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As is well known, a horseshoe map, i.e. a special injective reimbedding of the unit square $I^{2}$ in $ℝ^{2}$ (or more generally, of the cube $I^{m}$ in $ℝ^{m}$ ) as considered first by S. Smale [5], defines a shift dynamics on the maximal invariant subset of $I^{2}$ (or $I^{m}$ ). It is shown that this remains true almost surely for noninjective maps provided the contraction rate of the mapping in the stable direction is sufficiently strong, and bounds for this rate are given.

Simple and complex dynamics for circle maps.

Lluís Alsedà, Vladimir Fedorenko (1993)

Publicacions Matemàtiques

The continuous self maps of a closed interval of the real line with zero topological entropy can be characterized in terms of the dynamics of the map on its chain recurrent set. In this paper we extend this characterization to continuous self maps of the circle. We show that, for these maps, the chain recurrent set can exhibit a new dynamic behaviour which is specific of the circle maps of degree one.

Solution to the gradient problem of C.E. Weil.

Zoltán Buczolich (2005)

Revista Matemática Iberoamericana

In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set G ⊂ R2 we construct a differentiable function f: G → R for which there exists an open set Ω1 ⊂ R2 such that ∇f(p) ∈ Ω1 for a p ∈ G but ∇f(q) ∉ Ω1 for almost every q ∈ G. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.