Almost every interval translation map of three intervals is finite type

Abstract

Interval translation maps (ITMs) are a non-invertible generalization of interval exchange transformations (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. In this paper, we prove the finiteness conjecture for the ITMs of three intervals. Namely, the subset of ITMs of finite type contains an open, dense, and full Lebesgue measure subset of the space of ITMs of three intervals. For this, we show that any ITM of three intervals can be reduced either to a rotation or to a double rotation.1991 Mathematics Subject Classification. Primary: 37C05, 37C20, 37C70, 37D20, 37D45.

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

“…In particular, they demonstrated that orientation-preserving ITMs without periodic points can have at most n ergodic invariant probability measures where n is the number of intervals. D. Volk [14] demonstrated that almost every (w.r.t. the Lebesgue measure on the parameter set) 3-ITM is conjugated to either a rotation or a double rotation and is finite.…”

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

“…In particular, they demonstrated that orientationpreserving ITMs without periodic points can have at most n ergodic invariant probability measures where n is the number of intervals. D. Volk [31] demonstrated that almost every (w.r.t. the Lebesgue measure on the parameter set) 3-ITM is conjugated to either a rotation or a double rotation and is finite.…”

Invariant measures of torus piecewise isometries

“…The most interesting among them in a historical order belongs to M. Boshernitzan & I. Kornfeld (1995) [10], J. Buzzi & P. Hubert (2004) [5], H. Bruin (2007Bruin ( ,2012 [3], S. Marmi, P. Moussa, J-C. Yoccoz (2012) [14], D. Volk (2014) [22]. A further generalization when a general cover is used instead of a special partition (which inevitably leads to a multi-valued map) was considered in A. Skripchenko & S. Troubetzkoy (2015) [18].…”

“…A number of attempts to follow this idea in the multidimensional setting have been tried (see, e.g. [22,19] and further references therein). Unfortunately, in distinction to the one-dimensional case, the multidimensional dynamics is much more complicated and cannot be reduced to some version of ITM, which has been demonstrated by A. Goetz in Figure 3: Dynamics of a triangle piecewise isometry (from [9]).…”