The Triple Lattice PETs

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 of the triple lattice PETs in the interval (0, 1). We analyze the limit set Λ φ with respect to the parameter φ = −1+ √ 5 2. By renormalization, we show that Λ φ is the limit of embedded polygons in R 2 and its Hausdorff dimension satisfies the inequality 1 < dim H (Λ φ ) = log( √ 2 − 1)/ log(φ) < 2.

Renormalizing an infinite rational IET

“…There are numerous examples in the literature of such maps admitting an open dense set of periodic points but with interesting dynamics on the complementary sets. See for example [1], [7], [8], [9], [14], [16]. This sort of behavior is impossible for IETs formed by permuting finitely many intervals [11,Theorem 6.6].…”

Renormalizing an infinite rational IET

“…ere are numerous examples in the literature of such maps admi ing an open dense set of periodic points but with interesting dynamics on the complimentary sets. See for example [AH13], [Goe00], [Goe03], [Hoo13], [Sch14], [Yi18]. is sort of behaviour is impossible for IETs formed by permuting nitely many intervals [MT02,eorem 6.6].…”