Fractal diffusion coefficient from dynamical zeta functions

Dynamical zeta functions provide a powerful method to analyze low dimensional dynamical systems when the underlying symbolic dynamics is under control. On the other hand even simple one dimensional maps can show an intricate structure of the grammar rules that may lead to a non smooth dependence of global observable on parameters changes. A paradigmatic example is the fractal diffusion coefficient arising in a simple piecewise linear one dimensional map of the real line. Using the Baladi-Ruelle generalization of the Milnor-Thurnston kneading determinant we provide the exact dynamical zeta function for such a map and compute the diffusion coefficient from its smallest zero.Comment: 8 pages, 2 figure

Soil Behavior Under Dynamic Loadings

Stress-controlled cyclic triaxial tests were performed on clay samples obtained from the subgrades of existing highway pavements in the State of Michigan. These samples were unconsolidated or isotropically consolidated and then tested under the following conditions: 1) confining pressures of 34.5 psi and 172.4 kPa (5 and 25 psi), 2) a range of stress ratio from 2 to 4, 3) a constant frequency of one cycle per second, and 4) at the natural water contents. Test equipment included an MTS electrohydraulic closed-1oop system which applies the sustained and cyclic loads to the sample in a stress-controlled mode, a minicomputer which controls the actuator of the MTS system to yield a loading function similar to that applied, by tandem axle truck, to the pavement section, and measuring devices including two vertical and two radial LVDT(s), a load cell, a pore water pressure transducer, recording devices and an ascilloscope. The test results indicate that, for all samples, the plastic strain rate (in the logarithmic mode) assumes two values per test. The first of these values is constant from cycle number one to cycle number 100 while the second value is also constant and it controls the plastic strain beyond cycle number 100. This abrupt change in the strain rate values was observed in the unconsolidated and consolidated samples. Also, it was found that the ultimate stress ratio that it can be applied on the sample during cyclic loading is a function of the confining pressures and other sample parameters

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

The Effects of Time Dependent Stress-Path on The Plastic and Elastic Deformation of Sand and Clay Soils Subjected to Dynamic Loading

Cylindrical cyclic loading tests were used to study the effects of time dependent stress path on the plastic and elastic deformations of sand and clay soil samples. The test materials were obtained from the subgrade of existing highway pavements located throughout the state of Michigan. Approximately, 25 cylindrical soil samples (13.5 cm long, 5.5 cm in diameter) were tested under repeated loading conditions using different hydrostatic confining pressures and several time dependent stress-paths. In all tests, the first invariant of the stress tensor was cycled between two constant values and the first invariant of the stress deviator tensor was increased incremently and cycled such that its minimum value was kept greater than 6.895 kPa. at all times. Analysis of the test results indicated several findings, these include: 1) Increasing the first invariant of the stress deviator tensor with time,for sand samples, decreased the average rate and magnitude of the plastic deformation of the samples. 2) For clay samples, the rate and magnitude of the plastic and elastic deformations were found to be dependent on the first invariants of the stress tensor and stress deviator tensor, on the stresspath and on the sample parameters. For example, the elastic strains of two duplicate soil samples tested under the same stress conditions (same invariants of stress tensor and stress deviator tensor) were different by a factor of 2 to 100. The value of this factor was found to be a function of the time dependent stress path

On the spectrum of Farey and Gauss maps

In this paper we introduce Hilbert spaces of holomorphic functions given by generalized Borel and Laplace transforms which are left invariant by the transfer operators of the Farey map and its induced version, the Gauss map, respectively. By means of a suitable operator-valued power series we are able to study simultaneously the spectrum of both these operators along with the analytic properties of the associated dynamical zeta functions.Comment: 23 page

Ruelle-Perron-Frobenius spectrum for Anosov maps

We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension $d=2$ we show that the transfer operator associated to smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows to obtain easily very strong spectral stability results, which in turn imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite dimensional problem.Comment: 58 pages, LaTe

Stochastic stability versus localization in chaotic dynamical systems

We prove stochastic stability of chaotic maps for a general class of Markov random perturbations (including singular ones) satisfying some kind of mixing conditions. One of the consequences of this statement is the proof of Ulam's conjecture about the approximation of the dynamics of a chaotic system by a finite state Markov chain. Conditions under which the localization phenomenon (i.e. stabilization of singular invariant measures) takes place are also considered. Our main tools are the so called bounded variation approach combined with the ergodic theorem of Ionescu-Tulcea and Marinescu, and a random walk argument that we apply to prove the absence of ``traps'' under the action of random perturbations.Comment: 27 pages, LaTe

Resonances of the cusp family

We study a family of chaotic maps with limit cases the tent map and the cusp map (the cusp family). We discuss the spectral properties of the corresponding Frobenius--Perron operator in different function spaces including spaces of analytic functions. A numerical study of the eigenvalues and eigenfunctions is performed.Comment: 14 pages, 3 figures. Submitted to J.Phys.

Resonances of the Frobenius-Perron Operator for a Hamiltonian Map with a Mixed Phase Space

Resonances of the (Frobenius-Perron) evolution operator P for phase-space densities have recently attracted considerable attention, in the context of interrelations between classical and quantum dynamics. We determine these resonances as well as eigenvalues of P for Hamiltonian systems with a mixed phase space, by truncating P to finite size in a Hilbert space of phase-space functions and then diagonalizing. The corresponding eigenfunctions are localized on unstable manifolds of hyperbolic periodic orbits for resonances and on islands of regular motion for eigenvalues. Using information drawn from the eigenfunctions we reproduce the resonances found by diagonalization through a variant of the cycle expansion of periodic-orbit theory and as rates of correlation decay.Comment: 18 pages, 7 figure