivity''. The main theorem shows that the maximum is attained by a measure defined on a finite alphabet subsystem, and in particular on a compact subshift. This shows existence in a setting where little was known.