Topological pressure of simultaneous level sets

Multifractal analysis studies level sets of asymptotically defined quantities in a topological dynamical system. We consider the topological pressure function on such level sets, relating it both to the pressure on the entire phase space and to a conditional variational principle. We use this to recover information on the topological entropy and Hausdorff dimension of the level sets. Our approach is thermodynamic in nature, requiring only existence and uniqueness of equilibrium states for a dense subspace of potential functions. Using an idea of Hofbauer, we obtain results for all continuous potentials by approximating them with functions from this subspace. This technique allows us to extend a number of previous multifractal results from the $C^{1+\epsilon}$ case to the $C^1$ case. We consider ergodic ratios $S_n \phi/S_n \psi$ where the function $\psi$ need not be uniformly positive, which lets us study dimension spectra for non-uniformly expanding maps. Our results also cover coarse spectra and level sets corresponding to more general limiting behaviour.Comment: 32 pages, minor changes based on referee's comment

Ordering of magnetic impurities and tunable electronic properties of topological insulators

We study collective behavior of magnetic adatoms randomly distributed on the surface of a topological insulator. As a consequence of the spin-momentum locking on the surface, the RKKY-type interactions of two adatom spins depend on the direction of the vector connecting them, thus interactions of an ensemble of adatoms are frustrated. We show that at low temperatures the frustrated RKKY interactions give rise to two phases: an ordered ferromagnetic phase with spins pointing perpendicular to the surface, and a disordered spin-glass-like phase. The two phases are separated by a quantum phase transition driven by the magnetic exchange anisotropy. Ferromagnetic ordering occurs via a finite-temperature phase transition. The ordered phase breaks time-reversal symmetry spontaneously, driving the surface states into a gapped state, which exhibits an anomalous quantum Hall effect and provides a realization of the parity anomaly. We find that the magnetic ordering is suppressed by potential scattering. Our work indicates that controlled deposition of magnetic impurities provides a way to modify the electronic properties of topological insulators.Comment: 4+ pages, 2 figure

Exponential speed of mixing for skew-products with singularities

Let $f: [0,1]\times [0,1] \setminus {1/2} \to [0,1]\times [0,1]$ be the $C^\infty$ endomorphism given by $$f(x,y)=(2x- [2x], y+ c/|x-1/2|- [y+ c/|x-1/2|]),$$ where $c$ is a positive real number. We prove that $f$ is topologically mixing and if $c>1/4$ then $f$ is mixing with respect to Lebesgue measure. Furthermore we prove that the speed of mixing is exponential.Comment: 23 pages, 3 figure

The Lyapunov spectrum is not always concave

We characterize one-dimensional compact repellers having nonconcave Lyapunov spectra. For linear maps with two branches we give an explicit condition that characterizes non-concave Lyapunov spectra

Developmental Role and Regulation of cortex, a Meiosis-Specific Anaphase-Promoting Complex/Cyclosome Activator

During oogenesis in metazoans, the meiotic divisions must be coordinated with development of the oocyte to ensure successful fertilization and subsequent embryogenesis. The ways in which the mitotic machinery is specialized for meiosis are not fully understood. cortex, which encodes a putative female meiosis-specific anaphase-promoting complex/cyclosome (APC/C) activator, is required for proper meiosis in Drosophila. We demonstrate that CORT physically associates with core subunits of the APC/C in ovaries. APC/CCORT targets Cyclin A for degradation prior to the metaphase I arrest, while Cyclins B and B3 are not targeted until after egg activation. We investigate the regulation of CORT and find that CORT protein is specifically expressed during the meiotic divisions in the oocyte. Polyadenylation of cort mRNA is correlated with appearance of CORT protein at oocyte maturation, while deadenylation of cort mRNA occurs in the early embryo. CORT protein is targeted for degradation by the APC/C following egg activation, and this degradation is dependent on an intact D-box in the C terminus of CORT. Our studies reveal the mechanism for developmental regulation of an APC/C activator and suggest it is one strategy for control of the female meiotic cell cycle in a multicellular organism

Thermodynamic formalism for contracting Lorenz flows

We study the expansion properties of the contracting Lorenz flow introduced by Rovella via thermodynamic formalism. Specifically, we prove the existence of an equilibrium state for the natural potential $\hat\phi_t(x,y, z):=-t\log J_{(x, y, z)}^{cu}$ for the contracting Lorenz flow and for $t$ in an interval containing $[0,1]$. We also analyse the Lyapunov spectrum of the flow in terms of the pressure

Quantum Charge Fluctuations in a Superconducting Grain

We consider charge quantization in a small superconducting grain that is contacted by a normal-metal electrode and is controlled by a capacitively coupled gate. At zero temperature and zero conductance $G$ between the grain and the electrode, the charge $Q$ as a function of the gate voltage $V_g$ changes in steps. The step height is $e$ if $\Delta<E_c$, where $\Delta$ and $E_c$ are, respectively, the superconducting gap and the charging energy of the grain. Quantum charge fluctuations at finite conductance remove the discontinuity in the dependence of $Q$ on $V_g$ and lead to a finite step width $\propto G^2\Delta$. The resulting shape of the Coulomb blockade staircase is of a novel type. The grain charge is a continuous function of $V_g$ while the differential capacitance, $dQ/dV_g$, has discontinuities at certain values of the gate voltage. We determine analytically the shape of the Coulomb blockade staircase also at non-zero temperatures.Comment: 12 pages, 3 figure

Quantum spin Hall effect and spin-charge separation in a kagome lattice

A two-dimensional kagome lattice is theoretically investigated within a simple tight-binding model, which includes the nearest neighbor hopping term and the intrinsic spin-orbit interaction between the next nearest neighbors. By using the topological winding properties of the spin-edge states on the complex-energy Riemann surface, the spin Hall conductance is obtained to be quantized as $-e/2\pi$ ($e/2\pi$) in insulating phases. This result keeps consistent with the numerical linear-response calculation and the \textbf{Z}$_{2}$ topological invariance analysis. When the sample boundaries are connected in twist, by which two defects with $\pi$ flux are introduced, we obtain the spin-charge separated solitons at 1/3 (or 2/3) filling.Comment: 13 NJP pages, 7 figure

Geometry of expanding absolutely continuous invariant measures and the liftability problem

We show that for a large class of maps on manifolds of arbitrary finite dimension, the existence of a Gibbs-Markov-Young structure (with Lebesgue as the reference measure) is a necessary as well as sufficient condition for the existence of an invariant probability measure which is absolutely continuous measure (with respect to Lebesgue) and for which all Lyapunov exponents are positive.Comment: 21 pages; final versio

Delocalization of slowly damped eigenmodes on Anosov manifolds

We look at the properties of high frequency eigenmodes for the damped wave equation on a compact manifold with an Anosov geodesic flow. We study eigenmodes with spectral parameters which are asymptotically close enough to the real axis. We prove that such modes cannot be completely localized on subsets satisfying a condition of negative topological pressure. As an application, one can deduce the existence of a "strip" of logarithmic size without eigenvalues below the real axis under this dynamical assumption on the set of undamped trajectories.Comment: 28 pages; compared with version 1, minor modifications, add two reference