Iterated Functions Systems, Blenders, and Parablenders

Abstract

We recast the notion of parablender introduced in [B2] as a parametric IFS. This is done using the concept of open covering property and looking to parametric IFS as systems acting on jets. Iterated Functions SystemsDefinition 1. A (contracting) Iterated Functions System (IFS) is the data of a finite family (f b ) b∈B of contracting maps on R n . The IFS is of class C r , r ≥ 1, if each f b is of class C r . * This work is partially supported by the project BRNUH of Sorbonne Paris Cité University and the French-Brazilian network. arXiv:1603.01241v2 [math.DS] 30 May 2016The topology on the set of IFS of class C r (with Card B elements) is given by the product strong topology B C r (R n , R n ). The limit set of an IFS is:The limit set Λ is compact. One is usually interested in its geometry. Natural questions are:Question 2. Under which condition the limit set Λ has non-empty interior?Under which condition the limit set has C r -robustly non empty interior ?Let us recall that a system satisfies a property C r -robustly if the property holds also for any C r -perturbations of the system. Both questions are still open, although there are already partial answers to them. Let us state a classical sufficient property: Definition 3. The IFS (f b ) b∈B satisfies the covering property if there exists a non-empty open set U of R n such that: Closure(U ) ⊂ b∈B

Correction to: Generic family with robustly infinitely many sinks

“…Proof that (H 1 ) − (H 2 ) implies the fundamental property By proceeding as in the proof of Thm B. [1], (H 1 ) implies the existence of a C d -curve of points (Q a ) a in (W u loc (δ; f a )) a such that:…”

“…Let us define a category of C d -parablenders containing those of [1,2]. To this end, put I e = [−1, 1], Y e = I e × I e , ∂ s Y e = ∂ I e × I e and ∂ u Y e = I e × ∂ I e .…”

Homoclinic tangencies leading to robust heterodimensional cycles

“…Blenders with larger central dimensions were introduced in [37,8] to study instability problems in symplectic dynamics, in [6] to obtain robust heterodimensional cycles of large coindex, and in [7,2] to get robust tangencies of large codimension. Blenders of large central dimension also appear in the study of ergodicity of conservative partial hyperbolic systems [4], of holomorphic dynamics [51,26,13], and of parametric families of maps (endomorphisms in [11], [12] and diffeomorphisms [9]).…”

Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus

“…One can easily show that if IFS(F) satisfies the covering property with the open set D, then the limit set of the IFS contains D in C 1 -robust fashion (see [10]). In the following we extend this result to non-uniform contracting IFSs.…”

“…. , f 2m } defined in (11) can be followed by the inclusion given by (10). Hence, we must choose the mapping g in such a way that its largest eigenvalue λ 2 is close enough to 1.…”