It is well-known that the set of buried points of a Julia set of a rational function (also called the residual Julia set) is topologically “fat” in the sense that it is a dense $G_{\delta }$ if it is non-empty. We show that it is, in many cases, a full-measure subset of the Julia set with respect to conformal measure and the measure of maximal entropy. We also address Hausdorff dimension of buried points in the same cases, and discuss connectivity and topological dimension of the set of buried points. Finally,...

Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms $f:ℂ{\mathbb{P}}^k\to ℂ{\mathbb{P}}^k$, when k >1. Classical theory describes U(f) as the complement in $ℂ{\mathbb{P}}^k$ of the support of a dynamically defined closed positive (1,1) current. Given any closed positive (1,1) current S on $ℂ{\mathbb{P}}^k$, we give a definition of linking number between closed loops in $ℂ{\mathbb{P}}^k\setminus suppS$ and the current S. It has the property that if lk(γ,S) ≠ 0, then γ represents a non-trivial homology element in $H₁\left(ℂ{\mathbb{P}}^k\setminus suppS\right)$. As an application, we use these linking...

We describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable quadratic polynomials and prove that when they satisfy a priori bounds, the topology is rigid modulo combinatorial equivalence.

A technique is presented for multiplexing two ergodic measure preserving transformations together to derive a third limiting transformation. This technique is used to settle a question regarding rigidity sequences of weak mixing transformations. Namely, given any rigidity sequence for an ergodic measure preserving transformation, there exists a weak mixing transformation which is rigid along the same sequence. This establishes a wide range of rigidity sequences for weakly mixing dynamical systems....

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in ${ℝ}^n$ and an algebraic hypersurface. The answer is polynomial in the height (the magnitude of coefficients) of the equation and the size of the curve in the space-time, with the exponent depending only on the degree and the dimension.The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned...

It is known that for almost every (with respect to Lebesgue measure) a ∈ [√2,2] the forward trajectory of the turning point of the tent map $T_a$ with slope a is dense in the interval of transitivity of $T_a$. We prove that the complement of this set of parameters of full measure is σ-porous.

Let $U₁,...,U_d$ be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of $\mathbb{Z}{₊}^d$ and the associated collection of rectangular parallelepipeds in ${ℝ}^d$ with sides parallel to the axes and dimensions of the form $n₁\times \cdots \times n_d$ with $(n₁,...,n_d)\in \Gamma .$ The associated multiparameter geometric and ergodic maximal operators $M_{}$ and $M_{\Gamma }$ are defined respectively on $L¹\left({ℝ}^d\right)$ and L¹(Ω) by $M_{}g\left(x\right)=su{p}_{x\in R\in }1/\left|R\right|{\int }_R\left|g\left(y\right)\right|dy$ and $M_{\Gamma }f\left(\omega \right)=su{p}_{(n₁,...,n_d)\in \Gamma }1/n₁\cdots n_d{\sum }_{j₁=0}^{n₁-1}\cdots {\sum }_{j_d=0}^{n_d-1}\left|f\left(U{₁}^{j₁}\cdots U_d^{j_d}\omega \right)\right|$. Given a Young function Φ, it is shown that $M_{}$ satisfies the weak type estimate $|x\in {ℝ}^d:M_{}g\left(x\right)>\alpha |\le C_{}{\int }_{{ℝ}^d}\Phi (c_{}\left|g\right|/\alpha )$ for...