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, 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 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 Collections: Mathematics