is a Salem number of degree d = 2s, then it has two real conjugates,
and 
, and s-1 pairs of complex conjugates
for 
. All the numbers 
lie in the set 
. As is familiar from the geometry of numbers
[2], p.96, we can think of 
as a lattice 
in 
defined by mapping 
onto the point 
,
where 
denotes the conjugate of 
corresponding to the conjugation
.
The determinant of 
[2], p.99 is
, since there are d-2 nonreal
conjugates, where 
denotes not the discriminant of the field 
, but rather the discriminant of the order 
i.e.,
the discriminant of the minimal polynomial P of 
. Thus, the number of points
of 
in a large cube of volume V is asymptotically 
.
Now, consider the iterates 
as points in the lattice
. By definition, 
. The conjugate 
of 
corresponding to 
is 
, which satisfies
. For a typical conjugate 
with 
, the corresponding conjugate of 
is 
. However, it is reasonable to expect that 
may be
for some 
. In fact, it is plausible that 
, as we
argue in section 6.4 below. The following result contains the only completely rigorous argument
of this section.
![[Annotate]](/organics/icons/sannotate.gif){kind=icon}
![[Shownotes]](../gif/annotate/sshow-101.gif)
