Header

UZH-Logo

Maintenance Infos

The aperiodic complexities and connections to dimensions and Diophantine approximation


Schroeder, Viktor; Weil, Steffen (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.

Statistics

Citations

Altmetrics

Downloads

0 downloads since deposited on 15 Feb 2017
0 downloads since 12 months

Additional indexing

Item Type:Book Section, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2016
Deposited On:15 Feb 2017 08:06
Last Modified:26 Feb 2018 05:16
Publisher:American Mathematical Society
Series Name:Contemporary Mathematics
Number:669
ISSN:0271-4132
ISBN:9781470420208
OA Status:Closed
Publisher DOI:https://doi.org/10.1090/conm/669
Related URLs:https://arxiv.org/abs/1506.00955

Download