147 research outputs found

### Absolute Continuity Theorem for Random Dynamical Systems on $R^d$

In this article we provide a proof of the so called absolute continuity
theorem for random dynamical systems on $R^d$ which have an invariant
probability measure. First we present the construction of local stable
manifolds in this case. Then the absolute continuity theorem basically states
that for any two transversal manifolds to the family of local stable manifolds
the induced Lebesgue measures on these transversal manifolds are absolutely
continuous under the map that transports every point on the first manifold
along the local stable manifold to the second manifold, the so-called
Poincar\'e map or holonomy map. In contrast to known results, we have to deal
with the non-compactness of the state space and the randomness of the random
dynamical system.Comment: 46 page

### Lifting Measures to Inducing Schemes

In this paper we study the liftability property for piecewise continuous maps
of compact metric spaces, which admit inducing schemes in the sense of Pesin
and Senti [PS05, PS06]. We show that under some natural assumptions on the
inducing schemes - which hold for many known examples - any invariant ergodic
Borel probability measure of sufficiently large entropy can be lifted to the
tower associated with the inducing scheme. The argument uses the construction
of connected Markov extensions due to Buzzi [Buz99], his results on the
liftability of measures of large entropy, and a generalization of some results
by Bruin [Bru95] on relations between inducing schemes and Markov extensions.
We apply our results to study the liftability problem for one-dimensional cusp
maps (in particular, unimodal and multimodal maps) and for some
multidimensional maps.Comment: 28 pages. Final version. To appear in Ergodic Theory and Dynamical
System

### 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

### 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

### Well-posed infinite horizon variational problems on a compact manifold

We give an effective sufficient condition for a variational problem with
infinite horizon on a compact Riemannian manifold M to admit a smooth optimal
synthesis, i. e. a smooth dynamical system on M whose positive
semi-trajectories are solutions to the problem. To realize the synthesis we
construct a well-projected to M invariant Lagrange submanifold of the
extremals' flow in the cotangent bundle T*M. The construction uses the
curvature of the flow in the cotangent bundle and some ideas of hyperbolic
dynamics

### Ensemble averages and nonextensivity at the edge of chaos of one-dimensional maps

Ensemble averages of the sensitivity to initial conditions $\xi(t)$ and the
entropy production per unit time of a {\it new} family of one-dimensional
dissipative maps, $x_{t+1}=1-ae^{-1/|x_t|^z}(z>0)$, and of the known
logistic-like maps, $x_{t+1}=1-a|x_t|^z(z>1)$, are numerically studied, both
for {\it strong} (Lyapunov exponent $\lambda_1>0$) and {\it weak} (chaos
threshold, i.e., $\lambda_1=0$) chaotic cases. In all cases we verify that (i)
both $[\ln_q x \equiv (x^{1-q}-1)/(1-q); \ln_1 x=\ln x]$ and $<S_q
> [S_q \equiv (1-\sum_i p_i^q)/(q-1); S_1=-\sum_i p_i \ln p_i]$ {\it linearly}
increase with time for (and only for) a special value of $q$, $q_{sen}^{av}$,
and (ii) the {\it slope} of $and that of$ {\it coincide},
thus interestingly extending the well known Pesin theorem. For strong chaos,
$q_{sen}^{av}=1$, whereas at the edge of chaos, $q_{sen}^{av}(z)<1$.Comment: 5 pages, 5 figure

### A review of linear response theory for general differentiable dynamical systems

The classical theory of linear response applies to statistical mechanics
close to equilibrium. Away from equilibrium, one may describe the microscopic
time evolution by a general differentiable dynamical system, identify
nonequilibrium steady states (NESS), and study how these vary under
perturbations of the dynamics. Remarkably, it turns out that for uniformly
hyperbolic dynamical systems (those satisfying the "chaotic hypothesis"), the
linear response away from equilibrium is very similar to the linear response
close to equilibrium: the Kramers-Kronig dispersion relations hold, and the
fluctuation-dispersion theorem survives in a modified form (which takes into
account the oscillations around the "attractor" corresponding to the NESS). If
the chaotic hypothesis does not hold, two new phenomena may arise. The first is
a violation of linear response in the sense that the NESS does not depend
differentiably on parameters (but this nondifferentiability may be hard to see
experimentally). The second phenomenon is a violation of the dispersion
relations: the susceptibility has singularities in the upper half complex
plane. These "acausal" singularities are actually due to "energy
nonconservation": for a small periodic perturbation of the system, the
amplitude of the linear response is arbitrarily large. This means that the NESS
of the dynamical system under study is not "inert" but can give energy to the
outside world. An "active" NESS of this sort is very different from an
equilibrium state, and it would be interesting to see what happens for active
states to the Gallavotti-Cohen fluctuation theorem.Comment: 19 pages, 2 figure

### Non-uniqueness of ergodic measures with full Hausdorff dimension on a Gatzouras-Lalley carpet

In this note, we show that on certain Gatzouras-Lalley carpet, there exist
more than one ergodic measures with full Hausdorff dimension. This gives a
negative answer to a conjecture of Gatzouras and Peres

### 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

### Entropy and Poincar\'e recurrence from a geometrical viewpoint

We study Poincar\'e recurrence from a purely geometrical viewpoint. We prove
that the metric entropy is given by the exponential growth rate of return times
to dynamical balls. This is the geometrical counterpart of Ornstein-Weiss
theorem. Moreover, we show that minimal return times to dynamical balls grow
linearly with respect to its length. Finally, some interesting relations
between recurrence, dimension, entropy and Lyapunov exponents of ergodic
measures are given.Comment: 11 pages, revised versio

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