Trace formulas for stochastic evolution operators: Smooth conjugation method

The trace formula for the evolution operator associated with nonlinear stochastic flows with weak additive noise is cast in the path integral formalism. We integrate over the neighborhood of a given saddlepoint exactly by means of a smooth conjugacy, a locally analytic nonlinear change of field variables. The perturbative corrections are transfered to the corresponding Jacobian, which we expand in terms of the conjugating function, rather than the action used in defining the path integral. The new perturbative expansion which follows by a recursive evaluation of derivatives appears more compact than the standard Feynman diagram perturbation theory. The result is a stochastic analog of the Gutzwiller trace formula with the ``hbar'' corrections computed an order higher than what has so far been attainable in stochastic and quantum-mechanical applications.Comment: 16 pages, 1 figure, New techniques and results for a problem we considered in chao-dyn/980703

Holomorphic linearization of commuting germs of holomorphic maps

Let $f_1, ..., f_h$ be $h\ge 2$ germs of biholomorphisms of \C^n fixing the origin. We investigate the shape a (formal) simultaneous linearization of the given germs can have, and we prove that if $f_1, ..., f_h$ commute and their linear parts are almost simultaneously Jordanizable then they are simultaneously formally linearizable. We next introduce a simultaneous Brjuno-type condition and prove that, in case the linear terms of the germs are diagonalizable, if the germs commutes and our Brjuno-type condition holds, then they are holomorphically simultaneously linerizable. This answers to a multi-dimensional version of a problem raised by Moser.Comment: 24 pages; final version with erratum (My original paper failed to cite the work of L. Stolovitch [ArXiv:math/0506052v2]); J. Geom. Anal. 201

Scaling law in the Standard Map critical function. Interpolating hamiltonian and frequency map analysis

We study the behaviour of the Standard map critical function in a neighbourhood of a fixed resonance, that is the scaling law at the fixed resonance. We prove that for the fundamental resonance the scaling law is linear. We show numerical evidence that for the other resonances $p/q$, $q \geq 2$, $p \neq 0$ and $p$ and $q$ relatively prime, the scaling law follows a power--law with exponent $1/q$.Comment: AMS-LaTeX2e, 29 pages with 8 figures, submitted to Nonlinearit

Linearization in ultrametric dynamics in fields of characteristic zero - equal characteristic case

Let $K$ be a complete ultrametric field of charactersitic zero whose corresponding residue field $\Bbbk$ is also of charactersitic zero. We give lower and upper bounds for the size of linearization disks for power series over $K$ near an indifferent fixed point. These estimates are maximal in the sense that there exist exemples where these estimates give the exact size of the corresponding linearization disc. Similar estimates in the remaning cases, i.e. the cases in which $K$ is either a $p$-adic field or a field of prime characteristic, were obtained in various papers on the $p$-adic case (Ben-Menahem:1988,Thiran/EtAL:1989,Pettigrew/Roberts/Vivaldi:2001,Khrennikov:2001) later generalized in (Lindahl:2009 arXiv:0910.3312), and in (Lindahl:2004 http://iopscience.iop.org/0951-7715/17/3/001/,Lindahl:2010Contemp. Math) concerning the prime characteristic case