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The parameter space $S_p$ for monic centered cubic polynomial maps with a marked critical point of period p is a smooth affine algebraic curve whose genus increases rapidly with p. Each $S_p$ consists of a compact connectedness locus together with finitely many escape regions, each of which is biholomorphic to a punctured disk and is characterized by an essentially unique Puiseux series. This note with describe the topology of $S_p$, and of its smooth compactification, in terms of these escape regions. It concludes with a discussion of the real sub-locus of $S_p$.
We consider the family of holomorphic maps $e^z+c$ and show that fibers of postcritically finite parameters are trivial. This can be considered as the first and simplest class of non-escaping parameters for which we can obtain triviality of fibers in the exponential family.
We analyze a path-lifting algorithm for finding an approximate zero of a complex polynomial, and show that for any polynomial with distinct roots in the unit disk, the average number of iterates this algorithm requires is universally bounded by a constant times the log of the condition number. In particular, this bound is independent of the degree $d$ of the polynomial. The average is taken over initial values $z$ with $|z| = 1 + 1/d$ using uniform measure.
We study the affine orbifold laminations that were constructed in mishayair. An important question left open in mishayair is whether these laminations are always locally compact. We show that this is not the case.
The counterexample we construct has the property that the regular leaf space contains (many) hyperbolic leaves that intersect the Julia set; whether this can happen is itself a question raised in mishayair.
Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components. It also has a holomorphic interpretation, with the flavor of infinite dimensional Thurston--Teichmüller theory. We will discuss a topological theory of regluing, and trace a direction in which a holomorphic theory can develop.
We study the parameter space of unicritical polynomials $f_c:z\mapsto z^d+c$. For complex parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic, or Collet-Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the "principal nest" of parapuzzle pieces.
We study highly dissipative Hénon maps $$ F_{c,b}: (x,y) \mapsto (c-x^2-by, x) $$ with zero entropy. They form a region $\Pi$ in the parameter plane bounded on the left by the curve $W$ of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in $\Pi$, but there exist infinitely many different topological types of such maps (even away from $W$). We also prove that in the infinitely renormalizable case, the average Jacobian $b_F$ on the attracting Cantor set $\mathcal{O}_F$ is a topological invariant. These results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory. Along these lines, we show that the unstable manifolds of the periodic points form a lamination outside $\mathcal{O}_F$ if and only if there are no heteroclinic tangencies.
We prove Combinatorial rigidity for infinitely renormalizable unicritical polynomials, $f_c:z \mapsto z^d+c$, with a priori bounds and some "combinatorial condition". Combining with KL2, this implies local connectivity of the connectedness locus (the "Mandelbrot set" when $d=2$) at the corresponding parameter values.
We construct an entire function in the Eremenko-Lyubich class $\mathcal{B}$ whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in $\mathcal{B}$, in particular those of finite order, every escaping point can be connected to $\infty$ by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.