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DTSTART:20260329T030000
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DTSTAMP:20260614T101032Z
UID:1781703000@ist.ac.at
DTSTART:20260617T153000
DTEND:20260617T163000
DESCRIPTION:Speaker: Yunzhe Li\nhosted by Maksym Serbyn\nAbstract: We study
  spectral rigidity and nonrigidity phenomena in dynamical systems. The cen
 tral question is whether a dynamical system can be determined\, up to a na
 tural conjugacy\, by its spectrum.The first part of the talk focuses on st
 andard maps from the viewpoint of action spectra. We construct nontrivial 
 deformations of the standard map that preserve the symplectic actions of i
 nfinitely many periodic orbits accumulating onto an invariant curve. This 
 result can be viewed as a symplectic twist-map analogue of length-spectral
  nonrigidity phenomena for Riemannian manifolds and convex billiards\, mot
 ivating the problem of constructing analogous “partially length-isospect
 ral” deformations of strictly convex billiard tables. The proof combines
  a resonant normal form construction with Picard iteration schemes to prod
 uce a sequence of periodic orbits accumulating on an invariant curve with 
 a Liouville rotation number. The second part of the talk briefly explores
  rigidity questions for Liouville metrics on the two-dimensional torus. A 
 long-standing folklore conjecture asserts that Liouville metrics are the o
 nly integrable metrics on the torus. We give a length-spectral rigidity re
 sult for the class of trigonometric conformal deformations of Liouville me
 trics by exploiting the dynamical properties of rational tori\, which are 
 analogues of resonant convex caustics in billiards. We also establish a co
 mplementary classification result showing that marked-length-isospectral L
 iouville metrics are characterized by rearrangements of the one-dimensiona
 l functions appearing in their conformal factors\, generalizing a theorem 
 of Abbondandolo and Mazzucchelli. In particular\, this result yields many 
 nonrigidity examples within the class of Liouville metrics.
LOCATION:Office Bldg West / Ground floor / Heinzel Seminar Room (I21.EG.101
 ) and Zoom\, ISTA
ORGANIZER:
SUMMARY:Yunzhe Li: Thesis Defense: Spectral Rigidity and Nonrigidity of Dyn
 amical Systems
URL:https://talks-calendar.ista.ac.at/events/6506
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