Minimal Homeomorphisms on Low-dimension Tori

In this article we study minimal homeomorphisms(all orbits are dense) of the tori T , n < 5. The linear part of a homeomorphism φ of T n is the linear mapping L induced by φ on the first homology group of T . It follows from the Lefschetz fixed point theorem that 1 is an eigenvalue of L if φ minimal. We show that if φ is minimal and n < 5 then L is quasi-unipontent, i.e., all the eigenvalues of L are roots of unity and conversely if L ∈ GL(n, Z) is quasi-unipotent and 1 is an eigenvalue of L then there exists a C∞ minimal skew-product diffeomorphism φ of T n whose linear part is precisely L. We do not know if these results are true for n > 4. We give a sufficient condition for a smooth skew-product diffeomorphism of a torus of arbitrary dimension to be smoothly conjugate to an affine transformation. 1. Minimal homeomorphisms on low-dimension tori We first prove Proposition 1.1. Let φ be a minimal homeomorphism of a torus T n and L be the induced mapping on H1(T ,Z). Then the minimal polynomial p(x) of L can not be decomposed over Q[x], as p(x) = q(x)r(x) where all the roots of q(x) are roots of unity and r(x) is not constant with no roots in the unit circle. Proof. Assume that p(x) has such a decomposition. Then by the Primary Decomposition Theorem we have an invariant direct sum decomposition over Q (1) R = E ⊕ V where the restriction B of L to V is hyperbolic. Now Γ = V ∩ Z is a discrete cocompact subgroup of V and M = V/Γ is homeomorphic to a torus T , k < n. † Corresponding address: Rua Lopes Quintas, 225 ap. 401-A, Jardim Botânico, Rio de Janeiro, Cep 22460-010. Brazil. ‡ Partially supported by Chilean FONDECYT Grant N.1060977. 1 2 N. M. DOS SANTOS† AND R. URZÚA-LUZ‡ Let b be the hyperbolic diffeomorphism of M induced by B and φ be given on the covering by L+ F , where F : R → R is continuous and F (x+ l) = F (x) for all x ∈ R and l ∈ Z. We claim that b is a factor of φ. For, consider the continuous surjective mapping h : T n → M given on the covering R by (2) h(x) = P (x) +H(x) where P : R → V is the projection associated to the decomposition (1) and H : T n → V is a continuous solution of the cohomological equation (3) BH(x)−H(φ(x)) = P (F (x)) Now since B is hyperbolic then a continuous solution of (3) exists see [1] [Theorem 2.9.2 ] and since P ◦L = B ◦P then h ◦φ = b ◦h. Observing that h ◦ φ = b ◦ h for all l ∈ Z and since h is surjective we see that φ can not be minimal because b has periodic points. Theorem 1. Any minimal homeomorphism φ of a torus T , n < 5 is quasi-unipotent on the homology and 1 is an eigenvalue of its linear part. Proof. Minimality of φ and the Lefschetz fixed point Theorem shows that 1 is a root of the minimal polynomial p(x) of the linear part L of φ. Thus p(x) = (x− 1)s(x) where deg s(x) < 4, since deg p < 5. If deg s(x) = 3 then s(x) factors over Z [x] as (x ± 1)q(x) and by Proposition 1 all the roots of q are roots of unity. If deg s(x) < 3 again by Proposition 1 all the roots of s(x) are roots of unity. We do not know if the above Theorem is true if n > 4. There are irreducible polynomials in Q [x] with roots of absolute value 1 and roots of absolute value different of 1. Example 1.2. Eisenstein’s criterion shows that the polynomial p(x) = x + 4x − 6x + 4x+ 1 is irreducible over Q [x] and as p(x) = (x + 2(1− √ 3)x+ 1)(x + 2(1 + √ 3)x+ 1) we can see that ( √ 3− 1)± iλ, λ = √ (1− ( √ 3− 1)2) are roots of absolute value 1 and they are not roots of unity and the other two roots of p(x) have absolute value different from 1. MINIMAL HOMEOMORPHISMS ON LOW-DIMENSION TORI 3 2. Minimal skew-product transformations of the torus. In this section we show that every quasi-unipotent matrix L ∈ GL(n,Z), n < 5 with 1 as eigenvalue is the linear part of a smooth minimal skewproduct transformation of the torus T . Actually the skew products are of the particular type given in (5). Notice that T n−p acts freely on T p × T n−p by translation on the second factor. Thus if a homeomorphism ψ of T n commutes with this action it induces a homeomorphism ψ0 of the orbit space T /T n−p which is homeomorphic to the torus T p and we say that (T , ψ) is a free T -extension of (T , ψ0) [8] Theorem 2. Let L ∈ GL(n,Z), n < 5 be quasi-unipotent having 1 as an eigenvalue. Then there exists a minimal smooth skew-product diffeomorphism φ of the torus T n whose linear part is L. Proof. We may assume that by [Newman] [6] (4) L = (