Let (X, d) be a metric space and T: X → X a continuous map. If the sequence (T n)n∈ℕ of iterates of T is pointwise convergent in X, then for any x ∈ X, the limit $${\mu }_T\left(x\right)=\underset{n\to \infty }{lim}{T}^n\left(x\right)$$ is a fixed point of T. The problem of determining the form of µT leads to the invariance equation µT ○ T = µT, which is difficult to solve in general if the set of fixed points of T is not a singleton. We consider this problem assuming that X = I p, where I is a real interval, p ≥ 2 a fixed positive integer and T is the mean-type mapping...

Let ℳ be the set of pairs (T,g) such that T ⊂ ℝ is compact, g: T → T is continuous, g is minimal on T and has a piecewise monotone extension to convT. Two pairs (T,g),(S,f) from ℳ are equivalent if the map h: orb(minT,g) → orb(minS,f) defined for each m ∈ ℕ₀ by $h\left(g^m\left(minT\right)\right)=f^m\left(minS\right)$ is increasing on orb(minT,g). An equivalence class of this relation-a minimal (oriented) pattern A-is exhibited by a continuous interval map f:I → I if there is a set T ⊂ I such that (T,f|T) = (T,f) ∈ A. We define the forcing relation on...