The gauss map on a class of interval translation mappings

Abstract

Abstract. We study the dynamics of a class of interval translation map on three intervals. We show that in this class the typical ITM is of finite type (reduce to an interval exchange transformation) and that the complement contains a Cantor set. We relate our maps to substitution subshifts. Results on Hausdorff dimension of the attractor and on unique ergodicity are obtained.

Polygonal Billiards with One Sided Scattering

“…The term double rotations was introduced in [12], they have also been studied in [3], [4] and [5]. We show that up to a natural involution there exists a bijection between double rotations and billiard map we work with, and therefore almost all of our billiard map are of finite type due to the corresponding results for double rotations (Theorem 2.2 part 2) (see [12] and [5]).…”

Renormalization in a class of interval translation maps ofdbranches

“…Similar to [2], we prove the following theorem. Theorem 1.1: Let A d be the set of parameters such that T is of type 1.…”

“…Boshernitzan and Kornfeld showed, using a renormalization operator, that a specific ITM has an attracting Cantor set. Bruin and Troubetzkoy [2] extended this result to a 2-parameter family of ITMs with three pieces (or two pieces when considered on the circle), and showed that type 1 maps occur for an uncountable set of Lebesgue measure 0 in parameter space. In [3], it is shown that type 1 occurs with Lebesgue measure 0 in the full 3-parameter family of 2-piece ITMs on the circle.…”

“…In [3], it is shown that type 1 occurs with Lebesgue measure 0 in the full 3-parameter family of 2-piece ITMs on the circle. In addition, [2] gives estimates of the Hausdorff dimension of , and it gives conditions under which Tj is uniquely ergodic, or is not uniquely ergodic.…”

Одномерные Разбиения Фибоначчи И Индуцированные Двухцветные Повороты Окружности

“…История вопроса. Первоначально двухцветные сдвиги S (α,β,ε) (x) по-явились в работах [3], [4] и затем в общем виде (0.1) -в [2]. Двухцветные сдви-ги представляют собою частный случай IT-отображений (interval translation mappings [3]).…”