# The aperiodic complexities and connections to dimensions and Diophantine approximation

Schroeder, Viktor; Weiler, Stefan (2016). The aperiodic complexities and connections to dimensions and Diophantine approximation. In: Kolyada, Sergiǐ; Möller, Martin; Moree, Pieter; Ward, Thomas. Dynamics and numbers. Providence, Rhode Island: American Mathematical Society, 237-259.

## Abstract

In their earlier work (Ergodic Th. Dynam. Sys., 34: 1699 -1723, 10 2014), the authors introduced the so called F-aperiodic orbits of a dynamical system on a compact metric space X, which satisfy a quantitative condition measuring its recurrence and aperiodicity. Using this condition we introduce two new quantities $\mathcal{F, G}$, called the aperiodic complexities', of the system and establish relations between $\mathcal{F, G}$ with the topology and geometry of X. We compare them to well-know complexities such as the box-dimension and the topological entropy. Moreover, we connect our condition to the distribution of periodic orbits and we can classify an F-aperiodic orbit of a point x in X in terms of the collection of the introduced approximation constants of x. Finally, we discuss our results for several examples, in particular for the geodesic flow on hyperbolic manifolds. For each of our examples there is a suitable model of Diophantine approximation and we classify F-aperiodic orbits in terms of Diophantine properties of the point x. As a byproduct, we prove a metric version' of the closing lemma in the context of CAT(-1) spaces.

## Abstract

In their earlier work (Ergodic Th. Dynam. Sys., 34: 1699 -1723, 10 2014), the authors introduced the so called F-aperiodic orbits of a dynamical system on a compact metric space X, which satisfy a quantitative condition measuring its recurrence and aperiodicity. Using this condition we introduce two new quantities $\mathcal{F, G}$, called the aperiodic complexities', of the system and establish relations between $\mathcal{F, G}$ with the topology and geometry of X. We compare them to well-know complexities such as the box-dimension and the topological entropy. Moreover, we connect our condition to the distribution of periodic orbits and we can classify an F-aperiodic orbit of a point x in X in terms of the collection of the introduced approximation constants of x. Finally, we discuss our results for several examples, in particular for the geodesic flow on hyperbolic manifolds. For each of our examples there is a suitable model of Diophantine approximation and we classify F-aperiodic orbits in terms of Diophantine properties of the point x. As a byproduct, we prove a metric version' of the closing lemma in the context of CAT(-1) spaces.

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Item Type: Book Section, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2016 15 Feb 2017 08:06 15 Feb 2017 08:06 American Mathematical Society Contemporary Mathematics 669 0271-4132 9781470420208 https://doi.org/10.1090/conm/669 https://arxiv.org/abs/1506.00955

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