C1-Robustly Minimal IFS with Three Generators

Let M be a compact connected m-dimensional manifold. We denote by Diff(M) the space of all C1-diffeomorphisms from M to itself endowed with C1-topology. For a collection of diffeomorphisms L = {f1, . . . , fn} ⊂ Diff(M), the iterated function system (abbrivately IFS) G(M ; f1, . . . , fn) on M generated by L is given by iterates fi1o · · · ofik with ij ∈ {1, . . . , n}. Recall that an IFS G(M ; f1, . . . , fn) is called minimal if each closed subset X ⊂ M such that fi(X) ⊂ X, for all i, is empty or coincide with M . Gorodetski and Il’yashenko [2000] provided an example of a robust minimal iterated function system on the circle with two generators. In [Ghane et al., 2010], the first author with Homburg and Sarizadeh constructed an example of C1-robustly minimal IFS on each m-dimensional compact connected manifoldM withm+3 generators. They also proved that on the m-dimensional torus Tm and on compact surfaces, robust minimal IFSs with only two generators exist. This means that, in these cases the number of generators is optimal. Note that, ifM is a compact connected two-dimensional manifold and f : M → M is a continuous map and K ⊆ M is a minimal set of (M,f) then either K = M or K is a nowhere dense subset of M . Moreover, the former case is possible only if M is a torus or a Klein bottle [Blokh et al., 2005]. On manifolds of dimension greater than two, a general theorem by Katok [1972] and Fathi and Herman [1977] ties the existence of minimal diffeomorphisms to the existence of locally free diffeomorphisms. The classification of compact m-manifolds, m ≥ 3, admitting minimal maps is an open problem. These observations motivate to raise the question of the minimal number of generators of C1-robustly minimal IFS, as stated in [Ghane et al., 2010]. In this note, we provide a C1-robustly minimal IFS on any m-dimensional compact connected manifold with only three generators which improves the main result of [Ghane et al., 2010, Theorem 1.1]. Now, we formulate our main result as follows: