Denseness of zero entropy aperiodic ergodic measures
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2026
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| author | Crispin, Camila Díaz, Lorenzo J. |
| author_facet | Crispin, Camila Díaz, Lorenzo J. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_28398 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Denseness of zero entropy aperiodic ergodic measures Crispin, Camila Díaz, Lorenzo J. Dynamical Systems 37D30, 37D25, 37A35, 37C40 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. |
| title | Denseness of zero entropy aperiodic ergodic measures |
| topic | Dynamical Systems 37D30, 37D25, 37A35, 37C40 |
| url | https://arxiv.org/abs/2603.28398 |