Growth of heights in piecewise-affine planar maps

TL;DR

This paper investigates the asymptotic growth of heights of points under piecewise-affine planar maps with rational parameters, revealing uniform exponential growth in linear regions and complex fluctuations in $p$-adic heights, with numerical exploration of chaotic dynamics.

Contribution

It provides a detailed analysis of height growth rates in affine maps, including both global and local $p$-adic heights, and uncovers non-uniform convergence and chaotic behavior.

Findings

Almost all points in linear regions share the same exponential height growth rate.

Convergence of $p$-adic heights can be non-uniform with large fluctuations.

Numerical analysis of height behavior in chaotic regions shows complex dynamics.

Abstract

We consider the growth of heights of the points of the orbits of (piecewise) affine maps of the plane, with rational parameters. We analyse the asymptotic growth rate of both global and local ($p$-adic) heights, for the primes $p$ that divide the parameters. We show that almost all the points in a domain of linearity (such as an elliptic island in an area-preserving map) have the same exponential growth rate. We also show that the convergence of the $p$-adic height may be non-uniform, with arbitrarily large fluctuations occurring arbitrarily close to any point. We explore numerically the behaviour of heights in the chaotic regions, in both area-preserving and dissipative systems.