The Triple Lattice PETs. Issue 4 (2nd October 2019) - Yi, Ren
ABSTRACT: Polytope exchange transformations (PETs) are higher dimensional generalizations of interval exchange transformations (IETs) which have been well-studied for more than 40 years. A general method of constructing PETs based on multigraphs was described by R. Schwartz in 2013. In this paper, we describe a one-parameter family of multigraph PETs called the triple lattice PETs. We show that there exists a renormalization scheme for the triple lattice PETs on the parameter space (0, 1). We analyze the limit set Λφ with respect to the parameter ϕ = - 1 + 5 2 . By renormalization, we show that Λφ is the nested intersection of a countable sequence of finite unions of isosceles trapezoids. The Hausdorff dimension of Λφ satisfies the inequality 1 < dim H ( Λ ϕ ) = log ( 2 - 1 ) / log ( ϕ ) < 2 .