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### Open Access Journals

DCDS

We consider Birkhoff averages of an
observable $\phi$ along orbits of a continuous map $f:X \rightarrow X$
with respect to a non-invariant measure $m$. In the simple case
where the averages converge $m$-almost everywhere, one may discuss
the distribution of values of the average in a natural way. We
extend this analysis to the case where convergence does not hold
$m$-almost everywhere. The case that the averages converge
$m$-almost everywhere is shown to be related to the recently
defined notion of "predictable" behavior, which is a condition on
the existence of pointwise asymptotic measures (SRB measures). A
heteroclinic attractor is an example of a system which is not
predictable. We define a more general notion called
"statistically predictable" behavior which is weaker than
predictability, but is strong enough to allow meaningful
statistical properties, i.e. distribution of Birkhoff averages,
to be analyzed. Statistical predictability is shown to imply the
existence of an asymptotic measure, but not vice versa. We
investigate the relationship between the various notions of
asymptotic measures and distributions of Birkhoff average.
Analysis of the heteroclinic attractor is used to illustrate the
applicability of the concepts.

DCDS

We discuss one parameter families of unimodal maps,
with negative Schwarzian derivative, unfolding a
saddle-node bifurcation.
We show that there is a parameter set of positive but not full
Lebesgue density at the bifurcation, for which
the maps exhibit absolutely continuous
invariant measures which are supported on the largest possible
interval. We prove that these measures converge weakly to
an atomic measure supported on the orbit of the saddle-node point.
Using these measures we analyze the intermittent time series that result
from the destruction of the periodic attractor
in the saddle-node bifurcation and prove
asymptotic formulae for the frequency with which
orbits visit the region previously occupied by the periodic attractor.

DCDS

We study a saddle-node bifurcation in a Lipschitz family of
diffeomorphisms on a manifold, in the
case that the stable set and unstable set of the fixed point intersect transversally
in a countable collection of one-dimensional manifolds diffeomorphic to circles.
We formulate generic conditions on the circles stated in terms of
standard coordinates, a recently defined tool for the study of saddle-node
bifurcations. Under the conditions, it is shown that there is a
decreasing sequence of intervals $[\underline{\mu_j},\overline{\mu_j}]$ of parameter values for which the
diffeomorphism is semi-conjugated to shift dynamics on the space of binary sequences.
The semi-conjugacy is implied by a recent result in the Conley index theory.

DCDS

We study the saddle-node bifurcation of a partially hyperbolic
fixed point in a Lipschitz family of $C^{k}$
diffeomorphisms on a Banach manifold (possibly infinite dimensional) in the
case that the fixed point is a saddle along hyperbolic directions and has multiple curves of homoclinic orbits. We show that this bifurcation
results in an invariant set which consists of a countable collection of closed
invariant curves and an uncountable collection of nonclosed
invariant curves which are the topological limits of the closed curves.
In addition, it is shown that these curves are $C^k$-smooth and that this
invariant set is uniformly partially
hyperbolic.

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