Recall that Pisot numbers are algebraic integers greater than $1$, whose other Galois conjugates have modulus $<1$. The set of Pisot numbers is usually denoted $S$. It is known that $S$ is denumerable and closed (Salem). Alors, the sequence of derived sets $S',S'', ...$ does not terminate. The smallest accumulation point of $S$ is the golden ratio $\phi$.



> 
> Let $\lambda$ be an accumulation point in $S$. Therefore it is the limit of a sequence of Pisot numbers $\mu\_n\ne\lambda$. It is clear that the degree of $\mu\_n$ tends to $+\infty$, whereas its norm remains bounded as $n\rightarrow+\infty$. My question is about the Galois conjugates of $\mu\_n$; most of them must be close to the unit circle, because their product equals $N(\mu\_n)/\mu\_n$. Given a Galois conjugate $\tau$ of $\lambda$, does there exist a Galois conjugate $\tau\_n$ of $\mu\_n$ such that $\tau\_n\rightarrow\tau$ ? What is the statistics of the Galois conjugates of $\mu\_n$ as $n\rightarrow+\infty$ ?
> 
> 
> 


**Example**: the *multinacci* number of degree $d$ is the root $a\_d>1$ of the polynomial $X^d-X^{d-1}-\cdots-X-1$. When $d\rightarrow+\infty$, one has $a\_d\rightarrow2$. Then I have a proof that the empirical measure
$$m\_d:=\frac1d\sum\_{a\sim a\_d}\delta\_a$$
where the sum runs over all the Galois conjugates of $a\_d$ (including itself) converges vaguely towards the uniform measure over the unit circle.


**Edit**. The same property holds true in the situation depicted by Dufresnoy and Pisot. Let $P\in{\mathbb Z}[X]$ be the minimal polynomial of $\lambda$, unitary. Let $A\in{\mathbb Z}[X]$ be such that $|A(z)|\le|P(z)|$ on $\mathbb T$, the equality arising only at finitely many points. Define the polynomials $P\_n^\pm(X)=X^nP(X)\mp A(X)$. For infinitely many pairs $(n,\pm)$, $P\_n^\pm$ has only one root $\lambda\_n^\pm$ away from the unit disk, and none over $\mathbb T$. This Pisot number tends to $\lambda$, and the empirical measure defined as above tends to $\frac1{2\pi}d\theta$. It is thus tempting to conjecture that in every situation, the empirical measure tends to the uniform measure.



accumulation points within Pisot numbers 
The answer to the first question is "certainly not". Consider the polynomials $P\_d(X)=X^d-4X^{d-1}-X^{d-2}-1$. They have $d-1$ roots in the unit disk by Rouche, so their positive real roots are Pisot numbers. Also, the positive real roots of $P\_d$ tend to the larger root of $X^2-4X-1$, which is a Pisot number. However, any disk or radius $r<1$ is eventually free of roots of $P\_d$, so nothing tends to the conjugate. 


