368 research outputs found

### 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

### Statistical stability of equilibrium states for interval maps

We consider families of multimodal interval maps with polynomial growth of
the derivative along the critical orbits. For these maps Bruin and Todd have
shown the existence and uniqueness of equilibrium states for the potential
$\phi_t:x\mapsto-t\log|Df(x)|$, for $t$ close to 1. We show that these
equilibrium states vary continuously in the weak$^*$ topology within such
families. Moreover, in the case $t=1$, when the equilibrium states are
absolutely continuous with respect to Lebesgue, we show that the densities vary
continuously within these families.Comment: More details given and the appendices now incorporated into the rest
of the pape

### 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

### 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

### Press shaping of arched components by means of a mobile tool

The best tool motion in a press is considered, when producing U-shaped components from sheet. The elastoplastic properties of the deformed material are taken into account. © 2013 Allerton Press, Inc

### Typical Borel measures on $[0,1]d$ satisfy a multifractal formalism

In this article, we prove that in the Baire category sense, measures
supported by the unit cube of $\R^d$ typically satisfy a multifractal
formalism. To achieve this, we compute explicitly the multifractal spectrum of
such typical measures $\mu$. This spectrum appears to be linear with slope 1,
starting from 0 at exponent 0, ending at dimension $d$ at exponent $d$, and it
indeed coincides with the Legendre transform of the $L^q$-spectrum associated
with typical measures $\mu$.Comment: 17 pages. To appear in Nonlinearit

### 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

### Oseledets' Splitting of Standard-like Maps

For the class of differentiable maps of the plane and, in particular, for
standard-like maps (McMillan form), a simple relation is shown between the
directions of the local invariant manifolds of a generic point and its
contribution to the finite-time Lyapunov exponents (FTLE) of the associated
orbit. By computing also the point-wise curvature of the manifolds, we produce
a comparative study between local Lyapunov exponent, manifold's curvature and
splitting angle between stable/unstable manifolds. Interestingly, the analysis
of the Chirikov-Taylor standard map suggests that the positive contributions to
the FTLE average mostly come from points of the orbit where the structure of
the manifolds is locally hyperbolic: where the manifolds are flat and
transversal, the one-step exponent is predominantly positive and large; this
behaviour is intended in a purely statistical sense, since it exhibits large
deviations. Such phenomenon can be understood by analytic arguments which, as a
by-product, also suggest an explicit way to point-wise approximate the
splitting.Comment: 17 pages, 11 figure

### 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

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