The twisted baker map

TL;DR

The paper introduces the twisted baker map, a piecewise linear model for vortex dynamics, demonstrating the density and equidistribution of certain hyperbolic periodic points with diverse eigenvalue properties.

Contribution

It presents a new non-invertible map model that captures complex vortex dynamics and analyzes the distribution of hyperbolic periodic points within it.

Findings

Hyperbolic repelling periodic points are dense in the phase space.

Both types of hyperbolic points with different eigenvalues are equidistributed.

Lyapunov exponents show non-uniformity despite equidistribution.

Abstract

As a model to provide a hands-on, elementary understanding of "vortex dynamics", we introduce a piecewise linear non-invertible map called a twisted baker map. We show that the set of hyperbolic repelling periodic points with complex conjugate eigenvalues and that without complex conjugate eigenvalues are simultaneously dense in the phase space. We also show that these two sets equidistribute with respect to the normalized Lebesgue measure, in spite of a non-uniformity in their Lyapunov exponents.