Zeta functions and Dynamical Systems

In this brief note we present a very simple strategy to investigate dynamical determinants for uniformly hyperbolic systems. The construction builds on the recent introduction of suitable functional spaces which allow to transform simple heuristic arguments in rigorous ones. Although the results so obtained are not exactly optimal the straightforwardness of the argument makes it noticeable.Comment: 7 pages, no figuer

Lack of superstable trajectories in linear viscoelasticity: a numerical approach

Given a positive operator $A$ on some Hilbert space, and a nonnegative decreasing summable function $\mu$, we consider the abstract equation with memory $$\ddot u(t)+ A u(t)- \int_0^t \mu(s)Au(t-s) ds=0$$ modeling the dynamics of linearly viscoelastic solids. The purpose of this work is to provide numerical evidence of the fact that the energy \E(t)=\Big(1-\int_0^t\mu(s)ds\Big)\|u(t)\|^2_1+\|\dot u(t)\|^2 +\int_0^t\mu(s)\|u(t)-u(t-s)\|^2_1ds, of any nontrivial solution cannot decay faster than exponential, no matter how fast might be the decay of the memory kernel $\mu$. This will be accomplished by simulating the integro-differential equation for different choices of the memory kernel $\mu$ and of the initial data

Instability statistics and mixing rates

We claim that looking at probability distributions of \emph{finite time} largest Lyapunov exponents, and more precisely studying their large deviation properties, yields an extremely powerful technique to get quantitative estimates of polynomial decay rates of time correlations and Poincar\'e recurrences in the -quite delicate- case of dynamical systems with weak chaotic properties.Comment: 5 pages, 5 figure

Chains of infinite order, chains with memory of variable length, and maps of the interval

We show how to construct a topological Markov map of the interval whose invariant probability measure is the stationary law of a given stochastic chain of infinite order. In particular we caracterize the maps corresponding to stochastic chains with memory of variable length. The problem treated here is the converse of the classical construction of the Gibbs formalism for Markov expanding maps of the interval

A strong pair correlation bound implies the CLT for Sinai Billiards

For Dynamical Systems, a strong bound on multiple correlations implies the Central Limit Theorem (CLT) [ChMa]. In Chernov's paper [Ch2], such a bound is derived for dynamically Holder continuous observables of dispersing Billiards. Here we weaken the regularity assumption and subsequently show that the bound on multiple correlations follows directly from the bound on pair correlations. Thus, a strong bound on pair correlations alone implies the CLT, for a wider class of observables. The result is extended to Anosov diffeomorphisms in any dimension.Comment: 13 page

Upper bound on the density of Ruelle resonances for Anosov flows

Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the operator (-i)V, called Ruelle resonances, close to the real axis and for large real parts.Comment: 57 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

New methodology for diagnosis of orthopedic diseases through additive manufacturing models

Our purpose is to develop the preoperative diagnosis stage for orthopedic surgical treatments using additive manufacturing technology. Our methods involve fast implementations of an additive manufactured bone model, converted from CAT data, through appropriate software use. Then, additive manufacturing of the formed surfaces through special 3D-printers. With the structural model redesigned and printed in three dimensions, the surgeon is able to look at the printed bone and he can handle it because the model perfectly reproduces the real one upon which he will operate. We found that additive manufacturing models can precisely characterize the anatomical structures of fractures or lesions. The studied practice helps the surgeon to provide a complete preoperative valuation and a correct surgery, with minimized duration and risks. This structural model is also an effective device for communication between doctor and patient