Perfect retroreflectors and billiard dynamics
Pavel Bachurin Konstantin Khanin Jens Marklof Alexander Plakhov
Journal of Modern Dynamics 2011, 5(1): 33-48 doi: 10.3934/jmd.2011.5.33
We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability the exit velocity is exactly opposite to the entrance velocity, and the particle's exit point is arbitrarily close to its initial position. The method is based on asymptotic analysis of statistics of entrance times to a small interval for irrational circle rotations. The rescaled entrance times have a limiting distribution in the limit when the length of the interval vanishes. The proof of the main results follows from the study of related limiting distributions and their regularity properties.
keywords: billiards dynamical renormalization retroreflectors. homogeneous flow Recurrence circle rotation
Notes on a theorem of Katznelson and Ornstein
Habibulla Akhadkulov Akhtam Dzhalilov Konstantin Khanin
Discrete & Continuous Dynamical Systems - A 2017, 37(9): 4587-4609 doi: 10.3934/dcds.2017197

Let $\log f'$ be an absolutely continuous and $f"/f'∈ L_{p}(S^{1}, d\ell)$ for some $p>1, $ where $\ell$ is Lebesgue measure. We show that there exists a subset of irrational numbers of unbounded type, such that for any element $\widehat{ρ}$ of this subset, the linear rotation $R_{\widehat{ρ}}$ and the shift $f_{t}=f+t\mod 1, $ $t∈ [0, 1)$ with rotation number $\widehat{ρ}, $ are absolutely continuously conjugate. We also introduce a certain Zygmund-type condition depending on a parameter $γ$, and prove that in the case $γ>\frac{1}{2}$ there exists a subset of irrational numbers of unbounded type, such that every circle diffeomorphism satisfying the corresponding Zygmund condition is absolutely continuously conjugate to the linear rotation provided its rotation number belongs to the above set. Moreover, in the case of $γ> 1, $ we show that the conjugacy is $C^{1}$-smooth.

keywords: Circle diffeomorphisms rotation number Denjoy's inequality conjugating map

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