Stretched exponential estimates on growth of the number of periodic points for prevalent diffeomorphisms, I.

For diffeomorphisms of smooth compact finite-dimensional manifolds, we consider the problem of how fast the number of periodic points with period $n$ grows as a function of $n$. In many familiar cases (e.g., Anosov systems) the growth is exponential, but arbitrarily fast growth is possible; in fact, the first author has shown that arbitrarily fast growth is topologically (Baire) generic for $C^2$ or smoother diffeomorphisms. In the present work we show that, by contrast, for a measure-theoretic notion of genericity we call "prevalence", the growth is not much faster than exponential. Specifically, we show that for each $\rho, \delta > 0$, there is a prevalent set of $C^{1+\rho}$ (or smoother) diffeomorphisms for which the number of periodic $n$ points is bounded above by $\exp(C n^{1+\delta})$ for some $C$ independent of $n$. We also obtain a related bound on the decay of hyperbolicity of the periodic points as a function of $n$, and obtain the same results for $1$-dimensional endomorphisms. The contrast between topologically generic and measure-theoretically generic behavior for the growth of the number of periodic points and the decay of their hyperbolicity show this to be a subtle and complex phenomenon, reminiscent of KAM theory. Here in Part I we state our results and describe the methods we use. We complete most of the proof in the $1$-dimensional $C^2$-smooth case and outline the remaining steps, deferred to Part II, that are needed to establish the general case. The novel feature of the approach we develop in this paper is the introduction of Newton Interpolation Polynomials as a tool for perturbing trajectories of iterated maps

A multidimensional grid-adaptive relativistic magnetofluid code

A robust second order, shock-capturing numerical scheme for multi-dimensional special relativistic magnetohydrodynamics on computational domains with adaptive mesh refinement is presented. The base solver is a total variation diminishing Lax-Friedrichs scheme in a finite volume setting and is combined with a diffusive approach for controlling magnetic monopole errors. The consistency between the primitive and conservative variables is ensured at all limited reconstructions and the spatial part of the four velocity is used as a primitive variable. Demonstrative relativistic examples are shown to validate the implementation. We recover known exact solutions to relativistic MHD Riemann problems, and simulate the shock-dominated long term evolution of Lorentz factor 7 vortical flows distorting magnetic island chains.Comment: accepted for publication in Computer Physics Communication