ANALYTIC TOOLS FOR THE STUDY OF FLOWS AND INVERSE PROBLEMS

In this survey, we review recent results in hyperbolic dynamical systems and in geometric inverse problems using analytic tools, based on spectral theory and microlocal methods.

[1]  J. Hilgert,et al.  Classical and quantum resonances for hyperbolic surfaces , 2016, 1605.08801.

[2]  Gerhard Keller,et al.  Ruelle?Perron?Frobenius spectrum for Anosov maps , 2002 .

[3]  S. Dyatlov Resonance projectors and asymptotics for r -normally hyperbolic trapped sets , 2013, 1301.5633.

[4]  J. Sjöstrand,et al.  Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances , 2008, 0802.1780.

[5]  C. Croke Scattering rigidity with trapped geodesics , 2011, Ergodic Theory and Dynamical Systems.

[6]  Mark F. Demers,et al.  Exponential decay of correlations for finite horizon Sinai billiard flows , 2015, 1506.02836.

[7]  J. Sjöstrand,et al.  Upper Bound on the Density of Ruelle Resonances for Anosov Flows , 2010, 1003.0513.

[8]  T. Weich,et al.  Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces , 2017, International Mathematics Research Notices.

[9]  M. Tsujii Quasi-compactness of transfer operators for contact Anosov flows , 2008, 0806.0732.

[10]  G. Uhlmann,et al.  The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points , 2014, 1402.5559.

[11]  Boundary and lens rigidity of finite quotients , 2005 .

[12]  L. Tzou,et al.  Boundary and lens rigidity for non-convex manifolds , 2017, American Journal of Mathematics.

[13]  Charles Hadfield Ruelle and Quantum Resonances for Open Hyperbolic Manifolds , 2017, 1708.01200.

[14]  Calvin C. Moore,et al.  Exponential Decay of Correlation Coefficients for Geodesic Flows , 1987 .

[15]  S. Dyatlov,et al.  Pollicott–Ruelle Resonances for Open Systems , 2014, 1410.5516.

[16]  F. Faure PREQUANTUM CHAOS: RESONANCES OF THE PREQUANTUM CAT MAP , 2006, nlin/0606063.

[17]  S. Dyatlov,et al.  Power spectrum of the geodesic flow on hyperbolic manifolds , 2014, Analysis & PDE.

[18]  Plamen Stefanov,et al.  Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds , 2007, math/0701595.

[19]  Plamen Stefanov,et al.  Boundary rigidity and stability for generic simple metrics , 2004, math/0408075.

[20]  V. Sharafutdinov,et al.  Integral geometry of tensor fields on a manifold of negative curvature , 1988 .

[21]  Fredholm determinants, Anosov maps and Ruelle resonances , 2005, math/0505049.

[22]  René Michel,et al.  Sur la rigidité imposée par la longueur des géodésiques , 1981 .

[23]  J. Zukas Introduction to the Modern Theory of Dynamical Systems , 1998 .

[24]  D. Burago,et al.  Boundary rigidity and filling volume minimality of metrics close to a flat one , 2010 .

[25]  C. Croke,et al.  Lens rigidity with trapped geodesics in two dimensions , 2011, 1108.4938.

[26]  Marina Ratner,et al.  The rate of mixing for geodesic and horocycle flows , 1987, Ergodic Theory and Dynamical Systems.

[27]  Carlangelo Liverani,et al.  Banach spaces adapted to Anosov systems , 2005, Ergodic Theory and Dynamical Systems.

[28]  BY Frédéricnaud EXPANDING MAPS ON CANTOR SETS AND ANALYTIC CONTINUATION OF ZETA FUNCTIONS , 2005 .

[29]  G. Uhlmann,et al.  The geodesic X-ray transform with fold caustics , 2010, 1004.1007.

[30]  Horocyclic invariance of Ruelle resonant states for contact Anosov flows in dimension 3 , 2017, 1705.07965.

[31]  M. Tsujii Exponential mixing for generic volume-preserving Anosov flows in dimension three , 2016, 1601.00063.

[32]  M. Pollicott,et al.  Anosov flows and dynamical zeta functions , 2012, 1203.0904.

[33]  C. Croke,et al.  Rigidity and the distance between boundary points , 1991 .

[34]  Jean-Pierre Otal Le spectre marqué des longueurs des surfaces à courbure négative , 1990 .

[35]  Liverani Carlangelo On Contact Anosov Flows , 2003 .

[36]  C. Croke,et al.  The marked length-spectrum of a surface of nonpositive curvature☆ , 1992 .

[37]  G. Uhlmann,et al.  Inverting the local geodesic X-ray transform on tensors , 2014, Journal d'Analyse Mathématique.

[38]  M. Salo,et al.  Spectral rigidity and invariant distributions on Anosov surfaces , 2012, 1208.4943.

[39]  N. V. Dang,et al.  Spectral analysis of morse-smale gradient flows. , 2016, 1605.05516.

[40]  M. Tsujii,et al.  Prequantum transfer operator for symplectic Anosov diffeomorphism , 2012, Astérisque.

[41]  M. Tsujii,et al.  Band structure of the Ruelle spectrum of contact Anosov flows , 2013, 1301.5525.

[42]  C. Guillarmou,et al.  Marked boundary rigidity for surfaces , 2016, Ergodic Theory and Dynamical Systems.

[43]  A. Katok Four applications of conformal equivalence to geometry and dynamics , 1988, Ergodic Theory and Dynamical Systems.

[44]  M. Zworski,et al.  Decay of correlations for normally hyperbolic trapping , 2013, 1302.4483.

[45]  C. Croke,et al.  Spectral rigidity of a compact negatively curved manifold The first author was partly supported , 1998 .

[46]  M. Tsujii Contact Anosov flows and the Fourier–Bros–Iagolnitzer transform , 2011, Ergodic Theory and Dynamical Systems.

[47]  L. Stoyanov Spectra of Ruelle transfer operators for Axiom A flows , 2008, 0810.1126.

[48]  Gunther Uhlmann,et al.  Tensor tomography on surfaces , 2011, 1109.0505.

[49]  V. Guillemin,et al.  Some inverse spectral results for negatively curved 2-manifolds , 1980 .

[50]  A. Katok,et al.  Differentiability, rigidity and Godbillon-Vey classes for Anosov flows , 1990 .

[51]  Gunther Uhlmann,et al.  The inverse problem for the local geodesic ray transform , 2012, 1210.2084.

[52]  V. Baladi,et al.  Anisotropic hölder and sobolev spaces for hyperbolic diffeomorphisms , 2005, math/0505015.

[53]  N. S. Dairbekov Integral geometry problem for nontrapping manifolds , 2006 .

[54]  G. Uhlmann,et al.  On the microlocal analysis of the geodesic X-ray transform with conjugate points , 2015, 1502.06545.

[55]  J. Bourgain,et al.  Spectral gaps without the pressure condition , 2016, 1612.09040.

[56]  L. Stoyanov Pinching conditions, linearization and regularity of axiom a flows , 2010, 1010.1594.

[57]  Hai E Zhang,et al.  Sensitivity analysis of an inverse problem for the wave equation with caustics , 2012, 1211.6220.

[58]  Carlangelo Liverani,et al.  Smooth Anosov flows: Correlation spectra and stability , 2007 .

[59]  R. Bowen Ergodic theory of Axiom A flows , 1975 .

[60]  J. Journé On a regularity problem occurring in connection with Anosov diffeomorphisms , 1986 .

[61]  M. Tsujii,et al.  The semiclassical zeta function for geodesic flows on negatively curved manifolds , 2013, 1311.4932.

[62]  C. Guillarmou Invariant distributions and X-ray transform for Anosov flows , 2014, 1408.4732.

[63]  G. Uhlmann,et al.  Two dimensional compact simple Riemannian manifolds are boundary distance rigid , 2003, math/0305280.

[64]  C. Guillarmou Lens rigidity for manifolds with hyperbolic trapped set , 2014, 1412.1760.

[65]  R. Llave,et al.  Canonical perturbation theory of Anosov systems, and regularity results for the Livsic cohomology equation , 1985 .

[66]  Jean-Pierre Otal,et al.  Sur les longueurs des géodésiques d'une métrique à courbure négative dans le disque , 1990 .

[67]  D. Dolgopyat On decay of correlations in Anosov flows , 1998 .

[68]  G. Uhlmann,et al.  Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge , 2017, Annals of Mathematics.

[69]  V. Romanov,et al.  On uniqueness of determination of a form of first degree by its integrals along geodesics , 1997 .

[70]  M. Vignéras Varietes Riemanniennes Isospectrales et non Isometriques , 1980 .