Finite type approximations of Gibbs measures on sofic subshifts

Consider a H\"older continuous potential $\phi$ defined on the full shift A^\nn, where $A$ is a finite alphabet. Let X\subset A^\nn be a specified sofic subshift. It is well-known that there is a unique Gibbs measure $\mu_\phi$ on $X$ associated to $\phi$. Besides, there is a natural nested sequence of subshifts of finite type $(X_m)$ converging to the sofic subshift $X$. To this sequence we can associate a sequence of Gibbs measures $(\mu_{\phi}^m)$. In this paper, we prove that these measures weakly converge at exponential speed to $\mu_\phi$ (in the classical distance metrizing weak topology). We also establish a strong mixing property (ensuring weak Bernoullicity) of $\mu_\phi$. Finally, we prove that the measure-theoretic entropy of $\mu_\phi^m$ converges to the one of $\mu_\phi$ exponentially fast. We indicate how to extend our results to more general subshifts and potentials. We stress that we use basic algebraic tools (contractive properties of iterated matrices) and symbolic dynamics.Comment: 18 pages, no figure

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

Large deviations for non-uniformly expanding maps

We obtain large deviation results for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average decays to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. The rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states. The corrections added to the published version of this text appear in bold; see last section for a list of changesComment: 36 pages, 1 figure. After many PhD students and colleagues having pointed several errors in the statements and proofs, this is a correction to published article answering those comments. List of main changes in a new last sectio

Stimulation of Na⁺/H⁺ Exchanger Isoform 1 Promotes Microglial Migration

Regulation of microglial migration is not well understood. In this study, we proposed that Na+/H+ exchanger isoform 1 (NHE-1) is important in microglial migration. NHE-1 protein was co-localized with cytoskeletal protein ezrin in lamellipodia of microglia and maintained its more alkaline intracellular pH (pHi). Chemoattractant bradykinin (BK) stimulated microglial migration by increasing lamellipodial area and protrusion rate, but reducing lamellipodial persistence time. Interestingly, blocking NHE-1 activity with its potent inhibitor HOE 642 not only acidified microglia, abolished the BK-triggered dynamic changes of lamellipodia, but also reduced microglial motility and microchemotaxis in response to BK. In addition, NHE-1 activation resulted in intracellular Na+ loading as well as intracellular Ca2+ elevation mediated by stimulating reverse mode operation of Na+/Ca2+ exchange (NCXrev). Taken together, our study shows that NHE-1 protein is abundantly expressed in microglial lamellipodia and maintains alkaline pHi in response to BK stimulation. In addition, NHE-1 and NCXrev play a concerted role in BK-induced microglial migration via Na+ and Ca2+ signaling. © 2013 Shi et al

Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies

Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.Comment: 16 pages; revised and slightly expanded versio

Luminescent properties of Bi-doped polycrystalline KAlCl4

We observed an intensive near-infrared luminescence in Bi-doped KAlCl4 polycrystalline material. Luminescence dependence on the excitation wavelength and temperature of the sample was studied. Our experimental results allow asserting that the luminescence peaked near 1 um belongs solely to Bi+ ion which isomorphically substitutes potassium in the crystal. It was also demonstrated that Bi+ luminescence features strongly depend on the local ion surroundings

Close-packed dimers on the line: diffraction versus dynamical spectrum

The translation action of \RR^{d} on a translation bounded measure $\omega$ leads to an interesting class of dynamical systems, with a rather rich spectral theory. In general, the diffraction spectrum of $\omega$, which is the carrier of the diffraction measure, live on a subset of the dynamical spectrum. It is known that, under some mild assumptions, a pure point diffraction spectrum implies a pure point dynamical spectrum (the opposite implication always being true). For other systems, the diffraction spectrum can be a proper subset of the dynamical spectrum, as was pointed out for the Thue-Morse sequence (with singular continuous diffraction) in \cite{EM}. Here, we construct a random system of close-packed dimers on the line that have some underlying long-range periodic order as well, and display the same type of phenomenon for a system with absolutely continuous spectrum. An interpretation in terms of `atomic' versus `molecular' spectrum suggests a way to come to a more general correspondence between these two types of spectra.Comment: 14 pages, with some additions and improvement

Ultrastructural and functional fate of recycled vesicles in hippocampal synapses

Efficient recycling of synaptic vesicles is thought to be critical for sustained information transfer at central terminals. However, the specific contribution that retrieved vesicles make to future transmission events remains unclear. Here we exploit fluorescence and time-stamped electron microscopy to track the functional and positional fate of vesicles endocytosed after readily releasable pool (RRP) stimulation in rat hippocampal synapses. We show that most vesicles are recovered near the active zone but subsequently take up random positions in the cluster, without preferential bias for future use. These vesicles non-selectively queue, advancing towards the release site with further stimulation in an actin-dependent manner. Nonetheless, the small subset of vesicles retrieved recently in the stimulus train persist nearer the active zone and exhibit more privileged use in the next RRP. Our findings reveal heterogeneity in vesicle fate based on nanoscale position and timing rules, providing new insights into the origins of future pool constitution