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Towards a Dictionary for the Bargmann Transform
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
Finally in this section we mention that for any real θ the operator Uθ defined by Uθf(z) = f(eiθz) is clearly a unitary operator on F2. For θ = ±π/2 the resulting operators are unitarily equivalent to the Fourier and inverse Fourier transforms on L2(ℝ). When θ is a rational multiple of π, the structure of Uθ is relatively simple. However, if θ is an irrational multiple of π, the structure of such an “irrational rotation operator” Uθ is highly non-trivial.
Breakup of transport barriers in plasmas with flow described by symplectic maps
Published in Radiation Effects and Defects in Solids, 2022
Carolina A. Tafoya, Julio J. Martinell
Depending on the nature of the map the transport barriers have different properties. For non-degenerate maps (when the rotation number has a non-vanishing derivative), also known as twist maps, the last KAM surface to break up is the one with the most stable irrational rotation number (usually the golden mean). For non-twist maps which violate the non-degeneracy condition the most robust torus is the shearless surface. When there are no background plasma flows, transport barriers are not a characteristic feature of the map, although in (6) they were characterized for the kicked Harper model. However, the presence of flows produces clear transport barriers since the flow lines give rise to KAM torii that are resilient to chaos.
Non-rigid rank-one infinite measures on the circle
Published in Dynamical Systems, 2023
Hindy Drillick, Alonso Espinosa-Dominguez, Jennifer N. Jones-Baro, James Leng, Yelena Mandelshtam, Cesar E. Silva
It is not the case that if is any infinite ergodic measure preserving system with an irrational eigenvalue α and eigenfunction g, that is mutualy singular with respect to Lebesgue measure m. An example of such a system is the product of any infinite measure preserving weakly mixing system and the irrational rotation by α. This system is ergodic and has a factor to the irrational rotation by α. The pushforward measure of that factor is a measure equivalent to the Lebesgue measure that takes the value of infinity for any positive Lebesgue measure sets and 0 for all Lebesgue measure zero sets.