Feldman–Katok pseudometric and the GIKN non-hyperbolic ergodic measures

Abstract We introduce Feldman–Katok convergence for invariant measures of a topological dynamical system. This can be seen as a counterpart to the convergence with respect to the f -metric for finite-state stationary processes (shift-invariant measures on a symbolic space). Feldman–Katok convergence is based on a dynamically defined Feldman–Katok pseudometric. This convergence is stronger than weak ^* convergence. We prove that Feldman–Katok convergence preserves ergodicity and makes the Kolmogorov–Sinai entropy lower semicontinuous, thereby preserving zero entropy. We apply our findings to non-hyperbolic (having at least one vanishing Lyapunov exponent) ergodic measures constructed using the GIKN method as axiomatized by Bonatti, Díaz and Gorodetski Nonlinearity, 23 (2010), 687–705. The GIKN method, originally introduced by Gorodetski, Ilyashenko, Kleptsyn and Nalsky Functional Analysis and its Applications, 39 (2005), 21–30, has been widely adapted to produce non-hyperbolic ergodic measures for diffeomorphisms of compact manifolds. We prove that an ergodic measure satisfying the conditions provided by the axiomatized GIKN method is the Feldman–Katok limit of a sequence of periodic measures, which implies that it is either a periodic measure or a loosely Kronecker measure (a measure Kakutani equivalent to an aperiodic ergodic rotation on a compact group) and has zero entropy. This classifies all these measures up to Kakutani equivalence and confirms that geometric constructions of non-hyperbolic measures via periodic approximations based on the axiomatized GIKN method presented in Bonatti et al. op. cit. systematically produce zero-entropy systems.