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Let $P$ be a polynomial of degree $d$ with a Cremer point $p$ and no repelling or parabolic periodic bi-accessible points. We show that there are two types of such Julia sets $J_P$. The emph {red dwarf} $J_P$ are nowhere connected im kleinen and such that the intersection of all impressions of external angles is a continuum containing $p$ and the orbits of all critical images. The emph {solar} $J_P$ are such that every angle with dense orbit has a degenerate impression disjoint from other impressions and $J_P$ is connected im kleinen at its landing point. We study bi-accessible points and locally connected models of $J_P$ and show that such sets $J_P$ appear through polynomial-like maps for generic polynomials with Cremer points. Since known tools break down for $d>2$ (if $d>2$, it is not known if there are emph {small cycles} near $p$, while if $d=2$, this result is due to Yoccoz), we introduce emph {wandering ray continua} in $J_P$ and provide a new application of emph {Thurston laminations}. [ABSTRACT FROM AUTHOR]

We study topological dynamics on {\em unshielded} planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability. Then we deduce from this that a polynomial $f$ with a locally connected Julia set is backward stable outside any neighborhood of its attracting and neutral cycles. For a conformal measure $\mu$ this easily implies that one of the following holds: 1. for $\mu$-a.e. $x\in J(f)$, $\omega(x)=J(f)$; 2. for $\mu$-a.e. $x\in J(f)$, $\omega(x)=\omega(c(x))$ for a critical point $c(x)$ depending on $x$. [ABSTRACT FROM AUTHOR]

We call a rational map $f$ {\em graph critical} if any critical point either belongs to an invariant finite graph $G$, or has minimal limit set, or is non-recurrent and has limit set disjoint from $G$. We prove that, for any conformal measure, either for almost every point of the Julia set $J(f)$ its limit set coincides with $J(f)$, or for almost every point of $J(f)$ its limit set coincides with the limit set of a critical point of $f$. [ABSTRACT FROM AUTHOR]

We compare periodic orbits of circle rotations with their counterparts for interval maps. We prove that they are conjugate via a map of modality larger by at most 2 than the modality of the interval map. The proof is based on observation of trips of inhabitants of the Green Islands in the Black Sea. [ABSTRACT FROM AUTHOR]

We first give a geometric characterization of $\omega$-limit sets. We then use this characterization to prove that the family of $\omega$-limit sets of a continuous interval map is closed with respect to the Hausdorff metric. Finally, we apply this latter result to other dynamical systems. [ABSTRACT FROM AUTHOR]