Coupled skinny baker's maps and the Kaplan-Yorke conjecture

The Kaplan-Yorke conjecture states that for "typical" dynamical systems with a physical measure, the information dimension and the Lyapunov dimension coincide. We explore this conjecture in a neighborhood of a system for which the two dimensions do not coincide because the system consists of two uncoupled subsystems. We are interested in whether coupling "typically" restores the equality of the dimensions. The particular subsystems we consider are skinny baker's maps, and we consider uni-directional coupling. For coupling in one of the possible directions, we prove that the dimensions coincide for a prevalent set of coupling functions, but for coupling in the other direction we show that the dimensions remain unequal for all coupling functions. We conjecture that the dimensions prevalently coincide for bi-directional coupling. On the other hand, we conjecture that the phenomenon we observe for a particular class of systems with uni-directional coupling, where the information and Lyapunov dimensions differ robustly, occurs more generally for many classes of uni-directionally coupled systems (also called skew-product systems) in higher dimensions.Comment: 33 pages, 3 figure

Method for producing edge geometry superconducting tunnel junctions utilizing an NbN/MgO/NbN thin film structure

A method for fabricating an edge geometry superconducting tunnel junction device is discussed. The device is comprised of two niobium nitride superconducting electrodes and a magnesium oxide tunnel barrier sandwiched between the two electrodes. The NbN electrodes are preferably sputter-deposited, with the first NbN electrode deposited on an insulating substrate maintained at about 250 C to 500 C for improved quality of the electrode

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

The effect of projections on fractal sets and measures in Banach spaces

We study the extent to which the Hausdorff dimension of a compact subset of an infinite-dimensional Banach space is affected by a typical mapping into a finite-dimensional space. It is possible that the dimension drops under all such mappings, but the amount by which it typically drops is controlled by the ‘thickness exponent’ of the set, which was defined by Hunt and Kaloshin (Nonlinearity 12 (1999), 1263–1275). More precisely, let $X$ be a compact subset of a Banach space $B$ with thickness exponent $\tau$ and Hausdorff dimension $d$. Let $M$ be any subspace of the (locally) Lipschitz functions from B to $\mathbb{R}^{m}$ that contains the space of bounded linear functions. We prove that for almost every (in the sense of prevalence) function f in M, the Hausdorff dimension of f(X) is at least min{m,d/(1 + tau)}. We also prove an analogous result for a certain part of the dimension spectra of Borel probability measures supported on X. The factor 1/(1 + tau) can be improved to 1/(1 + tau/2) if B is a Hilbert space. Since dimension cannot increase under a (locally) Lipschitz function, these theorems become dimension preservation results when tau = 0. We conjecture that many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero. We also discuss the sharpness of our results in the case tau > 0

Edge geometry superconducting tunnel junctions utilizing an NbN/MgO/NbN thin film structure

An edge defined geometry is used to produce very small area tunnel junctions in a structure with niobium nitride superconducting electrodes and a magnesium oxide tunnel barrier. The incorporation of an MgO tunnel barrier with two NbN electrodes results in improved current-voltage characteristics, and may lead to better junction noise characteristics. The NbN electrodes are preferably sputter-deposited, with the first NbN electrode deposited on an insulating substrate maintained at about 250 to 500 C for improved quality of the electrode