A note on double rotations of infinite type

Abstract

We introduce a new renormalization procedure on double rotations, which is reminiscent of the classical Rauzy induction. Using this renormalization we prove that the set of parameters which induce infinite type double rotations has Hausdorff dimension strictly smaller than 3. Moreover, we construct a natural invariant measure supported on these parameters and show that, with respect to this measure, almost all double rotations are uniquely ergodic.MSC Classification: 37E05. Bibliography: 17 items.

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

“…In the latter case, as initially reported in [3] (for more details we refer to [4]), between two parameter intervals associated with K 1 -band and K 2 -band chaotic attractors there is a parameter interval associated with (K 1 + K 2 − 1)-band chaotic attractors. The same regularity can be observed in system (2). For example, in Fig.…”

“…3(b) one can see parameter intervals where the sets M (S) consist of 3 and 4 bands (connected components), and a parameter interval corresponding to sets M (S) consisting of 6 bands in-between, and so on. However, there is a major difference to the case described in the literature, as the bandcount adding structure has been so far reported for chaotic attractors only, while the sets M (S) in system (2) are not chaotic. To the best of our knowledge, a bandcount adding structure formed by non-chaotic attractors has never been reported before.…”

“…Furthermore, some conditions ensuring the existence of multiple ergodic invariant measures were provided. H. Bruin and G. Clark [8] studied the so-called double rotations (2-CTMs), see also [2], [28] and [33]. Almost all double rotations are finite.…”

“…It is easy to see that (x n , s n ) ∈ L, n ∈ N 0 . Thus, the strip L is the phase space for system (2). For the sake of convenience, we write system (2) as follows:…”

Non-Sturmian sequences of matrices providing the maximum growth rate of matrix products

“…4b, in Case 1 the angular function Φ(ϕ), considered as a mapping of the circle into itself, does not preserve orientation. In particular, this situation resembles the behavior of the so-called double rotations [4,19,46], defined by the equation…”