Year: 2025
Type: article
Abstract: We construct cocycles in T × S U ( 2 ) \mathbb {T} \times SU(2) over Diophantine rotations that are minimal and not uniquely ergodic. Such cocycles are dense in an open subset of cocycles over the fixed Diophantine rotation. By a standard argument, they are dense in the whole set of such cocycles if the rotation in the basis satisfies a full-measure arithmetic condition. Contrary to the examples existing in the literature, ours have vanishing Lyapunov exponents, like in the original construction by Furstenberg. The construction is explicit, without any need for a parameter exclusion argument, and can be carried out for any prescribed fibered rotation number, provided that the latter satisfies a necessary generic arithmetic condition with respect to the rotation in the basis T \mathbb {T} .
DOI: 10.1090/proc/17388
Author: Nikolaos Karaliolios
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4415293941