[Paper Review] Pseudo-rotations with sufficiently Liouvillean rotation number are C^0-rigid

Barney Bramham|arXiv (Cornell University)|May 29, 2012

Mathematical Dynamics and Fractals11 references4 citations

TL;DR

This paper establishes that irrational pseudo-rotations on the 2-disk with rotation numbers in a specific dense subset of Liouville numbers—denoted $\mathcal{L}_*$—are $C^0$-rigid, meaning some iterates converge uniformly to the identity. Using pseudoholomorphic curve techniques and $L^2$ to $C^0$ estimates, the authors show such systems cannot be mixing, resolving a key obstruction in smooth ergodic theory for zero entropy Hamiltonian systems.

ABSTRACT

It is an open question in smooth ergodic theory whether there exists a Hamiltonian disk map with zero topological entropy and (strong) mixing dynamics. Weak mixing has been known since Anosov and Katok first constructed examples in 1970. Currently all known examples with weak mixing are irrational pseudo-rotations with Liouvillean rotation number on the boundary. Our main result however implies that for a dense subset of Liouville numbers (strong) mixing cannot occur. Our approach involves approximating the flow of a suspension of the given disk map by pseudoholomorphic curves. Ellipticity of the Cauchy-Riemann equation allows quantitative L^2-estimates to be converted into C^0-estimates between the pseudoholomorphic curves and the trajectories of the flow on growing time scales. Arithmetic properties of the rotation number enter through these estimates.

Motivation & Objective

• To resolve the open question of whether mixing area-preserving disk maps with zero topological entropy exist.
• To investigate the dynamical behavior of irrational pseudo-rotations with Liouville-type rotation numbers.
• To determine whether weakly mixing examples constructed by Anosov and Katok can be upgraded to strong mixing.
• To characterize the dynamical rigidity of systems with highly non-Diophantine rotation numbers using symplectic geometry tools.

Proposed method

• Approximate the suspension flow of the disk map using pseudoholomorphic curves in symplectic manifolds.
• Apply elliptic regularity and $L^2$-estimates to control the $C^0$-distance between pseudoholomorphic curves and flow trajectories.
• Use arithmetic properties of rotation numbers in $\mathcal{L}_*$—defined by super-exponential rational approximations—to bound error terms.
• Establish a quantitative $L^2$ to $C^0$ embedding estimate via Sobolev-type inequalities on periodic or half-infinite cylinders.
• Lift functions from compact quotients $\mathbb{R} \times \mathbb{R}/n\mathbb{Z}$ to $\mathbb{R}^2$ using cut-off functions and apply embedding lemmas.
• Apply the Baire category theorem to show $\mathcal{L}_*$ is dense in $\mathbb{R}$, ensuring the result applies to a generic class of rotation numbers.

Experimental results

Research questions

• RQ1 Can a Hamiltonian disk map with zero topological entropy be mixing?
• RQ2 Do irrational pseudo-rotations with Liouville rotation numbers exhibit $C^0$-rigidity?
• RQ3 Is there a dynamical obstruction to mixing in systems with sufficiently Liouville rotation numbers?
• RQ4 Can pseudoholomorphic curve techniques be used to derive $C^0$-rigidity from $L^2$-estimates in symplectic dynamics?
• RQ5 What is the role of arithmetic properties of rotation numbers in determining the regularity and convergence of iterates?

Key findings

• Every irrational pseudo-rotation with boundary rotation number in $\mathcal{L}_*$ is $C^0$-rigid, i.e., some iterates $\varphi^{n_j}$ converge uniformly to the identity.
• $\mathcal{L}_*$ is a dense subset of $\mathbb{R}$, implying the result applies to a generic class of Liouville numbers.
• Such systems are not topologically mixing, and in particular not strongly mixing with respect to Lebesgue measure.
• The $C^0$-rigidity result rules out mixing dynamics for all pseudo-rotations with rotation numbers in $\mathcal{L}_*$, closing a major gap in the ergodic hierarchy.
• The proof relies on a sharp $L^2$ to $C^0$ estimate for functions on $\mathbb{R} \times \mathbb{R}/n\mathbb{Z}$, with a uniform constant independent of $n$.
• The key analytic tool is a Sobolev-type embedding inequality that bounds the $L^∞$-norm by the product of $L^2$ and $W^{1,∞}$-norms on periodic domains.