63 research outputs found

### Stabilization of heterodimensional cycles

We consider diffeomorphisms $f$ with heteroclinic cycles associated to
saddles $P$ and $Q$ of different indices. We say that a cycle of this type can
be stabilized if there are diffeomorphisms close to $f$ with a robust cycle
associated to hyperbolic sets containing the continuations of $P$ and $Q$. We
focus on the case where the indices of these two saddles differ by one. We
prove that, excluding one particular case (so-called twisted cycles that
additionally satisfy some geometrical restrictions), all such cycles can be
stabilized.Comment: 31 pages, 9 figure

### The Structure on Invariant Measures of $C^1$ generic diffeomorphisms

Let $\Lambda$ be an isolated non-trival transitive set of a $C^1$ generic
diffeomorphism f\in\Diff(M). We show that the space of invariant measures
supported on $\Lambda$ coincides with the space of accumulation measures of
time averages on one orbit. Moreover, the set of points having this property is
residual in $\Lambda$ (which implies the set of irregular$^+$ points is also
residual in $\Lambda$). As an application, we show that the non-uniform
hyperbolicity of irregular$^+$ points in $\Lambda$ with totally 0 measure
(resp., the non-uniform hyperbolicity of a generic subset in $\Lambda$)
determines the uniform hyperbolicity of $\Lambda$

### Collision, explosion and collapse of homoclinic classes

Homoclinic classes of generic $C^1$-diffeomorphisms are maximal transitive
sets and pairwise disjoint. We here present a model explaining how two
different homoclinic classes may intersect, failing to be disjoint. For that we
construct a one-parameter family of diffeomorphisms $(g_s)_{s\in [-1,1]}$ with
hyperbolic points $P$ and $Q$ having nontrivial homoclinic classes, such that,
for $s>0$, the classes of $P$ and $Q$ are disjoint, for $s<0$, they are equal,
and, for $s=0$, their intersection is a saddle-node.Comment: This is the final version, accepted in 200

### Non-hyperbolic ergodic measures with large support

We prove that there is a residual subset $\mathcal{S}$ in $\text{Diff}^1(M)$
such that, for every $f\in \mathcal{S}$, any homoclinic class of $f$ with
invariant one dimensional central bundle containing saddles of different
indices (i.e. with different dimensions of the stable invariant manifold)
coincides with the support of some invariant ergodic non-hyperbolic (one of the
Lyapunov exponents is equal to zero) measure of $f$

### Robust entropy expansiveness implies generic domination

Let $f: M \to M$ be a $C^r$-diffeomorphism, $r\geq 1$, defined on a compact
boundaryless $d$-dimensional manifold $M$, $d\geq 2$, and let $H(p)$ be the
homoclinic class associated to the hyperbolic periodic point $p$. We prove that
if there exists a $C^1$ neighborhood $\mathcal{U}$ of $f$ such that for every
$g\in {\mathcal U}$ the continuation $H(p_g)$ of $H(p)$ is entropy-expansive
then there is a $Df$-invariant dominated splitting for $H(p)$ of the form
$E\oplus F_1\oplus... \oplus F_c\oplus G$ where $E$ is contracting, $G$ is
expanding and all $F_j$ are one dimensional and not hyperbolic.Comment: 24 page

### 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

### Response operators for Markov processes in a finite state space: radius of convergence and link to the response theory for Axiom A systems

Using straightforward linear algebra we derive response operators describing the impact of small perturbations to finite state Markov processes. The results can be used for studying empirically constructedâ€”e.g. from observations or through coarse graining of model simulationsâ€”finite state approximation of statistical mechanical systems. Recent results concerning the convergence of the statistical properties of finite state Markov approximation of the full asymptotic dynamics on the SRB measure in the limit of finer and finer partitions of the phase space are suggestive of some degree of robustness of the obtained results in the case of Axiom A system. Our findings give closed formulas for the linear and nonlinear response theory at all orders of perturbation and provide matrix expressions that can be directly implemented in any coding language, plus providing bounds on the radius of convergence of the perturbative theory. In particular, we relate the convergence of the response theory to the rate of mixing of the unperturbed system. One can use the formulas derived for finite state Markov processes to recover previous findings obtained on the response of continuous time Axiom A dynamical systems to perturbations, by considering the generator of time evolution for the measure and for the observables. A very basic, low-tech, and computationally cheap analysis of the response of the Lorenz â€™63 model to perturbations provides rather encouraging results regarding the possibility of using the approximate representation given by finite state Markov processes to compute the systemâ€™s response

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