arXiv subjects

Melih Emin Can

Publications and source records attributed to Melih Emin Can.

4 recordsLinked to original sources

arXiv2026

Spectrum of invariant measures via generic points

We describe the spectrum of an ergodic invariant measure by examining the behaviour of its generic points. We define regular Wiener--Wintner generic points for a measure to generalise the characterisation of generic points for discrete spectrum measure from Lenz et al. [Ergodic Theory and Dynamical Systems vol. \textbf{44} (2024), no. 2, 524--568]. We also study limits of sequences of generic points with respect to the Besicovitch pseudometric. This translates to results about limits of measures with respect to the metric rho-bar $\barρ$ generalising Ornstein's d-bar metric. We study how the spectrum behaves when passing to the limit and we prove that points generic for discrete spectrum, totally ergodic, or (weakly) mixing measures, property K, zero entropy measures form a closed set with respect to the Besicovitch pseudometric. Hence, the same holds for corresponding measures with respect to the rho-bar metric. Our methods have already been used to prove existence of ergodic measures with desired properties, in particular with discrete spectrum. They also lead to a new proof of rational discrete spectrum of the Mirsky measure associated with a given set of $\mathscr B$-free numbers.

math.DS↗

arXiv2026

The interplay between partial specification, average shadowing, and Besicovitch completeness

Let $(X,T)$ be a compact dynamical system. This article proves that if $(X,T)$ has the partial specification property, then it has the average shadowing property. It is also proven that if $(X,T)$ is surjective and has the partial specification property, then the set of ergodic measures of $(X,T)$ is dense in the space of its invariant measures. An example of a compact dynamical system that is not Besicovitch complete is also given.

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arXiv2025

On the weakness of the vague specification property

We show that the vague specification property is strictly weaker than most of the specification-like properties, by establishing its equivalence with the asymptotic average shadowing property. In particular, we see that the weak specification property implies the vague specification property, but the converse does not hold, answering the question posed by Downarowicz and Weiss in [Ergod. Th. \& Dynam. Sys. 44(9) (2024), 2565--2580]. Additionally, we prove that, for surjective systems, the asymptotic average shadowing property is equivalent to the average shadowing property if the phase space is complete with respect to the dynamical Besicovitch pseudometric. We use the combination of both results to prove that the proximal and minimal shift spaces from [Ergod. Th. \& Dynam. Sys., 45(2) (2025), 396--426] possess the vague specification property (asymptotic average shadowing property). Our findings also allow us to address a couple of questions from [Fund. Math., 224(3) (2014), 241--278] about the asymptotic average shadowing property.

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arXiv2024

Minimal and proximal examples of $\bar{d}$-stable and $\bar{d}$-approachable shift spaces

We study shift spaces over a finite alphabet that can be approximated by mixing shifts of finite type in the sense of (pseudo)metrics connected to Ornstein's $\bar{d}$ metric ($\bar{d}$-approachable shift spaces). The class of $\bar{d}$-approachable shifts can be considered as a topological analog of measure-theoretical Bernoulli systems. The notion of $\bar{d}$-approachability together with a closely connected notion of $\bar{d}$-shadowing were introduced by Konieczny, Kupsa, and Kwietniak [in \emph{Ergodic Theory and Dynamical Systems}, vol. \textbf{43} (2023), issue 3, pp. 943--970]. These notions were developed with the aim to significantly generalize specification properties. Indeed, many popular variants of the specification property, including the classic one and almost/weak specification property ensure $\bar{d}$-approachability and $\bar{d}$-shadowing. Here, we study further properties and connections between $\bar{d}$-shadowing and $\bar{d}$-approachability. We prove that $\bar{d}$-shadowing implies $\bar{d}$-stability (a notion recently introduced by Tim Austin). We show that for surjective shift spaces with the $\bar{d}$-shadowing property the Hausdorff pseudodistance $\bar{d}^H$ between shift spaces induced by $\bar{d}$ is the same as the Hausdorff distance between their simiplices of invariant measures with respect to the Hausdorff distance induced by the Ornstein's metric $\bar{d}$ between measures. We prove that without $\bar{d}$-shadowing this need not to be true (it is known that the former distance always bounds the latter). We provide examples illustrating these results including minimal examples and proximal examples of shift spaces with the $\bar{d}$-shadowing property. The existence of such shift spaces was announced in our earlier paper [op. cit.]. It shows that $\bar{d}$-shadowing indeed generalises the specification property.

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