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We show that given a one parameter family $F_b$ of strongly dissipative infinitely renormalisable Hénon-like maps, parametrised by a quantity called the 'average Jacobian' b, the set of all parameters b such that $F_b$ has a Cantor set with unbounded geometry has full Lebesgue measure.
We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodel maps (with arbitrary combinatorics), in any even degree d. We then conclude that orbits of renormalization are asymptotic to the full renormalization horseshoe, which we construct. Our argument for exponential contraction is based on a precompactness property of the renormalization operator ("beau bounds"), which is leveraged in the abstract analysis of holomorphic iteration. Besides greater generality, it yields a unified approach to all combinatorics and degrees: there is no need to account for the varied geometric details of the dynamics, which were the typical source of contraction in previous restricted proofs.
In a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle. Distribution of the zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal-Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function $\mathcal{R}$ in two variables (the renormalization transformation). We prove that $\mathcal{R}$ is partially hyperbolic on an invariant cylinder $\mathcal{C}$. The Lee-Yang zeros are organized in a transverse measure for the central-stable foliation of $\mathcal{R}|\mathcal{C}$. Their distribution is absolutely continuous. Its density is $C^\infty$ (and non-vanishing) below the critical temperature. Above the critical temperature, it is $C^\infty$ on an open dense subset, but it vanishes on the complementary Cantor set of positive measure. This seems to be the first occasion of a complete rigorous description of the Lee-Yang distributions beyond 1D models.
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 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.
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 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 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.
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.