English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study the descriptive complexity of sets of points defined by placingrestrictions on statistical behaviour of their orbits in dynamical systems onPolish spaces. A particular examples of such sets are the set of generic pointsof a $T$-invariant Borel probability measure, but we also consider much moregeneral sets (for example, $\alpha$-Birkhoff regular sets and the irregular setappearing in multifractal analysis of ergodic averages of a continuousreal-valued function). We show that many of these sets are Borel. In fact, allthese sets are Borel when we assume that our space is compact. We provideexamples of these sets being non-Borel, properly placed at the first level ofthe projective hierarchy (they are complete analytic or co-analytic). Thisproves that the compactness assumption is in some cases necessary to obtainBorelness. When these sets are Borel, we use the Borel hierarchy to measuretheir descriptive complexity. We show that the sets of interest are located atmost at the third level of the hierarchy. We also use a modified version of thespecification property to show that for many dynamical systems these sets areproperly located at the third level. To demonstrate that the specificationproperty is a sufficient, but not necessary condition for maximal descriptivecomplexity of a set of generic points, we provide an example of a compactminimal system with an invariant measure whose set of generic points is$\Pi^0_3$-complete.

Borel complexity of sets of points with prescribed Birkhoff averages in Polish dynamical systems with a specification property