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Pisot expansion in self-inducing systems Shigeki Akiyama

Summary: Pisot expansion in self-inducing 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
- cd-1xd-1
- - 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, , T) be a measure theoretical dynamical system. For any subset
Y B of positive measure, an induced system (Y, B , , 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 , , T )
and (X, B, , T) are conjugate through . Then we say that it has a self-inducing structure.
An eigenvalue of the affine map (expansion constant) in a self-inducing system becomes a
Pisot number, moreover a Pisot unit, in many important examples.
1. Substitutive dynamical system. This is introduced as the simplest self-inducing
system. Let be a primitive substitution on the monoid {1, . . . , k}
with a fixed point
x {1, . . . , k}N


Source: Akiyama, Shigeki - Department of Mathematics, Niigata University


Collections: Mathematics