We discuss main properties of the dynamics on minimal attraction centers (σ-limit sets) of single trajectories for continuous maps of a compact metric space into itself. We prove that each nowhere dense nonvoid closed set in ${ℝ}^n$, n ≥ 1, is a σ-limit set for some continuous map.

On appelle échange d’intervalles l’application qui consiste à réordonner les intervalles d’une partition de $[0,1[$ suivant une permutation donnée. Dans le cas des partitions en trois intervalles, nous donnons une caractérisation combinatoire des suites codant, d’après la partition définissant l’échange, l’orbite d’un point de $[0,1[$ sous l’action de cette transformation.

In this paper we address the following question due to Marcy Barge: For what f:I → I is it the case that the inverse limit of I with single bonding map f can be embedded in the plane so that the shift homeomorphism $\widehat{f}$ extends to a diffeomorphism ([BB, Problem 1.5], [BK, Problem 3])? This question could also be phrased as follows: Given a map f:I → I, find a diffeomorphism $F:{ℝ}^2\to {ℝ}^2$ so that F restricted to its full attracting set, ${\bigcap }_{k\ge 0}{F}^k\left({ℝ}^2\right)$, is topologically conjugate to $\widehat{f}:(I,f)\to (I,f)$. In this situation, we say that the inverse...

Let f be a continuous map of the circle $S^1$ or the interval I into itself, piecewise $C^1$, piecewise monotone with finitely many intervals of monotonicity and having positive entropy h. For any ε > 0 we prove the existence of at least $e^{(h-\epsilon )n_k}$ periodic points of period $n_k$ with large derivative along the period, $|{\left(f^{n_k}\right)}^{\text{'}}|>e^{(h-\epsilon )n_k}$ for some subsequence $n_k$ of natural numbers. For a strictly monotone map f without critical points we show the existence of at least $(1-\epsilon )e^{hn}$ such points.