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A study of rational maps of the real or complex projective plane of degree two or more, concentrating on those which map an elliptic curve onto itself, necessarily by an expanding map. We describe relatively simple examples with a rich variety of exotic dynamical behaviors which are perhaps familiar to the applied dynamics community but not to specialists in several complex variables. For example, we describe smooth attractors with riddled or intermingled attracting basins, and we observe "blowout" bifurcations when the transverse Lyapunov exponent for the invariant curve changes sign. In the complex case, the elliptic curve (a topological torus) can never have a trapping neighborhood, yet it can have an attracting basin of large measure (perhaps even of full measure). We also describe examples where there appear to be Herman rings (that is topological cylinders mapped to themselves with irrational rotation number) with open attracting basin. In some cases we provide proofs, but in other cases the discussion is empirical, based on numerical computation.
In this paper geometric properties of infinitely renormalizable real Hénon-like maps $F$ in $\mathbb{R}^2$ are studied. It is shown that the appropriately defined renormalizations $R^n F$ converge exponentially to the one-dimensional renormalization fixed point. The convergence to one-dimensional systems is at a super-exponential rate controlled by the average Jacobian and a universal function $a(x)$. It is also shown that the attracting Cantor set of such a map has Hausdorff dimension less than 1, but contrary to the one-dimensional intuition, it is not rigid, does not lie on a smooth curve, and generically has unbounded geometry.
The punctured solenoid $\mathcal{H}$ is an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmüller space of $\mathcal{H}$ is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of $\mathcal{H}$. Furthermore, a point in the decorated Teichmüller space induces a polygonal decomposition of $\mathcal{H}$ giving a combinatorial description of its decorated Teichmüller space itself. This is used to obtain a non-trivial set of generators of the modular group of $\mathcal{H}$, which is presumably the main result of this paper. Moreover, each word in these generators admits a normal form, and the natural equivalence relation on normal forms is described. There is furthermore a non-degenerate modular group invariant two form on the Teichmüller space of $\mathcal{H}$. All of this structure is in perfect analogy with that of the decorated Teichmüller space of a punctured surface of finite type.
We prove that any unicritical polynomial $f_c:z\mapsto z^d+c$ which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. It implies that the connectedness locus (the "Multibrot set") is locally connected at the corresponding parameter values. It generalizes Yoccoz's Theorem for quadratics to the higher degree case.
This note is a shortened version of my dissertation thesis, defended at Stony Brook University in December 2004. It illustrates how dynamic complexity of a system evolves under deformations. The objects I considered are quartic polynomial maps of the interval that are compositions of two logistic maps. In the parameter space $P^{Q}$ of such maps, I considered the algebraic curves corresponding to the parameters for which critical orbits are periodic, and I called such curves left and right bones. Using quasiconformal surgery methods and rigidity I showed that the bones are simple smooth arcs that join two boundary points. I also analyzed in detail, using kneading theory, how the combinatorics of the maps evolves along the bones. The behavior of the topological entropy function of the polynomials in my family is closely related to the structure of the bone-skeleton. The main conclusion of the paper is that the entropy level-sets in the parameter space that was studied are connected.
We prove that the Julia set $J(f)$ of at most finitely renormalizable unicritical polynomial $f:z\mapsto z^d+c$ with all periodic points repelling is locally connected. (For $d=2$ it was proved by Yoccoz around 1990.) It follows from a priori bounds in a modified Principle Nest of puzzle pieces. The proof of a priori bounds makes use of new analytic tools developed in IMS Preprint #2005/02 that give control of moduli of annuli under maps of high degree.
We consider a Riemann surface $S$ of finite type containing a family of $N$ disjoint disks $D_i$, and prove the following Quasi-Additivity Law: If the total extremal width $\sum \mathcal{W}(S\smallsetminus D_i)$ is big enough (depending on $N$) then it is comparable with the extremal width $\mathcal{W} (S,\cup D_i)$ (under a certain ``separation assumption'') . We also consider a branched covering $f: U\rightarrow V$ of degree $N$ between two disks that restricts to a map $\Lambda\rightarrow B$ of degree $d$ on some disk $\Lambda \Subset U$. We derive from the Quasi-Additivity Law that if $\mod(U\smallsetminus \Lambda)$ is sufficiently small, then (under a ``collar assumption'') the modulus is quasi-invariant under $f$, namely $\mod(V\smallsetminus B)$ is comparable with $d^2 \mod(U\smallsetminus \Lambda)$. This Covering Lemma has important consequences in holomorphic dynamics which will be addressed in the forthcoming notes.
We investigate a class of area preserving non-uniformly hyperbolic maps of the two torus. First we establish some results on the regularity of the invariant foliations, then we use this knowledge to estimate the rate of mixing.
In the paper we discuss two questions about smooth expanding dynamical systems on the circle. (i) We characterize the sequences of asymptotic length ratios which occur for systems with Hölder continuous derivative. The sequence of asymptotic length ratios are precisely those given by a positive Hölder continuous function $s$ (solenoid function) on the Cantor set $C$ of $2$-adic integers satisfying a functional equation called the matching condition. The functional equation for the $2$-adic integer Cantor set is $$ s (2x+1)= \frac{s (x)} {s (2x)} \left( 1+\frac{1}{ s (2x-1)}\right)-1. $$ We also present a one-to-one correspondence between solenoid functions and affine classes of $2$-adic quasiperiodic tilings of the real line that are fixed points of the 2-amalgamation operator. (ii) We calculate the precise maximum possible level of smoothness for a representative of the system, up to diffeomorphic conjugacy, in terms of the functions $s$ and $cr(x)=(1+s(x))/(1+(s(x+1))^{-1})$. For example, in the Lipschitz structure on $C$ determined by $s$, the maximum smoothness is $C^{1+\alpha}$ for $0 < \alpha \le 1$ if, and only if, $s$ is $\alpha$-H\"older continuous. The maximum smoothness is $C^{2+\alpha}$ for $0 < \alpha \le 1$ if, and only if, $cr$ is $(1+\alpha)$-H\"older. A curious connection with Mostow type rigidity is provided by the fact that $s$ must be constant if it is $\alpha$-H\"older for $\alpha > 1$.
We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the hairiness phenomenon", there exist many Feigenbaum Julia sets $J(f)$ whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigenbaum Julia set, the Poincaré critical exponent $\delta_\mathrm{cr}$ is equal to the hyperbolic dimension $\operatorname{HD}_\mathrm{hyp}(J(f))$. Moreover, if $\operatorname{area} J(f)=0$ then $\operatorname{HD}_\mathrm{hyp} (J(f))=\operatorname{HD}(J(f))$. In the stationary case, the last statement can be reversed: if $\operatorname{area} J(f)> 0$ then $\operatorname{HD}_\mathrm{hyp} (J(f))< 2$. We also give a new construction of conformal measures on $J(f)$ that implies that they exist for any $\delta\in [\delta_\mathrm{cr}, \infty)$, and analyze their scaling and dissipativity/conservativity properties.