Search

Given a dynamical system, we say that a performance function has property P if its time averages along orbits are maximized at a periodic orbit. It is conjectured by several authors that for sufficiently hyperbolic dynamical systems, property P should be typical among sufficiently regular performance functions. In this paper we address this problem using a probabilistic notion of typicality that is suitable to infinite dimension: the concept of prevalence as introduced by Hunt, Sauer, and Yorke. For the one-sided shift on two symbols, we prove that property P is prevalent in spaces of functions with a strong modulus of regularity. Our proof uses Haar wavelets to approximate the ergodic optimization problem by a finite-dimensional one, which can be conveniently restated as a maximum cycle mean problem on a de Bruijin graph. [ABSTRACT FROM AUTHOR]

Calderón–Zygmund operators with noncommuting kernels may fail to be Lp-bounded for p≠2, even for kernels with good size/smoothness properties. In this paper, we obtain weak-type estimates for perfect dyadic CZO’s and cancellative Haar shifts associated to noncommuting kernels in terms of a row/column decomposition of the function. General CZO’s satisfy analogous H1→L1 type estimates. In conjunction with type estimates, we get similar row/column Lp estimates. Our approach also applies to paraproducts/martingale transforms with noncommuting symbols. [ABSTRACT FROM PUBLISHER]

Random contraction matrices obtained by truncating Haar distributed random unitary matrices were originally introduced and studied in the context of scattering theory. In this paper, we demonstrate a connection between ensembles of random contractions and the random-turns vicious walker model from statistical physics. In particular, we show that the moments of the trace of a random contraction enumerate configurations of vicious walkers. [ABSTRACT FROM PUBLISHER]