We introduce tessellation of the filled Julia sets for hyperbolic and parabolic quadratic maps. Then the dynamics inside their Julia sets are organized by tiles which work like external rays outside. We also construct continuous families of pinching semiconjugacies associated with hyperblic-to-parabolic degenerations without using quasiconformal deformation. Instead we use tessellation and inve...

A new Lax representation for the Bogoyavlensky lattice is found, its r–matrix interpretation is elaborated. The r–matrix structure turns out to be related to a highly nonlocal quadratic Poisson structure on a direct sum of asso-ciative algebras. The theory of such nonlocal structures is developed, the Poisson property of the monodromy map is worked out in the most general situation. Some proble...

We study the map 9: C!C defined by 9(w, z)=(z, z+w) and the associated collection of sequences [zj] satisfying the recurrence zj+2=zj+1+zj . Iteration of 9 is equivalent to the study of such sequences. We analyze growth rates of the sequences, positive sequences, periodic and asymptotically periodic sequences and establish the existence of doubly infinite homoclinic sequences, non-zero sequence...

The Plücker relations define a projective embedding of the Grassmann variety Gr(p, n). We give another finite set of quadratic equations which defines the same embedding, and whose elements all have rank 6. This is achieved by constructing a certain finite set of linear maps V

Abstract. We consider perturbations of quadratic maps fa admitting an absolutely continuous invariant probability measure, where a is in a certain positive measure set A of parameters, and show that in any neighborhood of any such an fa, we find a rich fauna of dynamics. There are maps with periodic attractors as well as non-periodic maps whose critical orbit is absorbed by the continuation of ...

We give examples of infinitely renormalizable quadratic polynomials Fc : z 7→ z 2 + c with stationary combinatorics whose Julia sets have Hausdorff dimension arbitrary close to 1. The combinatorics of the renormalization involved is close to the Chebyshev one. The argument is based upon a new tool, a “Recursive Quadratic Estimate” for the Poincaré series of an infinitely renormalizable map. Sto...

We prove that almost every non-regular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps have excellent ergodic properties, as exponential decay of correlations (Keller and Nowicki, Young) and stochastic stability in the strong sense (Baladi and Viana). This is an important step to get...

We give examples of infinitely renormalizable quadratic polynomials Fc : z 7→ z + c with stationary combinatorics whose Julia sets have Hausdorff dimension arbitrary close to 1. The combinatorics of the renormalization involved is close to the Chebyshev one. The argument is based upon a new tool, a “Recursive Quadratic Estimate” for the Poincaré series of an infinitely renormalizable map. Stony...

We continue the classiication, begun in 11], 14] and 12], of quadratic Artin-Schelter regular algebras of global dimension 4 which map onto a twisted homogeneous coordinate ring of a quadric hypersurface in P 3. In this paper, we consider those cases where the quadric has rank 3. We also give suucient conditions for the point scheme of any quadratic regular algebra of global dimension 4 to be t...

For parametrized families of dynamical systems, two major goals are classifying the systems up to topological conjugacy, and understanding the structure of the bifurcation locus. The family Fλ = z n + λ/z gives a 1-parameter, n+ d degree family of rational maps of the Riemann sphere, which arise as singular perturbations of the polynomial z. This work presents several results related to these g...