Scaling dynamics of a cubic interval-exchange transformation

摘要

We study the dynamics of renormalization of a specific interval-exchange transformation which features exact scaling (the cubic Arnoux-Yoccoz model). Using a symbolic space that describes both dynamics and scaling, we characterize the periodic points of the scaling map in terms of generalized decimal expansions, where the base is the reciprocal of a Pisot number and the digits are algebraic integers. The set of periodic points has a rich arithmetic and geometric structure: we establish rigorously some facts, and use extensive numerical experimentation to formulate a conjecture.