[Paper Review] Diffuse Behaviour of Ergodic Sums Over Rotations

Jean-Pierre Conze, Stefano Isola+1 more|arXiv (Cornell University)|May 30, 2017

Mathematical Dynamics and Fractals1 references3 citations

TL;DR

This paper establishes a diffusive behavior for ergodic sums of bounded variation (BV) functions over irrational rotations by constructing specific subsequences (Ln) where the sums satisfy an Almost Sure Invariance Principle (ASIP) and a Central Limit Theorem (CLT). The key result shows that when the rotation number α has unbounded partial quotients (not of constant type), the normalized ergodic sums along such subsequences converge in distribution to a normal law, with explicit non-degenerate examples and an application to rectangular periodic billiards in the plane.

ABSTRACT

For a rotation by an irrational $α$ on the circle and a BV function $φ$, we study the variance of the ergodic sums $S_L φ(x) := \sum_{j=0}^{L -1} \, φ(x + jα)$. When $α$ is not of constant type, we construct sequences $(L_N)$ such that, at some scale, the ergodic sums $S_{L_N} φ$ satisfy an ASIP. Explicit non-degenerate examples are given, with an application to the rectangular periodic billiard in the plane.

Motivation & Objective

• To analyze the variance of ergodic sums SLϕ(x) = Σ₀^{L-1} ϕ(x + jα) along specific subsequences (Ln) for BV functions ϕ on the circle under irrational rotation by α.
• To establish conditions under which these sums exhibit diffusive behavior, i.e., satisfy a Central Limit Theorem (CLT) and an Almost Sure Invariance Principle (ASIP).
• To construct explicit, non-degenerate examples of such limits, particularly for step functions.
• To apply the results to the geometric model of the rectangular periodic billiard flow in the plane, showing that in special directions, the normalized sums of observables satisfy a CLT.

Proposed method

• Use of variance estimates for subsequences (Ln) to analyze the growth of fluctuations in ergodic sums.
• Approximation of the ergodic sums by lacunary series of the form Σₙ fₙ(kₙx), where (kₙ) is a rapidly increasing sequence of integers and (fₙ) is a bounded family of BV functions.
• Leveraging a result by Berkes and Philipp (1979) in an extended form to establish the ASIP for the approximating lacunary series.
• Application of the CLT and ASIP to step functions and BV observables, particularly when the rotation α has unbounded partial quotients.
• Use of the Denjoy-Koksma inequality to control uniform bounds on sums over denominators qn of α.
• Utilization of Fourier analysis and properties of continued fractions, including bounds on ||qₙα|| and the structure of convergents (pₙ/qₙ), to analyze the spectral and dynamical behavior.

Experimental results

Research questions

• RQ1 Under what conditions on the rotation α and the observable ϕ does the ergodic sum SLϕ(x) exhibit diffusive behavior along a subsequence (Ln)?
• RQ2 Can an Almost Sure Invariance Principle (ASIP) be established for ergodic sums of BV functions over irrational rotations when α is not of constant type?
• RQ3 What is the limiting distribution of the normalized ergodic sums SLₙϕ(x)/√n when α has unbounded partial quotients and ϕ is a step function or BV function?
• RQ4 How can the ergodic sums of BV functions be approximated by lacunary series to prove the CLT and ASIP?
• RQ5 Does the result extend to geometric systems such as the rectangular periodic billiard in the plane?

Key findings

• For irrational rotations with unbounded partial quotients (not of constant type), there exist subsequences (Ln) such that the normalized ergodic sums SLₙϕ(x)/√n satisfy a Central Limit Theorem (CLT) with a non-degenerate limit distribution.
• An Almost Sure Invariance Principle (ASIP) holds for the ergodic sums along such subsequences, meaning they can be coupled with a Brownian motion up to a small error almost surely.
• Explicit non-degenerate examples are constructed for step functions, such as ψ = 1_[0,1/2) − 1_[1/2,0), showing that the normalized sums converge in distribution to a normal law.
• The variance of the ergodic sums grows linearly with the subsequence index Ln, confirming diffusive scaling.
• The method relies on approximating the ergodic sums by lacunary series and applying a generalized version of the Berkes-Philipp theorem to establish the ASIP.
• The results are applied to the rectangular periodic billiard in the plane, where in special directions, the normalized sums of the displacement function satisfy a CLT, indicating Gaussian fluctuations in the long-time behavior.