 
Summary: Pisot expansion in selfinducing systems
Shigeki Akiyama
1. Feb 2009: Number Theory and Ergodic Theory (Kanazawa)
Open problems for youngsters
A Pisot number is the real root > 1 of a polynomial xd
 cd1xd1
 · · ·  c0 with
ci Z, whose other roots are strictly within the unit circle. If c0 = ±1, then it is called
a Pisot unit. Let (X, B, µ, T) be a measure theoretical dynamical system. For any subset
Y B of positive measure, an induced system (Y, B , µ , T ) is canonically defined by the
first return map T (x) = Tm(x)
(x) Y . The induced system may behave quite differently
from the original. But in cases, there is an expansive affine map such that (Y, B , µ , T )
and (X, B, µ, T) are conjugate through . Then we say that it has a selfinducing structure.
An eigenvalue of the affine map (expansion constant) in a selfinducing system becomes a
Pisot number, moreover a Pisot unit, in many important examples.
1. Substitutive dynamical system. This is introduced as the simplest selfinducing
system. Let be a primitive substitution on the monoid {1, . . . , k}
with a fixed point
x {1, . . . , k}N
