# homoclinic points of toral automorphisms

Hi!

Can you help me with solving this problem:

Suppose that A is hyperbolic toral automorphism( represented with matrix A) with only one real eigenvalue \lambda >1 with geometric multiplicity 1. Suppose that v is right and w left eigenvector associated with \lambda and that =1. We construct a matrix H in the followig way. On position (i,j) matrix H has v_i w_j. I need to prove that set of homoclinic points to 0 of A is precisely p(Hm) where m belongs to Z^n and where p is a projection from R^n to n-dim torus T^n.

Recall that point is homoclinic to 0 if both positive and negative iterates of A converge to 0.

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sorry, it is missing that scalar product of v and w must be 1 – ivo May 21 '11 at 0:06
Feels like homework... – Nikita Sidorov May 21 '11 at 1:23
Is not Hm=cv for some real number c? Maybe I am making some terrible error. – Nishant Chandgotia May 21 '11 at 8:06
Nishant, this is true! This means that Hm will be in the unstable space of A in R^n. I would like to have that Hm plus some integer belongs to the stable space to. – ivo May 21 '11 at 11:06
Yes Nikita, it is a homework :) – ivo May 21 '11 at 11:07

See Example 3.3 of the paper ''Homoclinic points of algebraic Z^d-actions'' by Lind and Schmidt.

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