Let T α denote the rotation T α x = x + α (mod 1) by an irrational number α on the additive circleOne of the goals of this paper is to describe the ergodic properties of the cocycle (taking values in R d+1 ) generated over T α by the vectorial functionIt was already proved in [LeMeNa03] that Ψ 2 is regular for α with bounded partial quotients. In the present paper we show that Ψ 2 is regular for any irrational α. For higher dimensions, we give sufficient conditions for regularity. While the case d = 2 remains unsolved, for d = 3 we provide examples of non-regular cocycles Ψ 4 for certain values of the parameters β 1 , β 2 , β 3 .We also show that the problem of regularity for the cocycle Ψ d+1 reduces to the regularity of the cocycles of the formTherefore, a large part of the paper is devoted to the classification problems of step functions with values in R d .

No abstract

Funders, publishers, scholarly societies, universities, and other stakeholders need to be able to track the impact of programs and policies designed to advance data sharing and reuse. With the launch of the NIH data management and sharing policy in 2023, establishing a pre-policy baseline of sharing and reuse activity is critical for the biological and biomedical community. Toward this goal, we tested the utility of mentions of research resources, databases, and repositories (RDRs) as a proxy measurement of data sharing and reuse. We captured and processed text from methods sections of open access biological and biomedical research articles published in 2020 and 2021 and made available in PubMed Central. We used natural language processing to identify text strings to measure RDR mentions. In this article, we demonstrate our methodology, provide normalized baseline data sharing and reuse activity in this community, and highlight actions authors and publishers can take to encourage data sharing and reuse practices.