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Let f be an infinitely-renormalizable quadratic polynomial and J_infty be the intersection of forward orbits of small Julia sets of simple renormalizations of f. We prove that J_infty contains no hyperbolic sets.
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We analyse the intersection of positively and negatively sectional-hyperbolic sets for flows on compact manifolds. First we prove that such an intersection is hyperbolic if the intersecting sets are both transitive (this is false without such a hypothesis). Next we prove that, in general, such an intersection consists of a nonsingular hyperbolic set, finitely many singularities and regular orbits joining them. Afterward we exhibit a three-dimensional star flow with two homoclinic classes, one being positively (but not negatively) sectional-hyperbolic and the other negatively (but not positively) sectional-hyperbolic, whose intersection reduces to a single periodic orbit. This provides a counterexample to a conjecture by Shy, Zhu, Gan and Wen (cite{sgw}, cite{zgw}).
Let $f_0$ be a polynomial of degree $d_1+d_2$ with a periodic critical point $0$ of multiplicity $d_1-1$ and a Julia critical point of multiplicity $d_2$. We show that if $f_0$ is primitive, free of neutral periodic points and non-renormalizable at the Julia critical point, then the straightening map $chi_{f_0}:mathcal C(lambda_{f_0}) to mathcal C_{d_1}$ is a bijection. More precisely, $f^{m_0}$ has a polynomial-like restriction which is hybrid equivalent to some polynomial in $mathcal C_{d_1}$ for each map $f in mathcal C(lambda_{f_0})$, where $m_0$ is the period of $0$ under $f_0$. On the other hand, $f_0$ can be tuned with any polynomial $gin mathcal C_{d_1}$. As a consequence, we conclude that the straightening map $chi_{f_0}$ is a homeomorphism from $mathcal C(lambda_{f_0})$ onto the Mandelbrot set when $d_1=2$. This together with the main result in [SW] solve the problem for primitive tuning for cubic polynomials with connected Julia sets thoroughly.
We prove that every sectional-hyperbolic Lyapunov stable set contains a nontrivial homoclinic class.
In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression.
Let $mathcal{R}$ be a strongly compact $C^2$ map defined in an open subset of an infinite-dimensional Banach space such that the image of its derivative $D_F mathcal{R}$ is dense for every $F$. Let $Omega$ be a compact, forward invariant and partially hyperbolic set of $mathcal{R}$ such that $mathcal{R}colon Omegarightarrow Omega$ is onto. The $delta$-shadow $W^s_delta(Omega)$ of $Omega$ is the union of the sets $$W^s_delta(G)= {Fcolon dist(mathcal{R}^iF, mathcal{R}^iG) leq delta, for every igeq 0 },$$ where $G in Omega$. Suppose that $W^s_delta(Omega)$ has transversal empty interior, that is, for every $C^{1+Lip}$ $n$-dimensional manifold $M$ transversal to the distribution of dominated directions of $Omega$ and sufficiently close to $W^s_delta(Omega)$ we have that $Mcap W^s_delta(Omega)$ has empty interior in $M$. Here $n$ is the finite dimension of the strong unstable direction. We show that if $delta$ is small enough then $$cup_{igeq 0}mathcal{R}^{-i}W^s_{delta} (Omega)$$ intercepts a $C^k$-generic finite dimensional curve inside the Banach space in a set of parameters with zero Lebesgue measure, for every $kgeq 0$. This extends to infinite-dimensional dynamical systems previous studies on the Lebesgue measure of stable laminations of invariants sets.
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