Conditions for Equality between Lyapunov and Morse Decompositions

Let $Q\rightarrow X$ be a continuous principal bundle whose group $G$ is reductive. A flow $\phi$ of automorphisms of $Q$ endowed with an ergodic probability measure on the compact base space $X$ induces two decompositions of the flag bundles associated to $Q$. A continuous one given by the finest Morse decomposition and a measurable one furnished by the Multiplicative Ergodic Theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under pertubations leaving unchanged the flow on the base space

Towards a single-photon energy-sensitive pixel readout chip: pixel level ADCs and digital readout circuitry

Unlike conventional CMOS imaging, a single\ud photon imager detects each individual photon impinging on\ud a detector, accumulating the number of photons during a\ud certain time window and not the charge generated by the all\ud the photons hitting the detector during said time window.\ud The latest developments in the semiconductor industry\ud are allowing faster and more complex chips to be designed\ud and manufactured. With these developments in mind we are\ud working towards the next step in single photon X-ray imaging:\ud energy sensitive pixel readout chips. The goal is not only\ud to detect and count individual photons, but also to measure\ud the charge deposited in the detector by each photon, and\ud consequently determine its energy. Basically, we are aiming\ud at a spectrometer-in-a-pixel, or a “color X-ray camera”.\ud The approach we have followed towards this goal is the\ud design of small analog-to-digital-converters at the pixel level,\ud together with a very fast digital readout from the pixels to\ud the periphery of the chip, where the data will be transmitted\ud off-chip.\ud We will present here the design and measurement on prototype\ud chips of two different 4-bit pixel level ADCs. The\ud ADCs are optimized for very small area and low power, with\ud a resolution of 4-bits and a sample rate of 1 Msample/s. The\ud readout architecture is based around current-mode sense\ud amplifiers and asynchronous token-passing between the pixels.\ud This is done in order to achieve event-by-event readout\ud and, consequently, on-line imaging. We need to read eventby-\ud event (photon-by-photon), because we cannot have memory\ud on the pixels due to obvious size constraints. We use\ud current-mode sense amplifiers because they perform very\ud well in similar applications as very fast static-RAM readout