212 research outputs found

### Dominated Splitting and Pesin's Entropy Formula

Let $M$ be a compact manifold and $f:\,M\to M$ be a $C^1$ diffeomorphism on
$M$. If $\mu$ is an $f$-invariant probability measure which is absolutely
continuous relative to Lebesgue measure and for $\mu$ $a.\,\,e.\,\,x\in M,$
there is a dominated splitting $T_{orb(x)}M=E\oplus F$ on its orbit $orb(x)$,
then we give an estimation through Lyapunov characteristic exponents from below
in Pesin's entropy formula, i.e., the metric entropy $h_\mu(f)$ satisfies
$h_{\mu}(f)\geq\int \chi(x)d\mu,$ where
$\chi(x)=\sum_{i=1}^{dim\,F(x)}\lambda_i(x)$ and
$\lambda_1(x)\geq\lambda_2(x)\geq...\geq\lambda_{dim\,M}(x)$ are the Lyapunov
exponents at $x$ with respect to $\mu.$ Consequently, by using a dichotomy for
generic volume-preserving diffeomorphism we show that Pesin's entropy formula
holds for generic volume-preserving diffeomorphisms, which generalizes a result
of Tahzibi in dimension 2

### Pesin's Formula for Random Dynamical Systems on $R^d$

Pesin's formula relates the entropy of a dynamical system with its positive
Lyapunov exponents. It is well known, that this formula holds true for random
dynamical systems on a compact Riemannian manifold with invariant probability
measure which is absolutely continuous with respect to the Lebesgue measure. We
will show that this formula remains true for random dynamical systems on $R^d$
which have an invariant probability measure absolutely continuous to the
Lebesgue measure on $R^d$. Finally we will show that a broad class of
stochastic flows on $R^d$ of a Kunita type satisfies Pesin's formula.Comment: 35 page

### Random Walk with Shrinking Steps: First Passage Characteristics

We study the mean first passage time of a one-dimensional random walker with
step sizes decaying exponentially in discrete time. That is step sizes go like
$\lambda^{n}$ with $\lambda\leq1$ . We also present, for pedagogical purposes,
a continuum system with a diffusion constant decaying exponentially in
continuous time. Qualitatively both systems are alike in their global
properties. However, the discrete case shows very rich mathematical structure,
depending on the value of the shrinking parameter, such as self-repetitive and
fractal-like structure for the first passage characteristics. The results we
present show that the most important quantitative behavior of the discrete case
is that the support of the distribution function evolves in time in a rather
complicated way in contrast to the time independent lattice structure of the
ordinary random walker. We also show that there are critical values of
$\lambda$ defined by the equation $\lambda^{K}+2\lambda^{P}-2=0$ with
$\{K,N\}\in{\mathcal N}$ where the mean first passage time undergo transitions.Comment: Major Re-Editing of the article. Conclusions unaltere

### Induced topological pressure for countable state Markov shifts

We introduce the notion of induced topological pressure for countable state
Markov shifts with respect to a non-negative scaling function and an arbitrary
subset of finite words. Firstly, the scaling function allows a direct access to
important thermodynamical quantities, which are usually given only implicitly
by certain identities involving the classically defined pressure. In this
context we generalise Savchenko's definition of entropy for special flows to a
corresponding notion of topological pressure and show that this new notion
coincides with the induced pressure for a large class of H\"older continuous
height functions not necessarily bounded away from zero. Secondly, the
dependence on the subset of words gives rise to interesting new results
connecting the Gurevi{\vc} and the classical pressure with exhausting
principles for a large class of Markov shifts. In this context we consider
dynamical group extentions to demonstrate that our new approach provides a
useful tool to characterise amenability of the underlying group structure.Comment: 28 page

### Convolution of multifractals and the local magnetization in a random field Ising chain

The local magnetization in the one-dimensional random-field Ising model is
essentially the sum of two effective fields with multifractal probability
measure. The probability measure of the local magnetization is thus the
convolution of two multifractals. In this paper we prove relations between the
multifractal properties of two measures and the multifractal properties of
their convolution. The pointwise dimension at the boundary of the support of
the convolution is the sum of the pointwise dimensions at the boundary of the
support of the convoluted measures and the generalized box dimensions of the
convolution are bounded from above by the sum of the generalized box dimensions
of the convoluted measures. The generalized box dimensions of the convolution
of Cantor sets with weights can be calculated analytically for certain
parameter ranges and illustrate effects we also encounter in the case of the
measure of the local magnetization. Returning to the study of this measure we
apply the general inequalities and present numerical approximations of the
D_q-spectrum. For the first time we are able to obtain results on multifractal
properties of a physical quantity in the one-dimensional random-field Ising
model which in principle could be measured experimentally. The numerically
generated probability densities for the local magnetization show impressively
the gradual transition from a monomodal to a bimodal distribution for growing
random field strength h.Comment: An error in figure 1 was corrected, small additions were made to the
introduction and the conclusions, some typos were corrected, 24 pages,
LaTeX2e, 9 figure

### Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents

This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$.
One can associate to such an automorphism two currents $\mu^\pm$ and the
equilibrium measure $\mu=\mu^+\wedge\mu^-$. In this paper we study some
geometric and dynamical properties of these objects. First, we characterize
$\mu$ as the unique measure of maximal entropy. Then we show that the measure
$\mu$ has a local product structure and that the currents $\mu^\pm$ have a
laminar structure. This allows us to deduce information about periodic points
and heteroclinic intersections. For example, we prove that the support of $\mu$
coincides with the closure of the set of saddle points. The methods used
combine the pluripotential theory with the theory of non-uniformly hyperbolic
dynamical systems

### Orbits and phase transitions in the multifractal spectrum

We consider the one dimensional classical Ising model in a symmetric
dichotomous random field. The problem is reduced to a random iterated function
system for an effective field. The D_q-spectrum of the invariant measure of
this effective field exhibits a sharp drop of all D_q with q < 0 at some
critical strength of the random field. We introduce the concept of orbits which
naturally group the points of the support of the invariant measure. We then
show that the pointwise dimension at all points of an orbit has the same value
and calculate it for a class of periodic orbits and their so-called offshoots
as well as for generic orbits in the non-overlapping case. The sharp drop in
the D_q-spectrum is analytically explained by a drastic change of the scaling
properties of the measure near the points of a certain periodic orbit at a
critical strength of the random field which is explicitly given. A similar
drastic change near the points of a special family of periodic orbits explains
a second, hitherto unnoticed transition in the D_q-spectrum. As it turns out, a
decisive role in this mechanism is played by a specific offshoot. We
furthermore give rigorous upper and/or lower bounds on all D_q in a wide
parameter range. In most cases the numerically obtained D_q coincide with
either the upper or the lower bound. The results in this paper are relevant for
the understanding of random iterated function systems in the case of moderate
overlap in which periodic orbits with weak singularity can play a decisive
role.Comment: The article has been completely rewritten; the title has changed; a
section about the typical pointwise dimension as well as several references
and remarks about more general systems have been added; to appear in J. Phys.
A; 25 pages, 11 figures, LaTeX2

### Chains of infinite order, chains with memory of variable length, and maps of the interval

We show how to construct a topological Markov map of the interval whose
invariant probability measure is the stationary law of a given stochastic chain
of infinite order. In particular we caracterize the maps corresponding to
stochastic chains with memory of variable length. The problem treated here is
the converse of the classical construction of the Gibbs formalism for Markov
expanding maps of the interval

### Infinite ergodic theory and Non-extensive entropies

We bring into account a series of result in the infinite ergodic theory that
we believe that they are relevant to the theory of non-extensive entropie

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