in 1974, Sigmund proved that the specification property implies the density (in the weak-* topology) of periodic measures in the space of invariant measures. The goal of this work is to understand the structure of the space of ergodic measures in nonhyperbolic systems that do not satisfy the specification property. We prove global and semi-local versions of Sigmund's result for the set of nonhyperbolic ergodic measures, replacing periodic measures by a class of aperiodic ergodic ones with zero entropy called GIKN.

By the Kupka-Smale genericity Theorem, generically, there are no nonhyperbolic periodic measures. Therefore to obtain a result analogous to Sigmund’s one restricted to nonhyperbolic measures, we should seek measures that can serve as periodic measures.

Gorodetski, Ilyashenko, Kleptsyn, and Nalsky constructed nonhyperbolic ergodic measures as weak* limit of periodic ones, the so-called as GIKN measures. These measures have low complexity, as proven by Kwietniak and Łącka, and can thus be considered as natural replacements for periodic measures. We will show that, in partially hyperbolic contexts,GIKN measures form a dense subset in the space of nonhyperbolic ergodic measures.

Moreover, we adapt the construction of Gorodetski et al. to obtain hyperbolic GIKN measures and prove that, also in partially hyperbolic contexts, GIKN measures with a fixed Lyapunov exponent are dense in the set of hyperbolic measures with the same Lyapunov exponent.

We also state these results in the skew product and matrix cocycle settings.