On Max-Semistable Laws and Extremes for Dynamical Systems

Suppose (f,X,μ) is a measure preserving dynamical system and ϕ:X→R a measurable observable. Let Xi=ϕ∘fi−1 denote the time series of observations on the system, and consider the maxima process Mn:=max{X1,…,Xn}. Under linear scaling of Mn, its asymptotic statistics are usually captured by a three-parameter generalised extreme value distribution. This assumes certain regularity conditions on the measure density and the observable. We explore an alternative parametric distribution that can be used to model the extreme behaviour when the observables (or measure density) lack certain regular variation assumptions. The relevant distribution we study arises naturally as the limit for max-semistable processes. For piecewise uniformly expanding dynamical systems, we show that a max-semistable limit holds for the (linear) scaled maxima process.

Foliated hyperbolicity and foliations with hyperbolic leaves

Given a lamination in a compact space and a laminated vector field $X$ which is hyperbolic when restricted to the leaves of the lamination, we distinguish a class of $X$-invariant probabilities that describe the behavior of almost every $X$-orbit in every leaf, which we call Gibbs $u$-states. We apply this to the case of foliations in compact manifolds having leaves with negative curvature, using the foliated hyperbolic vector field on the unit tangent bundle to the foliation generating the leaf geodesics. When the Lyapunov exponents of such ergodic Gibbs $u$-states are negative, it is an SRB measure (having a positive Lebesgue basin of attraction). When the foliation is by hyperbolic leaves, this class of probabilities coincide with the classical harmonic measures introduced by Garnett. Furthermore, if the foliation is transversally conformal and does not admit a transverse invariant measure we show that there are finitely many ergodic Gibbs $u$-states, each supported in one minimal set of the foliation, each having negative Lyapunov exponents, and the union of their basins of attraction has full Lebesgue measure. The leaf geodesics emanating from a point have a proportion whose asymptotic statistics are described by each of these ergodic Gibbs $u$-states, giving rise to continuous visibility functions of the attractors. Reversing time, by considering $-X$, we obtain the existence of the same number of repellers of the foliated geodesic flow having the same harmonic measures as projections to $M$. In the case of only one attractor, we obtain a north to south pole dynamics.

Statistical theory for the compressible scalar turbulence with inviscid limitation

We developed the statistical theory for compressible scalar turbulence in the inviscid limit, in which the spatial structures of the precursors of velocity and scalar discontinuities (i.e. preshock and prefront) were discussed. By analyzing the asymptotic statistics of scalar turbulence in the strong compression region, it was found that the prefront structure of scalar influenced negligible to the statistics of velocity gradient, except certain Mach number (M) modulations such as M-2/3 and M-4/3.