scholarly journals Marked Length Spectrum, Homoclinic Orbits and the Geometry of Open Dispersing Billiards

2019 ◽  
Vol 374 (3) ◽  
pp. 1531-1575
Author(s):  
Péter Bálint ◽  
Jacopo De Simoi ◽  
Vadim Kaloshin ◽  
Martin Leguil
Keyword(s):  
Homoclinic Orbits ◽  
Length Spectrum ◽  
Marked Length Spectrum ◽  
Dispersing Billiards

scholarly journals Marked length rigidity for Fuchsian buildings

2018 ◽  
Vol 39 (12) ◽  
pp. 3262-3291
Author(s):  
DAVID CONSTANTINE ◽  
JEAN-FRANÇOIS LAFONT
Keyword(s):  
Automorphism Group ◽  
Riemannian Metrics ◽  
Length Spectrum ◽  
Negatively Curved ◽  
Spectrum Function ◽  
Marked Length Spectrum

We consider finite $2$-complexes $X$ that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT($-1$) metrics on $X$, which are piecewise hyperbolic and satisfy a natural non-singularity condition at vertices, are marked length spectrum rigid within certain classes of negatively curved, piecewise Riemannian metrics on $X$. As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of $X$.

Rigidity for non-compact surfaces of finite area and certain Kähler manifolds

1995 ◽  
Vol 15 (3) ◽  
pp. 475-516 ◽  
Author(s):  
Jianguo Cao
Keyword(s):  
Length Spectrum ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Finite Area ◽  
Marked Length Spectrum ◽  
Compact Surfaces

AbstractWe first consider the rigidity of the marked length spectrum for non-compact surfaces of finite area.

scholarly journals The marked length spectrum of Anosov manifolds

2019 ◽  
Vol 190 (1) ◽  
pp. 321 ◽  
Author(s):  
Guillarmou ◽  
Lefeuvre
Keyword(s):  
Length Spectrum ◽  
Marked Length Spectrum

Ergodic theory and rigidity on the symmetric space of non-compact type

2001 ◽  
Vol 21 (1) ◽  
pp. 93-114 ◽  
Author(s):  
INKANG KIM
Keyword(s):  
Ergodic Theory ◽  
Symmetric Spaces ◽  
Isometry Group ◽  
Length Spectrum ◽  
Compact Type ◽  
Negatively Curved ◽  
Rank One ◽  
Convex Cocompact ◽  
Marked Length Spectrum ◽  
Locally Symmetric Manifolds

In this paper we investigate the rigidity of symmetric spaces of non-compact type using ergodic theory such as Patterson–Sullivan measure and the marked length spectrum along with the cross ratio on the limit set. In particular, we prove that the marked length spectrum determines the Zariski dense subgroup up to conjugacy in the isometry group of the product of rank-one symmetric spaces. As an application, we show that two convex cocompact, negatively curved, locally symmetric manifolds are isometric if the Thurston distance is zero and the critical exponents of the Poincaré series are the same, and the same is true if the geodesic stretch is equal to one.

scholarly journals On the marked length spectrum of generic strictly convex billiard tables

2018 ◽  
Vol 167 (1) ◽  
pp. 175-209 ◽  
Author(s):  
Guan Huang ◽  
Vadim Kaloshin ◽  
Alfonso Sorrentino
Keyword(s):  
Length Spectrum ◽  
Strictly Convex ◽  
Marked Length Spectrum
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