84 research outputs found

### Lyapunov `Non-typical' Points of Matrix Cocycles and Topological Entropy

It follows from Oseledec Multiplicative Ergodic Theorem (or Kingman's
Sub-additional Ergodic Theorem) that the set of `non-typical' points for which
the Oseledec averages of a given continuous cocycle diverge has zero measure
with respect to any invariant probability measure. In strong contrast, for any
H$\ddot{o}$der continuous cocycles over hyperbolic systems, in this article we
show that either all ergodic measures have same Maximal Lyapunov exponents or
the set of Lyapunov `non-typical' points have full topological entropy and
packing topological entropy. Moreover, we give an estimate of Bowen Hausdorff
entropy from below.Comment: 23 pages. arXiv admin note: substantial text overlap with
arXiv:0808.0350 by other authors; text overlap with arXiv:0905.0739 by other
author

### Topological Pressure for the Completely Irregular Set of Birkhoff Averages

In this paper we mainly study the dynamical complexity of Birkhoff ergodic
average under the simultaneous observation of any number of continuous
functions. These results can be as generalizations of [6,35] etc. to study
Birkhorff ergodic average from one (or finite) observable function to any
number of observable functions from the dimensional perspective.
For any topological dynamical system with $g-$almost product property and
uniform separation property, we show that any {\it jointly-irregular set}(i.e.,
the intersection of a series of $\phi-$irregular sets over several continuous
functions) either is empty or carries full topological pressure. In particular,
if further the system is not uniquely ergodic, then the {\it
completely-irregular set}(i.e., intersection of all possible {\it nonempty
$\phi-$irregular} sets) is nonempty(even forms a dense $G_\delta$ set) and
carries full topological pressure. Moreover, {\it irregular-mix-regular sets}
(i.e., intersection of some $\phi-$irregular sets and $\varphi-$regular sets)
are discussed.
Similarly, the above results are suitable for the case of BS-dimension.
As consequences, these results are suitable for any system such as shifts of
finite type or uniformly hyperbolic diffeomorphisms, time-1 map of uniformly
hyperbolic flows, repellers, $\beta-$shifts etc..Comment: The title "Joint Birkhoff Ergodic Average and Topological Pressure"
is changed by "Topological Pressure for the Completely Irregular Set of
Birkhoff Averages

### Nonexistence of Lyapunov Exponents for Matrix Cocycles

It follows from Oseledec Multiplicative Ergodic Theorem that the
Lyapunov-irregular set of points for which the Oseledec averages of a given
continuous cocycle diverge has zero measure with respect to any invariant
probability measure. In strong contrast, for any dynamical system
$f:X\rightarrow X$ with exponential specification property and a
H$\ddot{\text{o}}$lder continuous matrix cocycle $A:X\rightarrow G
(m,\mathbb{R})$, we show here that if there exist ergodic measures with
different Lyapunov spectrum, then the Lyapunov-irregular set of $A$ is residual
(i.e., containing a dense $G_\delta$ set).Comment: arXiv admin note: substantial text overlap with arXiv:0808.0350 by
other author

### Distributional chaos in multifractal analysis, recurrence and transitivity

There are lots of results to study dynamical complexity on irregular sets and
level sets of ergodic average from the perspective of density in base space,
Hausdorff dimension, Lebesgue positive measure, positive or full topological
entropy (and topological pressure) etc.. However, it is unknown from the
viewpoint of chaos. There are lots of results on the relationship of positive
topological entropy and various chaos but it is known that positive topological
entropy does not imply a strong version of chaos called DC1 so that it is
non-trivial to study DC1 on irregular sets and level sets. In this paper we
will show that for dynamical system with specification property, there exist
uncountable DC1-scrambled subsets in irregular sets and level sets. On the
other hand, we also prove that several recurrent levels of points with
different recurrent frequency all have uncountable DC1-scrambled subsets. The
main technique established to prove above results is that there exists
uncountable DC1-scrambled subset in saturated sets.Comment: 23 page

### Diffeomorphisms with Liao-Pesin set

In this paper we mainly deal with an invariant (ergodic) hyperbolic measure
$\mu$ for a diffeomorphism $f,$ assuming that $f$ is just $C^1$ and for $\mu$
a.e. $x$, the sum of Oseledec spaces corresponding to negative Lyapunov
exponents (quasi-limit-)dominates the sum of Oseledec spaces corresponding to
positive Lyapunov exponents at $x$. We generalize a certain of results of Pesin
theory from $C^{1+\alpha}$ to the $C^1$ system $(f,\mu)$, including a
sufficient condition for existence of horseshoe, Livshitz theorem, exponential
growth of periodic points, distribution of periodic points, periodic measures,
horseshoes, nonuniform specification and lower semi-continuity of entropy
function etc. In particular, they are applied for $C^1$ partially hyperbolic
systems whose central bundle displays some non-uniform hyperbolicity, including
some robust systems. Moreover, for some $C^1$ partially hyperbolic (not
necessarily volume-preserving) systems, we get some information of Lebesgue
measure on Average-nonuniform hyperbolicityand volume-non-expanding.
A constructed machinery is developed for $C^1$ (not necessarily
$C^{1+\alpha}$) diffeomorphisms: new Pesin blocks is established topologically
(independent on measures) such that every block has stable manifold theorem and
simultaneously has exponential shadowing. The new construction, different with
classical $C^{1+\alpha}$ ones, is mainly inspired from Liao's
quasi-hyperbolicity and so here we call new blocks by Liao-Pesin blocks and
call the new established $C^1$ Pesin theory by $C^1$ Liao-Pesin Theory.
Liao-Pesin set not only exists for invariant measures, but also exists for
general probability measures, for example, Lebesgue measure (not assuming
invariant) in some partially hyperbolic systems.Comment: This version puts arXiv:1004.0486 \& arXiv:1011.6011 (year 2010)
together and updates some other new observation. 88 page

### Unstable entropies and Dimension Theory of Partially Hyperbolic Systems

In this paper we define unstable topological entropy for any subsets (not
necessarily compact or invariant) in partially hyperbolic systems as a
Carath\'{e}odory dimension characteristic, motivated by the work of Bowen and
Pesin etc. We then establish some basic results in dimension theory for Bowen
unstable topological entropy, including an entropy distribution principle and a
variational principle in general setting. As applications of this new concept,
we study unstable topological entropy of saturated sets and extend some results
in \cite{Bo, PS2007}. Our results give new insights to the multifractal
analysis for partially hyperbolic systems

### Existence and Distributional Chaos of Points that are Recurrent but Not Banach Recurrent

It is well-known that recurrence is typical from the probabilistic
perspective in the study of dynamical systems by Poincare recurrence theorem.
According to the recurrent frequency (i.e., the probability of finding the
orbit of an initial point entering in its neighborhood), many different levels
of recurrent points are found such as periodic points, almost periodic points,
weakly almost periodic points, quasi-weakly almost periodic points and Banach
recurrent points. From the probabilistic perspective, recurrent points between
Banach recurrence and genreal recurrence carry totally zero measure naturally.
In strong contrast, in this paper we will not only show the existence on six
refined levels of recurrent points between Banach recurrence and general
recurrence but also show that these levels carry a strong topological
complexity. On one hand, each new level of recurrent points is dense in the
whole space. On the other hand, from the perspective of distributional chaos,
we will prove that these levels all contain uncountable DC1-scrambled subsets.
These six recurrent levels of asymptotic behavior are discovered firstly in the
liturature of dynamical systems.Comment: 11 page

### Dominated Splitting, Partial Hyperbolicity and Positive Entropy

Let $f:M\rightarrow M$ be a $C^1$ diffeomorphism with a dominated splitting
on a compact Riemanian manifold $M$ without boundary.
We state and prove several sufficient conditions for the topological entropy
of $f$ to be positive. The conditions deal with the dynamical behaviour of the
(non-necessarily invariant) Lebesgue measure. In particular, if the Lebesgue
measure is $\delta$-recurrent then the entropy of $f$ is positive. We give
counterexamples showing that these sufficient conditions are not necessary.
Finally, in the case of partially hyperbolic diffeomorphisms, we give a
positive lower bound for the entropy relating it with the dimension of the
unstable and stable sub-bundles.Comment: 24page

### Multifractal Analysis of The New Level Sets

By an appropriate definition, we divide the irregular set into level sets.
Then we characterize the multifractal spectrum of these new pieces by
calculating their entropies. We also compute the entropies of various
intersections of the level sets of regular and irregular set which is rarely
studied in the literature. Moreover, our conclusions also hold for the
topological pressure. Finally, we consider the continuous case and use our
results to give a description for the suspension flow

### On the irregular points for systems with the shadowing property

We prove that when $f$ is a continuous selfmap acting on compact metric space
$(X,d)$ which satisfies the shadowing property, then the set of irregular
points (i.e. points with divergent Birkhoff averages) has full entropy.
Using this fact we prove that in the class of $C^0$-generic maps on
manifolds, we can only observe (in the sense of Lebesgue measure) points with
convergant Birkhoff averages. In particular, the time average of atomic
measures along orbit of such points converges to some SRB-like measure in the
weak$^*$ topology. Moreover, such points carry zero entropy. In contrast,
irregular points are non-observable but carry infinite entropy

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