In the space of polynomial maps of R2 of degree at least two, there are codimension-3 laminations of maps with at least 3 period doubling Cantor attractors. The leafs of the laminations are real-analytic and they have uniform diameter. The closure of each lamination contains the codimension-one tangency locus of a saddle point. Asymptotically, the leafs of each lamination align with the leafs of the eigenvalue foliation. This is an example of general coexistence theorems valid for higher dimensional real-analytic unfoldings of homoclinic tangencies. This reveals further universal and global aspects of the bifurcation pattern.

We study the classical H & eacute;non family fa,b : (x, y) i? (1 - ax(2) + y, bx), 0 < a < 2, 0 < b < 1, and prove that given an integer k = 1, there is a set of parameters Ek of positive two-dimensional Lebesgue measure so that fa,b, for (a, b) ? E-k, has at least k attractive periodic orbits and one strange attractor of the type studied in Benedicks and Carleson (Ann Math (2) 133(1):73-169, 1991). A corresponding statement also holds for the H & eacute;non-like families of Mora and Viana (Acta Math 171:1-71, 1993), and we use the techniques of Mora and Viana (1993) to study homoclinic unfoldings also in the case of the original H & eacute;non maps. The final main result of the paper is the existence, within the classical H & eacute;non family, of a positive Lebesgue measure set of parameters whose corresponding maps have two coexisting strange attractors.

A general invariant manifold theorem is needed to study the topological classes of smooth dynamical systems. These classes are often invariant under renormalization. The classical invariant manifold theorem cannot be applied, because the renormalization operator for smooth systems is not differentiable and sometimes does not have an attractor. Examples are the renormalization operator for general smooth dynamics, such as unimodal dynamics, circle dynamics, Cherry dynamics, Lorenz dynamics, Henon dynamics, etc. A general method to construct invariant manifolds of non-differentiable non-linear operators is presented. An application is that the C4+epsilon Fibonacci Cherry maps form a C-1 codimension one manifold.

We consider generalized interval exchange transformations, or briefly GIETs, that is bijections of the interval which are piecewise increasing homeomorphisms with finitely many branches. When all continuous branches are translations, such maps are classical interval exchange transformations, or briefly IETs. The well-known Rauzy renormalization procedure extends to a given GIET and a Rauzy renormalization path is defined, provided that the map is infinitely renormalizable. We define full families of GIETs, that is optimal finite dimensional parameter families of GIETs such that any prescribed Rauzy renormalization path is realized by some map in the family. In particular, a GIET and a IET with the same Rauzy renormalization path are semi-conjugated. This extends a classical result of Poincaré relating circle homeomorphisms and irrational rotations.

A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps with a break point. More recent developments show that under similar topological conditions, rigidity does not hold for slightly more general systems. In this paper we state a conjecture which describes how topological classes are organized into rigidity classes.

Studies of the physical measures for Cherry flows were initiated in Saghin and Vargas (2013). While the non-positive divergence case was resolved, the positive divergence case still lacked a complete description. Some conjectures were put forward. In this paper we make a contribution in this direction. Namely, under mild technical assumptions we solve some conjectures stated in Saghin and Vargas (2013) by providing a description of the physical measures for Cherry flows in the positive divergence case.