Typical uniqueness in ergodic optimization

Abstract For ergodic optimization on any topological dynamical system, with real-valued potential function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> belonging to any separable Banach space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> of continuous functions, we show that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> -maximizing measure is typically unique, in the strong sense that a countable collection of hypersurfaces contains the exceptional set of those <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> with non-unique maximizing measure. This strengthens previous results asserting that the uniqueness set is both residual and prevalent. For various other topological vector spaces of continuous functions, we show that uniqueness of the maximizing measure is either generic, or prevalent, or corresponds to the complement of a sparse set.