Eigenvalues for toral Anosov automorphisms - MathOverflow

Eigenvalues for toral Anosov automorphisms

2012-03-18T15:58:56Z

It is well known that on every $d$-dimensional torus there exists linear Anosov automorphisms.

My question is the following:

Given $k< d$ does there exists a linear Anosov automorphism of $\mathbb{T}^d$ with exactly $k$ eigenvalues smaller than $1$? If true (which I expect), does there exists an \emph{irreducible} linear Anosov automorphism of $\mathbb{T}^d$ with exactly $k$ eigenvalues smaller than $1$?

This can be phrased in terms of matrices with integer coeficients (please add the corresponding relevant tags) as:

Given $k< d$ does there exists a matrix in $SL(d,\mathbb{Z})$ such that all eigenvalues have modulus different from $1$ and $k$ of them are of modulus smaller than $1$? What about if the characteristic polynomial is irreducible over $\mathbb{Q}$?.

Some relevant related information can be found in this paper (http://arxiv.org/pdf/1009.2994v2.pdf) where some results of W. Duke, Z. Rudnick, P. Sarnak as well as of Nevo and Sarnak are refered to.

Answer by Denis Serre for Eigenvalues for toral Anosov automorphisms

2012-03-18T17:12:28Z

This is only a partial answer, which I shall delete if I find a better one. Every pair $(k,d)$ of the form $$d=\frac12\phi(n),\qquad k={\rm card}(\frac{n}{6}\le j \le\frac{n}{2},j\wedge n=1)$$ is OK: take the cyclotomic polynomial $\Phi_n$ and form the irreducible polynomial $P_n\in{\mathbb Z}[X]$ defined by $$\Phi_n(t)=t^{\frac{n}{2}}P_n\left(t+\frac1t\right).$$ The roots of $P_n$ are the numbers $2\cos\frac{2j\pi}{n}$ with $j\wedge n=1$, smaller than $1$ if and only if $\frac{n}{6}\le j \le\frac{n}{2}$.

If instead $k=d-1$, take any Pisot number. Edit (after Nikita's comment below): One may take the companion matrix of $X^d-X^{d-1}-\cdots-X-1$. Its only root of modulus greter than $1$ is a Pisot number, also called a multinacci number. If $d=2$, this is just the golden ratio, at the basis of the Fibonacci sequence, hence the `word' multinacci.

Answer by Nikita Sidorov for Eigenvalues for toral Anosov automorphisms

2012-03-18T17:25:06Z

Given $k < d$, one can always construct a monic polynomial irreducible over $\mathbb Q$ with exactly $k$ roots less than 1 in modulus and $d-k$ roots greater than 1 in modulus. This follows from the general construction of algebraic units, namely, each group of units of an algebraic field contains a unit with a given $k$ -- see, e.g., [Borevich and Shafarevich].

Then you can simply take the companion matrix of such a polynomial.

Or do you need an explicit construction?