Let denote the circle, parameterized by . For every irrational number , let denote the irrational rotation by :
For any function , one can consider the skew product extension given by
from now on called cylinder flow. In this series of posts I intend to discuss the basic results regarding this class of transformations. Following the established notation in this blog, the titles will have the form “CF” (where CF stands for “Cylinder flows” and “” stands for the number of the lecture).
There is a natural invariant measure for : it is , where and are the Lebesgue measure of and , respectively. This measure is infinite and so the ergodic theoretical properties of cylinder flows have to be investigated in a different perspective from that of probability measure-preserving systems. For example, in the infinite measure situation the Birkhoff averages converge to zero almost surely, independent of the function.
In some situations, the map takes values in and in this case we also call the cylinder flow a -extension. When we want to include both situations, we will write instead of or .
I have no intention in covering all the subjects that arise in this area. To this purpose, I refer the reader to the book An introduction to infinite ergodic theory of Jon Aaronson in which one chapter is devoted to the study of cylinder flows. Instead, this series of posts will discuss some of these topics as well as new results:
- Uniform distribution and Denjoy-Koksma’s inequality: in the first post we discuss the uniform distribution of and Denjoy-Koksma’s inequality which roughly speaking says that, along the denominators of the continued fraction expansion of , the orbits of are highly equidistributed. Ergodic cylinder flows arise at this point as a tool to guarantee that the opposite phenomena of Denjoy-Koksma’s inequality, i.e. the irregularity of equidistribution, appears.
- Ergodicity and essential values: because has infinite Lebesgue measure, ergodicity for cylinder flows has not many characterizations as in the finite measure situation. Nevertheless, to each cylinder flow one can associate a subgroup of that characterizes ergodicity in the sense that is ergodic iff . Each element is called an essential value and so is ergodic iff every element of is an essential value.
- Basics in infinite ergodic theory: the foregoing discussion requires a basic understanding of classical results in infinite ergodic theory, such as Hopf’s ratio ergodic theorem, the non-existence of Birkhoff’s theorem, Aaronson’s theorem, rational ergodicity and law of large numbers.
- Law of large numbers for some cylinder flows: I will discuss a result in collaboration with Patricia Cirilo and Enrique Pujals, in which we construct new examples of cylinder flows that have law of large numbers. We prove that, along a subsequence of iterates, the map is rationally ergodic, and this is enough to have law of large numbers.
- Further results: recently there has been a growing interest in more general cylinder flows, such as -extensions and -covers of translation surfaces. See e.g the paper Ergodicity of Extension of Irrational Rotation on the Circle of Yuqing Zhang and the recent preprints Ergodic infinite group extensions of geodesic flows on translation surfaces of David Ralston and Serge Troubetzkoy and Ergodicity for Infinite Periodic Translation Surfaces of Pascal Hubert and Barak Weiss.