Irregular sets are residual

Abstract

For shifts with weak specification, we show that the set of points for which the Birkhoff averages of a continuous function diverge is residual. This includes topologically transitive topological Markov chains, sofic shifts and more generally shifts with specification. In addition, we show that the set of points for which the Birkhoff averages of a continuous function have a prescribed set of accumulation points is also residual. The proof consists of bridging together strings of sufficiently large length corresponding to a dense set of limits of Birkhoff averages. Finally, we consider intersections of finitely many irregular sets and show that they are again residual. As an application, we show that the set of points for which the Lyapunov exponents on a conformal repeller are not limits is residual.