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### Open Access Journals

DCDS

We study families of hyperbolic skew products with the
transversality condition and in particular, the Hausdorff dimension
of their fibers, by using thermodynamical formalism. The maps we
consider can be non-invertible, and the study of their dynamics is
influenced greatly by this fact.

We introduce and employ probability measures (constructed from equilibrium measures on the natural extension), which are supported on the fibers of the skew product. A stronger condition, that of Uniform Transversality is then considered in order to obtain a general formula for Hausdorff dimension of fibers for all base points and almost all parameters.

In the end we study a large class of examples of transversal hyperbolic families which locally depend linearly on the parameters, and also another class of examples related to complex dynamics.

We introduce and employ probability measures (constructed from equilibrium measures on the natural extension), which are supported on the fibers of the skew product. A stronger condition, that of Uniform Transversality is then considered in order to obtain a general formula for Hausdorff dimension of fibers for all base points and almost all parameters.

In the end we study a large class of examples of transversal hyperbolic families which locally depend linearly on the parameters, and also another class of examples related to complex dynamics.

DCDS

We derive the multifractal analysis of the conformal measure
(or, equivalently, of the invariant measure)
associated to a family of weights imposed upon a graph
directed Markov system
(GDMS) using balls as the filtration. Our analysis is done over
a subset of the limit set, a subset which is often large. In particular, this
subset is the entire limit set when the GDMS under scrutiny satisfies a boundary
separation condition. Our analysis also applies to more general situations such
as real and complex continued fractions.

DCDS

We introduce and explore random conformal graph directed Mar-kov
systems governed by measure-preserving ergodic dynamical
systems. We first develop the symbolic thermodynamic
formalism for random finitely primitive subshifts of finite type
with a countable alphabet (by establishing tightness in
a narrow topology). We then construct fibrewise conformal and
invariant measures along with fibrewise topological pressure.
This enables us to define the expected topological pressure
$\mathcal EP(t)$ and to prove a variant of Bowen's formula
which identifies the Hausdorff dimension of almost every
limit set fiber with $\inf\{t:\mathcal EP(t)\leq0\}$, and is the
unique zero of the expected pressure if the alphabet is finite or
the system is regular. We introduce the class of essentially
random systems and we show that in the realm of systems with
finite alphabet their limit set fibers are never homeomorphic in a
bi-Lipschitz fashion to the limit sets of deterministic systems;
they thus make up a drastically new world. We also provide a
large variety of examples, with exact computations of Hausdorff
dimensions, and we study in detail the small random perturbations
of an arbitrary elliptic function.

DCDS

We estimate the Bowen parameters and the Hausdorff dimensions of the
Julia sets of expanding finitely generated rational semigroups.
We show that the Bowen parameter is larger than or equal to the ratio of the
entropy of the skew product map $\tilde{f}$ and the Lyapunov exponent of $\tilde{f}$ with
respect to the maximal entropy measure for $\tilde{f}$. Moreover, we show that the equality holds if and only if
the generators are simultaneously conjugate to the form $a_{j}z^{\pm d}$ by a M\"{o}bius transformation.
Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that
the Bowen parameter is strictly larger than $2$.

DCDS

We deal with the Fatou functions $f_\lambda(z)=z+e^{-z}+\lambda$, Re$\lambda\ge
1$. We consider the set $J_r(f_\lambda)$ consisting of those points
of the Julia set of $f_\lambda$ whose real parts do not escape to infinity under
positive iterates of $f_\lambda$. Our ultimate result is that the function
$\lambda\mapsto$HD$(J_r(f_\lambda))$ is real-analytic. In order to prove it we
develop the thermodynamic formalism of potentials of the form
$-t$log$|F_\lambda'|$, where $F_\lambda$ is the projection of $f_\lambda$ to the
infinite cylinder. It includes appropriately defined
topological pressure, Perron-Frobenius operators, geometric and
invariant generalized conformal measures (Gibbs states). We show that
our Perron-Frobenius operators are quasicompact, that they embed into a
family of operators depending holomorphically on an appropriate parameter
and we obtain several
other properties of these operators. We prove an appropriate
version of Bowen's formula that the Hausdorff dimension of the set
$J_r(f_\lambda)$ is equal to the unique zero of the pressure
function. Since the formula
for the topological pressure is independent of the set $J_r(f_\lambda)$,
Bowen's formula also indicates that $J_r(f_\lambda)$ is the right set to deal
with. What concerns geometry of the set $J_r(f_\lambda)$ we also prove
that the HD$(J_r(f_\lambda))$-dimensional Hausdorff measure of the set $J_r(F_\lambda)$
is positive and finite whereas its HD$(J_r(f_\lambda))$-dimensional
packing measure is locally infinite. This last property allows us to
conclude that HD$(J_r(f_\lambda))<2$.
We also study in detail the properties of
quasiconformal conjugations between the maps $f_\lambda$. As a byproduct of
our main course
of reasoning we prove stochastic properties of the dynamical system
generated by $F_\lambda$ and the invariant Gibbs states $\mu_t$ such as the
Central Limit Theorem and the exponential decay of correlations.

DCDS

We consider the dynamics of semi-hyperbolic semigroups generated by
finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds
it is proved that there exists a geometric measure on the Julia set with exponent
$h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff
and packing measures are finite and positive on the Julia set and are mutually equivalent
with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal
dimensions, Hausdorff, packing and box counting are equal. It is also proved that for
the canonically associated skew-product map there exists a unique $h$-conformal measure.
Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely
continuous invariant (under the skew-product map) measure. In fact these two measures are
equivalent, and the invariant measure is metrically exact, hence ergodic.

DCDS

For points $x$ belonging to a basic set $\Lambda$ of an Axiom A
holomorphic endomorphism of $\mathbb P^2$,
one can construct the local stable manifold $W_{\varepsilon_0}^s(x)$
and the local unstable manifolds $W_{\varepsilon_0}^u(\hat x)$.
A priori, $W_{\varepsilon_0}^u(\hat x)$ should depend on the entire
prehistory $\hat x$ of $x$.
However, all known examples have all their local unstable manifolds
depending only on the base point $x$.
Therefore a natural problem is to give actual examples where, for
different prehistories of points in the basic sets of holomorphic
Axiom A maps, we obtain different unstable manifolds.
We solve this problem by considering the map $(z^4+\varepsilon w^2, w^4)$
and then also show that, by perturbing $(z^2+c, w^2)$, one can get also maps
$f_\varepsilon$ which are injective on $\Lambda_\varepsilon$, their corresponding basic
sets, hence the cardinality of the set $(f_\varepsilon|_{\Lambda_\varepsilon})^{-1}(x), x
\in \Lambda_\varepsilon$, is not stable under perturbation.

DCDS

The conference "Dynamical Systems II, Denton 2009" held in the
University of North Texas in Denton from May 17, 2009 through May 23,
2009 gathered approximately forty participants working on various
subbranches of dynamical systems such as holomorphic and conformal dynamics,
transcendental dynamics, random dynamical systems, thermodynamic formalism,
and iterated function systems, including random behavior of
deterministic systems,
the theory of fractal sets, including dimension theory. Apart from
stimulating, highly informative mathematical talks delivered by nearly
all participants of the conference, its outcome resulted also in
thirteen outstanding research articles presented in this volume. The
subject of these papers varies from author to author reflecting their
scientific interests. The articles were written by A. Badeńska (Warsaw
University of Technology), D. Hensley (Texas A&M), P. Haissinsky
(Université de Provence) ,
M. Kesseboehmer (University of Bremen), D. Mayer (TU Clausthal),
E. Mihailescu (Romanian Academy),
T. Mühlenbruch (FernUniversität in Hagen), F. Naud (Université
d'Avignon), S. Munday
(University of St Andrews), K. Pilgrim (Indiana University), Mario Roy
(York University),
H. H. Rugh, D. Simmons (University of North Texas), B. Stratmann
(University of Bremen), F. Strömberg (TU Darmstadt),
H. Sumi (University of Osaka), and M. Urbański (University of North Texas).

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DCDS

Developing the pioneering work of Lars Olsen [14], we deal with the
question of continuity of the numerical value of
Hausdorff measures of some natural families of conformal dynamical
systems endowed with an appropriate natural topology. In particular,
we prove such
continuity for hyperbolic polynomials from the
Mandelbrot set, and more generally for the space of hyperbolic rational
functions of a fixed degree. We go beyond hyperbolicity by proving
continuity for maps including parabolic rational functions, for
example that the parameter $1/4$ is such a continuity point for
quadratic polynomials $z\mapsto z^2+c$ for $c\in [0,1/4]$. We prove
the continuity of the numerical value of Hausdorff measures
also for the spaces of conformal expanding repellers and parabolic
ones, more generally for parabolic Walters conformal maps. We also
prove some partial continuity results for all conformal Walters maps;
these are in general of infinite degree. In order to
do this, as one of our tools, we provide a detailed local analysis,
uniform with respect to
the parameter, of the behavior of conformal maps around parabolic
fixed points in any dimension. We also establish continuity of
numerical values of
Hausdorff measures for some families of infinite $1$-dimensional
iterated function systems.

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