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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.
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.
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.
We study stability and bifurcations in holomorphic families of polynomial automorphisms of $\mathbb{C}^2$. We say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there. We first show that this defines a meaningful notion of stability, which parallels in many ways the classical notion of J-stability in one-dimensional dynamics. In the second part of the paper, we prove that under an assumption of moderate dissipativity, the parameters displaying homoclinic tangencies are dense in the bifurcation locus. This confirms one of Palis' Conjectures in the complex setting. The proof relies on the formalism of semi-parabolic bifurcation and the construction of "critical points" in semiparabolic basins (which makes use of the classical Denjoy-Carleman-Ahlfors and Wiman Theorems).
We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive (1,1)-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.
Fatou components for rational functions in the Riemann sphere are very well understood and play an important role in our understanding of one-dimensional dynamics. In higher dimensions the situation is less well understood. In this work we give a classification of invariant Fatou components for moderately dissipative Hénon maps. Most of our methods apply in a much more general setting. In particular we obtain a partial classification of invariant Fatou components for holomorphic endomorphisms of projective space, and we generalize Fatou's Snail Lemma to higher dimensions.
We show that repelling periodic points are landing points of periodic rays for exponential maps whose singular value has bounded orbit. For polynomials with connected Julia sets, this is a celebrated theorem by Douady, for which we present a new proof. In both cases we also show that points in hyperbolic sets are accessible by at least one and at most finitely many rays. For exponentials this allows us to conclude that the singular value itself is accessible.
We consider infinitely renormalizable Lorenz maps with real critical exponent $\alpha>1$ and combinatorial type which is monotone and satisfies a long return condition. For these combinatorial types we prove the existence of periodic points of the renormalization operator, and that each map in the limit set of renormalization has an associated unstable manifold. An unstable manifold defines a family of Lorenz maps and we prove that each infinitely renormalizable combinatorial type (satisfying the above conditions) has a unique representative within such a family. We also prove that each infinitely renormalizable map has no wandering intervals and that the closure of the forward orbits of its critical values is a Cantor attractor of measure zero.
Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of existence of a conservative renormalization fixed point. Furthermore, it has been shown that infinitely renormalizable maps in a neighborhood of this fixed point admit invariant Cantor sets on which the dynamics is "stable" - the Lyapunov exponents vanish on these sets.
Infinite renormalizability exists in several settings in dynamics, for example, in unimodal maps, dissipative Hénon-like maps, and conservative Hénon-like maps. All of these types of maps have associated invariant Cantor sets. The unimodal Cantor sets are rigid: the restrictions of the dynamics to the Cantor sets for any two maps are $C^{1+\alpha}$-conjugate. Although, strongly dissipative Hénon maps can be seen as perturbations of unimodal maps, surprisingly the rigidity breaks down. The Cantor attractors of Hénon maps with different average Jacobians are not smoothly conjugated. It is conjectured that the average Jacobian determines the rigidity class. This conjecture holds when the Jacobian is identically zero, and in this paper we prove that the conjecture also holds for conservative maps close to the conservative renormalization fixed point.
Rigidity is a consequence of an interplay between the decay of geometry and the convergence rate of renormalization towards the fixed point. Therefore, to demonstrate rigidity, we prove that the upper bound on the spectral radius of the action of the renormalization derivative on infinitely renormalizable maps is sufficiently small.