We prove that almost every real quadratic map is either regular or Collet-Eckmann with polynomial recurrence of the critical orbit. 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 the same results for more ge...

Given any rational map f , there is a lamination by Riemann surfaces associated to f . Such laminations were constructed in general by Lyubich and Minsky. In this paper, we classify laminations associated to quadratic polynomials with periodic critical point. In particular, we prove that the topology of such laminations determines the combinatorics of the parameter. We also describe the topolog...

We report exact analytical expressions locating the 0 → 1, 1 → 2 and 2 → 4 bifurcation curves for a prototypical system of two linearly coupled quadratic maps. Of interest is the precise location of the parameter sets where Naimark-Sacker bifurcations occur, starting from a non-diagonal period-2 orbit. This result is the key to understand the onset of synchronization in networks of quadratic maps.

Inspired by work done for polynomial automorphisms, we apply pluripotential theory to study iteration of birational maps of P2. A major theme is that success of pluripotential theoretic constructions depends on separation between orbits of the forward and backward indeterminacy sets. In particular, we show that a very mild separation hypothesis guarantees the existence of a plurisubharmonic esc...

In this paper we prove that the Newton method applied to the generalized equation y ∈ f(x) + F (x) with a C function f and a set-valued map F acting in Banach spaces, is locally convergent uniformly in the parameter y if and only if the map (f +F ) is Aubin continuous at the reference point. We also show that the Aubin continuity actually implies uniform Q-quadratic convergence provided that th...

Stony Brook IMS Preprint #1997/8 July, 1997 Abstract. We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it that the set of infinitely renormalizable parameter values in the real quadratic family Pc : x 7→ x 2 + c has zero measure. This yields the statement in the title (where “ regular” means to have an attracting cycle and ...

In this paper we explore the dynamical behaviour of the qdeformed versions of widely studied 1D nonlinear map-the Gaussian map and another famous 2D nonlinear map-the Henon map. The Gaussian map is perhaps the only 1D nonlinear map which exhibits the co-existing attractors. In this study we particularly, compare the dynamical behaviour of the Gaussian map and q-deformed Gaussian map with a spec...

We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of R. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and is also the general form of a quadratic three-dimensional map with a quadratic inverse. Much of the nontrivial dynamics of this map occurs when its two fixed points are saddle-foci with...

Computingmaximum a posteriori (MAP) estimation in graphical models is an important inference problem with many applications. We present message-passing algorithms for quadratic programming (QP) formulations of MAP estimation for pairwise Markov random fields. In particular, we use the concaveconvex procedure (CCCP) to obtain a locally optimal algorithm for the non-convex QP formulation. A simil...