Counting Berg partitions

TL;DR

This paper characterizes all Berg partitions, special Markov partitions with two rectangles, for hyperbolic toral automorphisms, providing a complete classification and counting of these partitions based on their connectivity matrices.

Contribution

It offers a complete description and enumeration of Berg partitions for hyperbolic toral automorphisms, including a formula for their count based on connectivity matrices.

Findings

Exactly (k + n + l + m)/2 non-equivalent Berg partitions per connectivity matrix

Complete classification of Berg partitions for given automorphisms

Explicit enumeration formula for Berg partitions

Abstract

We call a Markov partition of a two dimensional hyperbolic toral automorphism a Berg partition if it contains just two rectangles. We describe all Berg partitions for a given hyperbolic toral automorphism. In particular there are exactly (k + n + l + m)/2 nonequivalent Berg partitions with the same connectivity matrix (k, l, m, n).