A Sampling Theorem for Rotation Numbers of Linear Processes in R2{\R}^{2}

We prove an ergodic theorem for the rotation number of the composition of a sequence os stationary random homeomorphisms in $S^{1}$. In particular, the concept of rotation number of a matrix $g\in Gl^{+}(2,{\R})$ can be generalized to a product of a sequence of stationary random matrices in $Gl^{+}(2,{\R})$. In this particular case this result provides a counter-part of the Osseledec's multiplicative ergodic theorem which guarantees the existence of Lyapunov exponents. A random sampling theorem is then proved to show that the concept we propose is consistent by discretization in time with the rotation number of continuous linear processes on ${\R}^{2}.

Mechanical and microscopic properties of the reversible plastic regime in a 2D jammed material

At the microscopic level, plastic flow of a jammed, disordered material consists of a series of particle rearrangements that cannot be reversed by subsequent deformation. An infinitesimal deformation of the same material has no rearrangements. Yet between these limits, there may be a self-organized plastic regime with rearrangements, but with no net change upon reversing a deformation. We measure the oscillatory response of a jammed interfacial material, and directly observe rearrangements that couple to bulk stress and dissipate energy, but do not always give rise to global irreversibility.Comment: 5 pages, 4 figures. A supplemental PDF detailing methods, and movies corresponding to Fig. 2(a, b, f), are availabl

A family of rotation numbers for discrete random dynamics on the circle

We revisit the problem of well-defining rotation numbers for discrete random dynamical systems on the circle. We show that, contrasting with deterministic systems, the topological (i.e. based on Poincar\'{e} lifts) approach does depend on the choice of lifts (e.g. continuously for nonatomic randomness). Furthermore, the winding orbit rotation number does not agree with the topological rotation number. Existence and conversion formulae between these distinct numbers are presented. Finally, we prove a sampling in time theorem which recover the rotation number of continuous Stratonovich stochastic dynamical systems on $S^1$ out of its time discretisation of the flow.Comment: 15 page

Robustness of the O(NN) universality class

We calculate the critical exponents for Lorentz-violating O($N$) $\lambda\phi^{4}$ scalar field theories by using two independent methods. In the first situation we renormalize a massless theory by utilizing normalization conditions. An identical task is fulfilled in the second case in a massive version of the same theory, previously renormalized in the BPHZ method in four dimensions. We show that although the renormalization constants, the $\beta$ and anomalous dimensions acquire Lorentz-violating quantum corrections, the outcome for the critical exponents in both methods are identical and furthermore they are equal to their Lorentz-invariant counterparts. Finally we generalize the last two results for all loop levels and we provide symmetry arguments for justifying the latter