Denseness of zero entropy aperiodic ergodic measures

Abstract: We study partially hyperbolic homoclinic classes of $C^1$-generic diffeomorphisms with a one-dimensional central bundle, so that the central Lyapunov exponent $χ^c(μ)$ is well defined for any ergodic measure $μ$ supported on the class. We focus on nonhyperbolic homoclinic classes supporting ergodic measures with positive, zero, and negative central exponents. For each $α$ and a nontrivial homoclinic class $H$ of a $C^1$-generic diffeomorphism $f$, we consider the level set of measures [ \mathcal{M}^α_{\mathrm{erg}}(f,H)= \left{\text{$μ$ ergodic, supported on $H$, with } χ^c(μ)=α\right}. ] In this generic setting, the range of $α$ for which $\mathcal{M}^α_{\mathrm{erg}}(f,H)$ is nonempty forms a nontrivial closed interval $I$. Since the set of periodic measures is countable, most of these sets contain no periodic measures. We show that for every $α$ in the interior of $I$, the so-called Axiomatized GIKN measures, a class of low-complexity, zero-entropy measures, are dense in $\mathcal{M}^α_{\mathrm{erg}}(f,H)$. This result can be viewed as an analogue of Sigmund's classical density of periodic measures for systems with the specification property, obtained here in a setting where the specification property does not hold and periodic measures are typically absent (in the considered level sets). We also present a similar result for the open class of blender-minimal diffeomorphisms, contained in the class of $C^1$-robustly transitive ones.