Stable resonances and signal propagation in a chaotic network of coupled units

We apply the linear response theory developed in \cite{Ruelle} to analyze how a periodic signal of weak amplitude, superimposed upon a chaotic background, is transmitted in a network of non linearly interacting units. We numerically compute the complex susceptibility and show the existence of specific poles (stable resonances) corresponding to the response to perturbations transverse to the attractor. Contrary to the poles of correlation functions they depend on the pair emitting/receiving units. This dynamic differentiation, induced by non linearities, exhibits the different ability that units have to transmit a signal in this network.Comment: 10 pages, 3 figures, to appear in Phys. rev.

Characterizing dynamics with covariant Lyapunov vectors

A general method to determine covariant Lyapunov vectors in both discrete- and continuous-time dynamical systems is introduced. This allows to address fundamental questions such as the degree of hyperbolicity, which can be quantified in terms of the transversality of these intrinsic vectors. For spatially extended systems, the covariant Lyapunov vectors have localization properties and spatial Fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the Lyapunov exponents.Comment: 4 pages, 3 figures, submitted to Physical Review letter

On the susceptibility function of piecewise expanding interval maps

We study the susceptibility function Psi(z) associated to the perturbation f_t=f+tX of a piecewise expanding interval map f. The analysis is based on a spectral description of transfer operators. It gives in particular sufficient conditions which guarantee that Psi(z) is holomorphic in a disc of larger than one. Although Psi(1) is the formal derivative of the SRB measure of f_t with respect to t, we present examples satisfying our conditions so that the SRB measure is not Lipschitz.*We propose a new version of Ruelle's conjectures.* In v2, we corrected a few minor mistakes and added Conjectures A-B and Remark 4.5. In v3, we corrected the perturbation (X(f(x)) instead of X(x)), in particular in the examples from Section 6. As a consequence, Psi(z) has a pole at z=1 for these examples.Comment: To appear Comm. Math. Phy

Linear response formula for piecewise expanding unimodal maps

The average R(t) of a smooth function with respect to the SRB measure of a smooth one-parameter family f_t of piecewise expanding interval maps is not always Lipschitz. We prove that if f_t is tangent to the topological class of f_0, then R(t) is differentiable at zero, and the derivative coincides with the resummation previously proposed by the first named author of the (a priori divergent) series given by Ruelle's conjecture.Comment: We added Theorem 7.1 which shows that the horizontality condition is necessary. The paper "Smooth deformations..." containing Thm 2.8 is now available on the arxiv; see also Corrigendum arXiv:1205.5468 (to appear Nonlinearity 2012

Space and time from translation symmetry

We show that the notions of space and time in algebraic quantum field theory arise from translation symmetry if we assume asymptotic commutativity. We argue that this construction can be applied to string theory.Comment: 10 pages, Essential changes and additions, in particular, in the discussion of string field theor

Extending the definition of entropy to nonequilibrium steady states

We study the nonequilibrium statistical mechanics of a finite classical system subjected to nongradient forces $\xi$ and maintained at fixed kinetic energy (Hoover-Evans isokinetic thermostat). We assume that the microscopic dynamics is sufficiently chaotic (Gallavotti-Cohen chaotic hypothesis) and that there is a natural nonequilibrium steady state $\rho_\xi$. When $\xi$ is replaced by $\xi+\delta\xi$ one can compute the change $\delta\rho$ of $\rho_\xi$ (linear response) and define an entropy change $\delta S$ based on energy considerations. When $\xi$ is varied around a loop, the total change of $S$ need not vanish: outside of equilibrium the entropy has curvature. But at equilibrium (i.e. if $\xi$ is a gradient) we show that the curvature is zero, and that the entropy $S(\xi+\delta\xi)$ near equilibrium is well defined to second order in $\delta\xi$.Comment: plain TeX, 10 pagesemacs ded

Note on nonequilibrium stationary states and entropy

In transformations between nonequilibrium stationary states, entropy might be a not well defined concept. It might be analogous to the ``heat content'' in transformations in equilibrium which is not well defined either, if they are not isochoric ({\it i.e.} do involve mechanical work). Hence we conjecture that un a nonequilbrium stationary state the entropy is just a quantity that can be transferred or created, like heat in equilibrium, but has no physical meaning as ``entropy content'' as a property of the system.Comment: 4 page

Bowen Measure From Heteroclinic Points

We present a new construction of the entropy-maximizing, invariant probability measure on a Smale space (the Bowen measure). Our construction is based on points that are unstably equivalent to one given point, and stably equivalent to another: heteroclinic points. The spirit of the construction is similar to Bowen's construction from periodic points, though the techniques are very different. We also prove results about the growth rate of certain sets of heteroclinic points, and about the stable and unstable components of the Bowen measure. The approach we take is to prove results through direct computation for the case of a Shift of Finite type, and then use resolving factor maps to extend the results to more general Smale spaces

Automorphisms of the affine SU(3) fusion rules

We classify the automorphisms of the (chiral) level-k affine SU(3) fusion rules, for any value of k, by looking for all permutations that commute with the modular matrices S and T. This can be done by using the arithmetic of the cyclotomic extensions where the problem is naturally posed. When k is divisible by 3, the automorphism group (Z_2) is generated by the charge conjugation C. If k is not divisible by 3, the automorphism group (Z_2 x Z_2) is generated by C and the Altsch\"uler--Lacki--Zaugg automorphism. Although the combinatorial analysis can become more involved, the techniques used here for SU(3) can be applied to other algebras.Comment: 21 pages, plain TeX, DIAS-STP-92-4