Linear parabolic maps on the torus

Abstract

We investigate linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of these systems is equal to zero, their dynamics is non-trivial due to folding of the image of the unit square into the torus. We study the structure of the maximal invariant set, and in a generic case we prove the sensitive dependence on the initial conditions. We study the decay of correlations and the diffusion in the corresponding system on the plane. We also demonstrate how the rationality of the real numbers defining the map influences the dynamical properties of the system.

Introduction

Linear area-preserving maps on the torus T² are often analyzed in the theory of ergodic Hamiltonian systems 1, 2. The map M, defined through

$$\text{x′=ax+by}\text{|}\text{mod}\text{1}\text{y′=cx+dy}\text{|}\text{mod}\text{1,}$$

can be represented by the Jacobian matrix

$$\text{g=}\text{a}\text{b}\text{c}\text{d}\text{.}$$

Due to the area-preserving condition ad−bc=1 the matrix g pertains to the non-commutative group $\text{SL(2,}\text{R}\text{)}$. Let us emphasize here that the area-preserving property holds for the map when unfolded on the plane. If the dynamics takes place on the torus, folding causes overlap of some parts of the image of the unit square, so the area is not conserved.

For each matrix g there exist its inverse g⁻¹=[d,−b;−c,a], which defines the inverse map M⁻¹ when the dynamics takes place on the plane (x,y). However, if the dynamics takes place on the torus T² (modulo 1 restriction in Eq. (1)), one has to wrap the image of the basic square back into the torus, and some parts of this image may overlap. Consequently, some points in the torus may not have any pre-images, and the map (1) may not be invertible on the torus. On the other hand, the systems defined by matrices with all integer elements are invertible. The set of these matrices G_Int forms a discrete subgroup of $\text{SL(2,}\text{R}\text{)}$. Also matrices of the type [1,t₁;0,1] and [1,0;t₂,0] represent invertible (but not continuous) maps on the torus for any non-integer parameters t₁ and t₂. These matrices correspond to the horocyclic flows [1] and form two continuous subgroups of $\text{SL(2,}\text{R}\text{)}$, denoted by G_H1 and G_H2.

Dynamical properties of the map (1) can be characterized by the trace T=Tr(g)=a+d. For |T|>2 the map is hyperbolic, eigenvalues of g are real; z>1 and 1/z (see e.g. Berry in [2]). There exist infinitely many periodic orbits and all of them are unstable with the same stability exponent λ₁=ln(z). The Arnold cat map defined by the matrix g=[1,1;1,2] is one of the most celebrated examples of such chaotic dynamical system. Since all elements of g are integer, this map is invertible on the torus T²(it is an automorphism on the torus). For |T|<2 the map is called elliptic. The eigenvalues of g are complex and both Lyapunov exponents are equal to zero. Dynamics of such a system is regular and corresponds to a rotation. Properties of the elliptic maps on the torus and some features of the elliptic-hyperbolic transition were studied by Amadasi and Casartelli [3]

The intermediate case, |T|=2 is called parabolic. Although this case is called by Berry (see Ref. [2]) `special, non-generic, set-of-measure-zero, infinitely-improbable-unless-you-deliberately-set-out-to-create-them' case, we believe that it is worth analyzing, for two reasons. On one hand, these maps are interesting, as they lie in between the elliptic and the hyperbolic cases, which are very important for several physical applications. On the other, the parabolic maps display several remarkable properties. In particular, we show that a generic linear, area-preserving, parabolic map on the torus displays sensitivity on initial conditions. Furthermore, we demonstrate how the rationality of numbers defining the map affects the dynamics of the system. A related study of the systems leading to the interval exchange maps were discussed in [4].

This paper is organized as follows. In Section 2, we discuss the general properties of parabolic maps on the torus and show corresponding families of 1D maps. In Section 3and 4, we analyze the case of rational and irrational maps, respectively. In Section 5, we analyze the property of sensitive dependence on the initial conditions. Decay of correlations and the diffusion rate are analyzed in Section 6.