arXiv subjects
Publications and source records attributed to Alexey Teplinsky.
We show the equivalence of two possible definitions of a rotational interval exchange transformation: by the first one, it is a first return map for a circle rotation onto a union of finite number of circle arcs, and by the second one, it is an interval exchange with a scheme (in the sense of interval rearrangement ensembles), whose dual is an interval exchange scheme as well.
We introduce a new concept of interval rearrangement ensembles (IRE), which is a generalization of interval exchange transformations (IET). This construction expands the space of IETs in accordance with the natural duality that we pinpoint. Induction of Rauzy\,--\,Veech kind is applicable to IREs. It is conjugate to the reverse operation by the duality mentioned above. A natural extension of an IRE is associated with two transversal flows on a flat translation surface with branching points.
In this paper we answer positively a question of whether it is possible for a circle diffeomorphism with breaks to be smoothly conjugate to a rigid rotation in the case when its breaks are lying on pairwise distinct trajectories. An example constructed is a piecewise linear circle homeomorphism that has four break points lying on distinct trajectories, and whose invariant measure is absolutely continuous w.r.t.\ the Lebesgue measure. The irrational rotation number for our example can be chosen a Roth number, but not of bounded type.
We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs. Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals.
We prove asymptotic estimates for the cross-ratio distortion with respect to a smooth or holomorphic function in terms of its Schwarz derivative.
We prove that a $C^{3+β}$-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class $D_δ$, $0<β<δ<1$, is $C^{2+β-δ}$-smoothly conjugate to a rigid rotation.
We prove that a $C^{2+α}$-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class $D_δ$, $0<δ<α\le1$, is $C^{1+α-δ}$-smoothly conjugate to a rigid rotation. We also derive the most precise version of Denjoy's inequality for such diffeomorphisms.