Eur. Phys. J. B 29, 339-343 (2002)
https://doi.org/10.1140/epjb/e2002-00313-8

Symmetry–breaking in local exponents

School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India

Corresponding author: ^a r.ramaswamy@mail.jnu.ac.in

Received: 25 October 2001
Revised: 8 December 2001
Published online: 2 October 2002

Abstract

Integrable dynamical systems, namely those having as many independent conserved quantities as freedoms, have all exponents equal to zero. Locally, the instantaneous or finite time exponents are nonzero, but owing to a symmetry, their global averages vanish. When the system becomes nonintegrable, this symmetry is broken. A parallel to this phenomenon occurs in mappings which derive from quasiperiodic Schrödinger problems in 1–dimension. For values of the energy such that the eigenstate is extended, the exponent is zero, while if the eigenstate is localized, the exponent becomes negative. This occurs by a breaking of the quasiperiodic symmetry of local exponents, and corresponds to a breaking of a symmetry of the wavefunction in extended and critical states.