Arithmetic exponents in piecewise-affine planar maps

Abstract

We consider the growth of some indicators of arithmetical complexity of rational orbits of (piecewise) affine maps of the plane, with rational parameters. The exponential growth rates are expressed by a set of exponents; one exponent describes the growth rate of the so-called logarithmic height of the points of an orbit, while the others describe the growth rate of the size of such points, measured with respect to the p-adic metric. Here p is any prime number which divides the parameters of the map. 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 set of exponents. We also show that the convergence of the p-adic exponents may be non-uniform, with arbitrarily large fluctuations occurring arbitrarily close to any point. We explore numerically the behaviour of these quantities in the chaotic regions, in both area-preserving and dissipative systems. In the former case, we conjecture that wherever the Lyapunov exponent is zero, the arithmetical exponents achieve a local maximum.