Spectral Properties of the Ruelle Operator for Product Type Potentials on Shift Spaces

We study a class of potentials $f$ on one sided full shift spaces over finite or countable alphabets, called potentials of product type. We obtain explicit formulae for the leading eigenvalue, the eigenfunction (which may be discontinuous) and the eigenmeasure of the Ruelle operator. The uniqueness property of these quantities is also discussed and it is shown that there always exists a Bernoulli equilibrium state even if $f$ does not satisfy Bowen's condition. We apply these results to potentials $f:\{-1,1\}^\mathbb{N} \to \mathbb{R}$ of the form $$f(x_1,x_2,\ldots) = x_1 + 2^{-\gamma} \, x_2 + 3^{-\gamma} \, x_3 + ...+n^{-\gamma} \, x_n + \ldots$$ with $\gamma >1$. For $3/2 < \gamma \leq 2$, we obtain the existence of two different eigenfunctions. Both functions are (locally) unbounded and exist a.s. (but not everywhere) with respect to the eigenmeasure and the measure of maximal entropy, respectively.Comment: To appear in the Journal of London Mathematical Societ

Ergodicity of avalanche transformations

PublishedIn this paper, we study dynamical systems of product type and some particular inducing scheme motivated by neural dynamics (called avalanche transformation). We derive the distribution of avalanche sizes and give sufficient conditions such that the avalanche transformation is ergodic. Moreover, we deduce a multivariate central limit theorem as a corollary.We would like to thank Ira Gessel and Wlodek Bryc for some helpful remarks concerning Section 2. The research of M. Denker was supported by the National Science Foundation grant DMS- 1008538. The research of A. Rodrigues is supported by the Swedish Research Council (VR Grant 2010/5905). The authors would like to thank the Goran Gustafsson Foundation UU/KTH for the ¨ financial support

The Thermal Environment of the Fiber Glass Dome for the New Solar Telescope at Big Bear Solar Observatory

The New Solar Telescope (NST) is a 1.6-meter off-axis Gregory-type telescope with an equatorial mount and an open optical support structure. To mitigate the temperature fluctuations along the exposed optical path, the effects of local/dome-related seeing have to be minimized. To accomplish this, NST will be housed in a 5/8-sphere fiberglass dome that is outfitted with 14 active vents evenly spaced around its perimeter. The 14 vents house louvers that open and close independently of one another to regulate and direct the passage of air through the dome. In January 2006, 16 thermal probes were installed throughout the dome and the temperature distribution was measured. The measurements confirmed the existence of a strong thermal gradient on the order of 5 degree Celsius inside the dome. In December 2006, a second set of temperature measurements were made using different louver configurations. In this study, we present the results of these measurements along with their integration into the thermal control system (ThCS) and the overall telescope control system (TCS).Comment: 12 pages, 11 figures, submitted to SPIE Optics+Photonics, San Diego, U.S.A., 26-30 August 2007, Conference: Solar Physics and Space Weather Instrumentation II, Proceedings of SPIE Volume 6689, Paper #2

Utilizing weak pump depletion to stabilize squeezed vacuum states

We propose and demonstrate a pump-phase locking technique that makes use of weak pump depletion (WPD) - an unavoidable effect that is usually neglected - in a sub-threshold optical parametric oscillator (OPO). We show that the phase difference between seed and pump beam is imprinted on both light fields by the non-linear interaction in the crystal and can be read out without disturbing the squeezed output. Our new locking technique allows for the first experimental realization of a pump-phase lock by reading out the pre-existing phase information in the pump field. There is no degradation of the detected squeezed states required to implement this scheme.Comment: 11 pages, 7 figure

Two-Dimensional Spectroscopy of Photospheric Shear Flows in a Small delta Spot

In recent high-resolution observations of complex active regions, long-lasting and well-defined regions of strong flows were identified in major flares and associated with bright kernels of visible, near-infrared, and X-ray radiation. These flows, which occurred in the proximity of the magnetic neutral line, significantly contributed to the generation of magnetic shear. Signatures of these shear flows are strongly curved penumbral filaments, which are almost tangential to sunspot umbrae rather than exhibiting the typical radial filamentary structure. Solar active region NOAA 10756 was a moderately complex, beta-delta sunspot group, which provided an opportunity to extend previous studies of such shear flows to quieter settings. We conclude that shear flows are a common phenomenon in complex active regions and delta spots. However, they are not necessarily a prerequisite condition for flaring. Indeed, in the present observations, the photospheric shear flows along the magnetic neutral line are not related to any change of the local magnetic shear. We present high-resolution observations of NOAA 10756 obtained with the 65-cm vacuum reflector at Big Bear Solar Observatory (BBSO). Time series of speckle-reconstructed white-light images and two-dimensional spectroscopic data were combined to study the temporal evolution of the three-dimensional vector flow field in the beta-delta sunspot group. An hour-long data set of consistent high quality was obtained, which had a cadence of better than 30 seconds and sub-arcsecond spatial resolution.Comment: 23 pages, 6 gray-scale figures, 4 color figures, 2 tables, submitted to Solar Physic

Limit theorems for von Mises statistics of a measure preserving transformation

For a measure preserving transformation $T$ of a probability space $(X,\mathcal F,\mu)$ we investigate almost sure and distributional convergence of random variables of the form $$x \to \frac{1}{C_n} \sum_{i_1<n,...,i_d<n} f(T^{i_1}x,...,T^{i_d}x),\, n=1,2,...,$$ where $f$ (called the \emph{kernel}) is a function from $X^d$ to $\R$ and $C_1, C_2,...$ are appropriate normalizing constants. We observe that the above random variables are well defined and belong to $L_r(\mu)$ provided that the kernel is chosen from the projective tensor product $$L_p(X_1,\mathcal F_1, \mu_1) \otimes_{\pi}...\otimes_{\pi} L_p(X_d,\mathcal F_d, \mu_d)\subset L_p(\mu^d)$$ with $p=d\,r,\, r\ \in [1, \infty).$ We establish a form of the individual ergodic theorem for such sequences. Next, we give a martingale approximation argument to derive a central limit theorem in the non-degenerate case (in the sense of the classical Hoeffding's decomposition). Furthermore, for $d=2$ and a wide class of canonical kernels $f$ we also show that the convergence holds in distribution towards a quadratic form $\sum_{m=1}^{\infty} \lambda_m\eta^2_m$ in independent standard Gaussian variables $\eta_1, \eta_2,...$. Our results on the distributional convergence use a $T$--\,invariant filtration as a prerequisite and are derived from uni- and multivariate martingale approximations