Title: Feldman-Katok pseudometric and the GIKN construction of nonhyperbolic ergodic measures,
Abstract: The GIKN construction was introduced by Gorodetski, Ilyashenko, Kleptsyn, and
Nalsky in [Functional Analysis and its Applications, 39 (2005), 21--30]. It
gives a nonhyperbolic ergodic measure which is a weak$^*$ limit of a special
sequence of measures supported on periodic orbits. This method was later
adapted by numerous authors and provided examples of nonhyperbolic invariant
measures in various settings. We prove that the result of the GIKN construction
is always a loosely Kronecker measure in the sense of Ornstein, Rudolph, and
Weiss (equivalently, standard measure in the sense of Katok, another name is
loosely Bernoulli measure with zero entropy). For a proof we introduce and
study the Feldman-Katok pseudometric $\bar{F_{K}}$. The pseudodistance
$\bar{F_{K}}$ is a topological counterpart of the $\bar f$ metric for
finite-state stationary stochastic processes introduced by Feldman and,
independently, by Katok, later developed by Ornstein, Rudolph, and Weiss. We
show that every measure given by the GIKN construction is the
$\bar{F_{K}}$-limit of a sequence of periodic measures. On the other hand we
prove that a measure which is the $\bar{F_{K}}$-limit of a sequence of ergodic
measures is ergodic and its entropy is smaller or equal than the lower limit of
entropies of measures in the sequence. Furthermore we demonstrate that
$\bar{F_{K}}$-Cauchy sequence of periodic measures tends in the weak$^*$
topology either to a periodic measure or to a loosely Kronecker measure.