Related papers: Almost blenders and parablenders
Blenders are special hyperbolic sets used to produce various robust dynamical phenomena which appear fragile at first glance. We prove for $C^r$ diffeomorphisms ($r=2,\dots,\infty,\omega$) that blenders naturally exist (without…
We give an explicit family of polynomial maps called center unstable H\'enon-like maps and prove that they exhibits blenders for some parametervalues. Using this family, we also prove the occurrence of blenders near certain non-transverse…
A blender is a closed convex cone of real homogeneous polynomials that is also closed under linear changes of variable. Non-trivial blenders only occur in even degree. Examples include the cones of psd forms, sos forms, convex forms and…
Heterodimensional cycles are heteroclinic cycles that connect periodic orbits whose unstable manifolds have different dimensions. This is a source of nonhyperbolic dynamics and unstable dimension variability. For smooth invertible maps…
A blender-horseshoe is a locally maximal transitive hyperbolic set that appears in dimension at least three carrying a distinctive geometrical property: its local stable manifold "behaves" as a manifold of topological dimension greater than…
The aim of this paper is twofold. First, we introduce standard blenders (special hyperbolic sets) and their variations, and prove their fundamental properties on the generation of $C^1$-robust tangencies. In particular, these blenders…
We study partially-hyperbolic skew-product maps over the Bernoulli shift with H\"older dependence on the base points. In the case of contracting fiber maps, symbolic blender-horseshoe is defined as an invariant set which meets any almost…
In this paper, we study stable ergodicity of the action of groups of diffeomorphisms on smooth manifolds. Such actions are known to exist only on one-dimensional manifolds. The aim of this paper is to introduce a geometric method to…
We present a characterisation of blenders based on mapping properties of certain sets of curves that can be rigorously verified by computer-assisted methods. We develop an algorithm to construct these sets of curves that requires only a…
This paper is devoted to the study of a certain class of principal bundles on del Pezzo surfaces, which were introduced and studied by Friedman and Morgan in \cite{FMdP}: The two authors showed that there exists a unique principal bundle…
In this paper we show that if $p$ is a polynomial which bifurcates then the product map $(z,w)\mapsto(p(z),q(w))$ can be approximated by polynomial skew products possessing special dynamical objets called blenders. Moreover, these objets…
A blender is a hyperbolic basic set such that the projection of its stable/unstable set onto some center subspace has a higher topological dimension than the set itself. We prove that, for any $C^s$ symplectic diffeomorphism (where…
We study stable conditional measures for a certain equilibrium measure for hyperbolic endomorphisms, on basic sets with overlaps; we show that these conditional measures are geometric probabilities and measures of maximal stable dimension.…
We present a characterisation of a blender based on the topological alignment of certain sets in phase space in combination with cone conditions. Importantly, the required conditions can be verified by checking properties of a single…
We show that a homeomorphism of Euclidean space is quasiconformal if and only if at each point there exists a sequence of uncentered open sets with bounded eccentricity shrinking to that point whose images also have bounded eccentricity.…
In this work we obtain a new criterion to establish ergodicity and non-uniform hyperbolicity of smooth measures of diffeomorphisms. This method allows us to give a more accurate description of certain ergodic components. The use of this…
A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least…
A ring $R$ is said to be an almost Armendariz ring if whenever product of two polynomials in $R[x]$ is zero, then product of their coefficients are in $N_{*}(R)$. In this article, for an endomorphism $\alpha$ on $R$, we define an…
This article is mostly a writeup of two talks, the first given in the Besse Seminar at the Ecole Polytechnique in 1998 and the second given at the 2000 International Congress on Differential Geometry in memory of Alfred Gray in Bilbao,…
Given a locally maximal compact invariant hyperbolic set $\Lambda$ for a $C^1$ flow or diffeomorphism on a Riemann manifold with $C^1$ unstable laminations, we construct an invariant continuous bundle of tangent vectors to local unstable…