# American Institute of Mathematical Sciences

• Previous Article
Construction of ergodic cocycles that are fundamental solutions to linear systems of a special form
• JMD Home
• This Issue
• Next Article
Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data
April  2007, 1(2): 255-285. doi: 10.3934/jmd.2007.1.255

## Prequantum chaos: Resonances of the prequantum cat map

 1 Institut Fourier 100, rue des Maths, BP 74, 38402 St Martin d'Heres, France

Received  June 2006 Revised  December 2006 Published  January 2007

Prequantum dynamics was introduced in the 70s by Kostant, Souriau and Kirillov as an intermediate between classical and quantum dynamics. In common with the classical dynamics, prequantum dynamics transports functions on phase space, but adds some phases which are important in quantum interference effects. In the case of hyperbolic dynamical systems, it is believed that the study of the prequantum dynamics will give a better understanding of the quantum interference effects for large time, and of their statistical properties. We consider a linear hyperbolic map $M$ in SL $(2,\mathbb{Z})$ which generates a chaotic dynamical system on the torus. The dynamics is lifted to a prequantum fiber bundle. This gives a unitary prequantum (partially hyperbolic) map. We calculate its resonances and show that they are related to the quantum eigenvalues. A remarkable consequence is that quantum dynamics emerges from long-term behavior of prequantum dynamics. We present trace formulas, and discuss perspectives of this approach in the nonlinear case.
Citation: Frédéric Faure. Prequantum chaos: Resonances of the prequantum cat map. Journal of Modern Dynamics, 2007, 1 (2) : 255-285. doi: 10.3934/jmd.2007.1.255
 [1] John Erik Fornæss. Infinite dimensional complex dynamics: Quasiconjugacies, localization and quantum chaos. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 51-60. doi: 10.3934/dcds.2000.6.51 [2] Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems & Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009 [3] Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492 [4] Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883 [5] Simon Scott. Relative zeta determinants and the geometry of the determinant line bundle. Electronic Research Announcements, 2001, 7: 8-16. [6] Michiko Yuri. Polynomial decay of correlations for intermittent sofic systems. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 445-464. doi: 10.3934/dcds.2008.22.445 [7] Vincent Lynch. Decay of correlations for non-Hölder observables. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 19-46. doi: 10.3934/dcds.2006.16.19 [8] Ioannis Konstantoulas. Effective decay of multiple correlations in semidirect product actions. Journal of Modern Dynamics, 2016, 10: 81-111. doi: 10.3934/jmd.2016.10.81 [9] Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453 [10] J. Alberto Conejero, Francisco Rodenas, Macarena Trujillo. Chaos for the Hyperbolic Bioheat Equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 653-668. doi: 10.3934/dcds.2015.35.653 [11] Yu. Dabaghian, R. V. Jensen, R. Blümel. Integrability in 1D quantum chaos. Conference Publications, 2003, 2003 (Special) : 206-212. doi: 10.3934/proc.2003.2003.206 [12] Harald Markum, Rainer Pullirsch. Classical and quantum chaos in fundamental field theories. Conference Publications, 2003, 2003 (Special) : 596-603. doi: 10.3934/proc.2003.2003.596 [13] Stefano Galatolo, Pietro Peterlongo. Long hitting time, slow decay of correlations and arithmetical properties. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 185-204. doi: 10.3934/dcds.2010.27.185 [14] C. Bonanno, G. Menconi. Computational information for the logistic map at the chaos threshold. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 415-431. doi: 10.3934/dcdsb.2002.2.415 [15] Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665 [16] Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287-313. doi: 10.3934/jmd.2008.2.287 [17] Priyanjana M. N. Dharmawardane. Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity. Conference Publications, 2013, 2013 (special) : 197-206. doi: 10.3934/proc.2013.2013.197 [18] Thinh Tien Nguyen. Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1651-1684. doi: 10.3934/dcds.2019073 [19] Karla Díaz-Ordaz. Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-like maps. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 159-176. doi: 10.3934/dcds.2006.15.159 [20] Jérôme Buzzi, Véronique Maume-Deschamps. Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 639-656. doi: 10.3934/dcds.2005.12.639

2018 Impact Factor: 0.295