Arithmetic, Hamiltonian, and data applications

Core idea

Follow recurrence to arithmetic progressions, volume preservation to Hamiltonian Koopman dynamics, and finite observations to data-driven operator approximations.

Prerequisite doorway

• Symplectic manifold
• Hamiltonian vector field
• Hamiltonian flow
• Poisson bracket on a symplectic manifold
• Eigenvalue

Reading path

• Furstenberg multiple recurrence theorem
• Upper Banach density
• Furstenberg correspondence principle
• Szemerédi theorem
• Liouville theorem for Hamiltonian volume
• Koopman dynamics of a Hamiltonian flow
• Dynamic mode decomposition
• Extended dynamic mode decomposition

Return to Ergodic theory.