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### Open Access Journals

DCDS

We study the case of a smooth noninvertible map $f$ with Axiom A, in
higher dimension. In this paper, we look first
at the unstable dimension (i.e the Hausdorff dimension of the intersection between
local unstable manifolds and a basic set $\Lambda$), and prove that it is given
by the zero
of the pressure function of the unstable potential, considered on the natural
extension $\hat\Lambda$ of the basic set $\Lambda$; as a consequence,
the unstable dimension is independent of the prehistory $\hat x$.
Then we take a closer look at the theorem of construction for the local
unstable
manifolds of a perturbation $g$ of $f$, and for the conjugacy $\Phi_g$ defined
on $\hat \Lambda$.
If the map $g$ is holomorphic, one can prove some special estimates of the
Hölder
exponent of $\Phi_g$ on the liftings of the local unstable manifolds.
In this way we obtain a new estimate of the speed of convergence of the unstable
dimension of $g$, when $g \rightarrow f$.
Afterwards we prove the real analyticity of the unstable dimension when
the map $f$ depends on a real analytic parameter.
In the end we show that there exist Gibbs measures on the intersections between
local unstable manifolds and basic sets,
and that they are in fact geometric measures; using this, the unstable dimension
turns out to be equal to the upper box dimension. We notice also that in the
noninvertible case, the Hausdorff dimension of basic sets does not vary continuously with respect to the perturbation $g$
of $f$. In the case of noninvertible Axiom A maps on $\mathbb P^2$,
there can exist an infinite number of local unstable manifolds passing
through the
same point $x$ of the basic set $\Lambda$, thus there is no unstable
lamination. Therefore many of the methods used in the case of diffeomorphisms break down and new phenomena and methods of proof must appear. The results in this paper answer to some questions of Urbanski
([21]) about the extension
of one dimensional theory of Hausdorff dimension of fractals to the
higher dimensional case. They also improve some results and
estimates from [7].

keywords:
Gibbs states.
,
unstable manifolds
,
topological pressure
,
Smale spaces
,
Hausdorff dimension

DCDS

We give approximations for the Gibbs
states of arbitrary Hölder potentials $\phi$,
with the help of weighted sums of atomic measures on preimage sets, in the case of smooth

*non-invertible*maps hyperbolic on*folded basic sets*$\Lambda$. The endomorphism may have also stable directions on $\Lambda$ and is non-expanding in general. Folding of the phase space means that we do not have a foliation structure for the local unstable manifolds (instead they depend on the whole past and may intersect each other both inside and outside $\Lambda$). We consider here*simultaneously all*$n$-preimages in $\Lambda$ of a point, instead of the usual way of taking only the consecutive preimages from some given prehistory. We thus obtain the weighted distribution of consecutive preimage sets, with respect to various equilibrium measures on the saddle-type folded set $\Lambda$. In particular we obtain the distribution of preimage sets on $\Lambda$, with respect to the measure of maximal entropy. Our result is not a direct application of Birkhoff Ergodic Theorem on the inverse limit $\hat \Lambda$, since the set of prehistories of a point is uncountable in general, and the speed of convergence may vary for different prehistories in $\hat \Lambda$. For hyperbolic toral endomorphisms, we obtain the distribution of the consecutive preimage sets towards an*inverse SRB measure*, for Lebesgue-almost all points.
DCDS

The dynamics of endomorphisms (i.e non-invertible smooth maps)
presents many significant differences from that of diffeomorphisms, as
well as from the dynamics of expanding maps. There are numerous
concrete examples of hyperbolic endomorphisms. Many methods cannot
be used here due to overlappings in the fractal set and to the
existence of (possibly infinitely) many local unstable manifolds
going through the same point. First we will present the general problems and
explain how to construct
certain useful limit measures for atomic measures supported on various
prehistories. These limit measures are in many cases shown to be
equal to certain equilibrium measures for Hölder potentials. We
obtain thus an analogue of the SRB measure, namely an inverse SRB
measure in the case of a hyperbolic repeller, or of an Anosov
endomorphism. We study then the 1-sided Bernoullicity (or lack of
it) for certain measures invariant to endomorphisms, and give a
Classification Theorem for the ergodic and metric types of
behaviour of perturbations of a class of maps on their respective
basic sets, in terms of the values of the stable dimension. We
give also relations between thermodynamic formalism and fractal
dimensions (Hausdorff dimension of stable/unstable intersections
with basic sets, stable/unstable box dimensions, dimension of the
global unstable set for endomorphisms). Applications to certain nonlinear evolution models are also given in the end.

DCDS

In this paper, we review several notions from thermodynamic formalism,
like topological pressure and entropy and show how they can be employed, in order
to obtain information about stable and unstable sets of holomorphic endomorphisms
of $\mathbb P^2$ with Axiom A.

In particular, we will consider the non-wandering set of such a mapping and its "saddle" part $S_1$, i.e the subset of points with both stable and unstable directions. Under a derivative condition, the stable manifolds of points in S1 will have a very "thin" intersection with $S_1$, from the point of view of Hausdorff dimension. While for diffeomorphisms there is in fact an equality between $HD(W^s_\varepsilon(x)\cap S_1)$ and the unique zero of $P(t \cdot \phi^s)$ (Verjovsky-Wu [VW]) in the case of endomorphisms this will not be true anymore; counterexamples in this direction will be provided. We also prove that the unstable manifolds of an endomorphism depend Hölder continuously on the corresponding prehistory of their base point and employ this in the end to give an estimate of the Hausdorff dimension of the global unstable set of $S_1$. This set could be à priori very large, since, unlike in the case of H´enon maps, there is an uncountable collection of local unstable manifolds passing through each point of $S_1$.

In particular, we will consider the non-wandering set of such a mapping and its "saddle" part $S_1$, i.e the subset of points with both stable and unstable directions. Under a derivative condition, the stable manifolds of points in S1 will have a very "thin" intersection with $S_1$, from the point of view of Hausdorff dimension. While for diffeomorphisms there is in fact an equality between $HD(W^s_\varepsilon(x)\cap S_1)$ and the unique zero of $P(t \cdot \phi^s)$ (Verjovsky-Wu [VW]) in the case of endomorphisms this will not be true anymore; counterexamples in this direction will be provided. We also prove that the unstable manifolds of an endomorphism depend Hölder continuously on the corresponding prehistory of their base point and employ this in the end to give an estimate of the Hausdorff dimension of the global unstable set of $S_1$. This set could be à priori very large, since, unlike in the case of H´enon maps, there is an uncountable collection of local unstable manifolds passing through each point of $S_1$.

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

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.

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