Bijective coding of automorphisms of the torus and binary quadratic forms

Abstract

We consider special maps from a symbolic compactum to a two-dimensional torus that connect a bidirectional translation with a given hyperbolic automorphism of a two-dimensional torus and preserve the maximum entropy measure. In contrast to the known geometric coding method with the aid of Markovian partitions (see references in [2] and [9]), the coding under consideration uses the arithmetic and group theoretic structure. This makes it possible to connect the arithmetic of the corresponding algebraic extension of Q and the properties of a certain binary quadratic form with the dynamics of the automorphism. The transformations can be parametrized by the units of the quadratic field or by the homoclinic points of some specific orbit. Let T be a hyperbolic automorphism ofT2 given by the matrixM = a b c d ∈ GL(2,Z). We put r = TrM,σ = detM,D = r2 − 4σ. Hyperbolicity implies that r 6= 0 for σ = −1, and |r| ≥ 3 for σ = +1. For simplicity, we shall assume that r > 0. Let λ = (r+ √ D)/2 be the greatest eigenvalue of M . As a symbolic compactum X we choose either a stationary Markov compactum Xr with state space 0,1, . . . , r and pairwise prohibitions {en = r ⇒ en+1 = 0, n ∈Z} for σ = −1, or a sofic compactumYr = {{en}−∞ : 0 ≤ en ≤ r−1, (en . . . en+s) 6= (r−1)(r−2)s−2(r−1) for any n ∈Z and any s ≥ 2} for σ = +1. Therefore, each of the two compacta is fully determined by the trace and is a β-compactum for β = λ (see [7]). Let τ denote a bidirectional translation on X, that is, (τe)n = en+1, and let μ be a measure of maximum entropy. We define the normalization operation as a map n : Q∞ −∞{0,1, . . . , 2r} → X such that if {xk} is a finite sequence, then n({xk}) = {ek} ′ l is the sequence of coefficients of a Markovian (sofic) β-decomposition with β = λ of the number x = P k xkλ −k. More precisely, l = [− logλ x] + 1, el = [λlx], ek = [λSk−l(λlx)], k > l, where Sx = {λx}. For an arbitrary sequence the normalization is defined as the weak limit of normalizations of finite approximations. On the well-posedness of this definition, see [4] and [9]. The sum of two sequences in X is defined to be the normalization of their coordinatewise sum. Similarly, one can define the inverse element with respect to addition. It can be shown that the sum and difference are defined for a.a. pairs of sequences under the measure μ×μ. After the natural arithmetic factorization mod 0, the maximum entropy measure p : X → X′ defines a group structure on the quotient space X′, with respect to which the translation τ is a group automorphism. We recall that a point s ∈T2 is said to be homoclinic to zero (briefly: homoclinic) if it belongs to the intersection of the leaves of the expanding and the contracting foliations passing through 0. Each homoclinic point s can be obtained in a unique way as the projection of a lattice point z(s) ∈ Z2 onto the one-dimensional eigenspace Lu corresponding to the eigenvalue λ along the other one-dimensional eigenspace Ls, followed by projection of the resulting point t = t(s) onto the torus (see [1]). The point t = t(s) will be called the plane coordinate of the homoclinic point s.