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

Cell migration requires both ion translocation and cytoskeletal anchoring by the Na-H exchanger NHE1

Directed cell movement is a multi-step process requiring an initial spatial polarization that is established by asymmetric stimulation of Rho GTPases, phosphoinositides (PIs), and actin polymerization. We report that the Na-H exchanger isoform 1 (NHE1), a ubiquitously expressed plasma membrane ion exchanger, is necessary for establishing polarity in migrating fibroblasts. In fibroblasts, NHE1 is predominantly localized in lamellipodia, where it functions as a plasma membrane anchor for actin filaments by its direct binding of ezrin/radixin/moesin (ERM) proteins. Migration in a wounding assay was impaired in fibroblasts expressing NHE1 with mutations that independently disrupt ERM binding and cytoskeletal anchoring or ion transport. Disrupting either function of NHE1 impaired polarity, as indicated by loss of directionality, mislocalization of the Golgi apparatus away from the orientation of the wound edge, and inhibition of PI signaling. Both functions of NHE1 were also required for remodeling of focal adhesions. Most notably, lack of ion transport inhibited de-adhesion, resulting in trailing edges that failed to retract. These findings indicate that by regulating asymmetric signals that establish polarity and by coordinating focal adhesion remodeling at the cell front and rear, cytoskeletal anchoring by NHE1 and its localized activity in lamellipodia act cooperatively to integrate cues for directed migration

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

Curves with rational chord-length parametrization

It has been recently proved that rational quadratic circles in standard Bezier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the analogue in bipolar coordinates of nonparametric curves. This interpretation furnishes a compact explicit expression for all planar curves with rational chord-length parametrization. In addition to straight lines and circles in standard form, they include remarkable curves, such as the equilateral hyperbola, Lemniscate of Bernoulli and Limacon of Pascal. The extension to 3D rational curves is also tackled

Flow Field Evolution of a Decaying Sunspot

We study the evolution of the flows and horizontal proper motions in and around a decaying follower sunspot based on time sequences of two-dimensional spectroscopic observations in the visible and white light imaging data obtained over six days from June~7 to~12, 2005. During this time period the sunspot decayed gradually to a pore. The spectroscopic observations were obtained with the Fabry-P\'{e}rot based Visible-Light Imaging Magnetograph (VIM) in conjunction with the high-order adaptive optics (AO) system operated at the 65 cm vacuum reflector of the Big Bear Solar Observatory (BBSO). We apply local correlation tracking (LCT) to the speckle reconstructed time sequences of white-light images around 600 nm to infer horizontal proper motions while the Doppler shifts of the scanned \FeI line at 630.15 nm are used to calculate line-of-sight (LOS) velocities with sub-arcsecond resolution. We find that the dividing line between radial inward and outward proper motions in the inner and outer penumbra, respectively, survives the decay phase. In particular the moat flow is still detectable after the penumbra disappeared. Based on our observations three major processes removed flux from the sunspot: (a) fragmentation of the umbra, (b) flux cancelation of moving magnetic features (MMFs; of the same polarity as the sunspot) that encounter the leading opposite polarity network and plages areas, and (c) flux transport by MMFs (of the same polarity as the sunspot) to the surrounding network and plage regions that have the same polarity as the sunspot.Comment: 9 pages, 7 figures, The Astrophysical Journal, accepted September, 200

Periods implying almost all periods, trees with snowflakes, and zero entropy maps

Let $X$ be a compact tree, $f$ be a continuous map from $X$ to itself, $End(X)$ be the number of endpoints and $Edg(X)$ be the number of edges of $X$. We show that if $n>1$ has no prime divisors less than $End(X)+1$ and $f$ has a cycle of period $n$, then $f$ has cycles of all periods greater than $2End(X)(n-1)$ and topological entropy $h(f)>0$; so if $p$ is the least prime number greater than $End(X)$ and $f$ has cycles of all periods from 1 to $2End(X)(p-1)$, then $f$ has cycles of all periods (this verifies a conjecture of Misiurewicz for tree maps). Together with the spectral decomposition theorem for graph maps it implies that $h(f)>0$ iff there exists $n$ such that $f$ has a cycle of period $mn$ for any $m$. We also define {\it snowflakes} for tree maps and show that $h(f)=0$ iff every cycle of $f$ is a snowflake or iff the period of every cycle of $f$ is of form $2^lm$ where $m\le Edg(X)$ is an odd integer with prime divisors less than $End(X)+1$