Renormalization in a class of interval translation maps ofdbranches

Abstract

Bruin and Troubetzkoy's 2003 results are generalized to a class of interval translation maps with arbitrarily many pieces. It is shown that there is an uncountable set of parameters leading to type 1 interval translation maps (ITMs), but that the Lebesgue measure of these parameters is 0. Furthermore, conditions are given that imply that the ITMs have multiple ergodic invariant measures.

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]).…”

Almost every interval translation map of three intervals is finite type

“…Until now, the question was answered (positively) only for some specific families of ITMs: by Bruin and Troubetzkoy [4] for a 2-parameter family in ITM(3), by Bruin [2] for a d-parameter family in ITM(d + 1), and by Suzuki, Ito, Aihara [11] and Bruin, Clack [3] for so-called double rotations, see Subsection 1.3. Schmeling and Troubetzkoy [10] established a number of related topological results, in particular, that the interval translation mappings of infinite type form a G δ subset of the set of all interval translation mappings, whereas the interval translation mappings of finite type contain an open subset.…”

Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

“…H. Bruin and S. Troubetzkoy [7,8] studied ITMs with three segments and demonstrated that in this case, a typical 3-ITM is finite. In the general case, they estimated the Hausdorff dimension of attractors and obtained sufficient conditions for the existence of a unique ergodic invariant measure.…”