*$\lambda$-Lemma for families of Riemann surfaces and the critical loci of complex Hénon map*

We prove a version of the classical $\lambda$-lemma for holomorphic families of Riemann surfaces. We then use it to show that critical loci for complex Hénon maps that are small perturbations of quadratic polynomials with Cantor Julia sets are all quasiconformally equivalent.

*Error Diffusion on Simplices: Invariant Regions, Tessellations and Acuteness*

The error diffusion algorithm can be considered as a time dependent dynamical system that transforms a sequence of *inputs*; into a sequence of *inputs*;. That dynamical system is a time dependent translation acting on a partition of the phase space $\mathbb{A}$, a finite dimensional real affine space, into the Voronoï regions of the set $C$ of vertices of some polytope $\mathbf {P}$ where the inputs all belong.

Given a sequence $g(i)$ of inputs that are point in $\mathbb{A}$, $g(i)$ gets added to the *error vector* $e(i)$, the total vector accumulated so far, that belongs to the (Euclidean) vector space mofelling $\mathbb{A}$. The sum $g(i)+e(i)$ is then again in $\mathbb{A}$, thus in a well defined element of the partition of $\mathbb{A}$ that determines in turns one vertex $v(i)$. The point $v(i)$ of $\mathbb{A}$ is the $i^\textrm{th}$ output, and the new error vector to be used next is $e(i+1)\,=\, g(i)+e(i)-v(i)$. The maps $e(i)\mapsto e(i+1)$ and $g(i)+e(i)\mapsto g(i+1)+e(i+1)$ are two form of error diffusion, respectively in the vector space and affine space. Long term behavior of the algorithm can be deduced from the asymptotic properties of invariant sets, especially from the absorbing ones that serve as traps to all orbits. The existence of invariant sets for arbitrary sequence of inputs has been established in full generality, but in such a context, the invariant sets that are shown to exist are arbitrarily large and only few examples of minimal invariant sets can be described. Since the case of constant input (that corresponds to a time independent translation) has its own interest, we study here the invariant set for constant input for special polytopes that contain the $n$-dimensional regular simplices.

In that restricted context of interest in number theory, we study the properties of the minimal absorbing invariant set and prove that typically those sets are bounded fundamental sets for a discrete lattice generated by the simplex and that the intersections of those sets with the elements of the partition are fundamental sets for specific derived lattices.

*Quasisymmetries of Sierpinski carpet Julia sets*

We prove that if $\xi$ is a quasisymmetric homeomorphism between Sierpinski carpets that are the Julia sets of postcritically-finite rational maps, then $\xi$ is the restriction of a Mobius transformation to the Julia set. This implies that the group of quasisymmetric homeomorphisms of a Sierpinski carpet Julia set of a postcritically-finite rational map is finite.

*A structure theorem for semi-parabolic Henon maps*

Consider the parameter space $\mathcal{P}_{\lambda}\subset \mathbb{C}^{2}$ of complex Hénon maps \[ H_{c,a}(x,y)=(x^{2}+c+ay, ax),\ a\neq 0 \] which have a semi-parabolic fixed point with one eigenvalue $\lambda=e^{2\pi i p/q}$. We give a characterization of those Hénon maps from the curve $\mathcal{P}_{\lambda}$ that are small perturbations of a quadratic polynomial $p$ with a parabolic fixed point of multiplier $\lambda$. We prove that there is an open disk of parameters in $\mathcal{P}_{\lambda}$ for which the semi-parabolic Hénon map has connected Julia set $J$ and is structurally stable on $J$ and $J^{+}$. The Julia set $J^{+}$ has a nice local description: inside a bidisk $\mathbb{D}_{r}\times \mathbb{D}_{r}$ it is a trivial fiber bundle over $J_{p}$, the Julia set of the polynomial $p$, with fibers biholomorphic to $\mathbb{D}_{r}$. The Julia set $J$ is homeomorphic to a quotiented solenoid.

*Poly-time computability of the Feigenbaum Julia set*