MathDB
Smooth map can not avoid fixed points for all large n

Source: Alibaba Global Math Competition 2021, Problem 12

July 4, 2021
geometrytopologyFixed point

Problem Statement

Let A=(aij)A=(a_{ij}) be a 5×55 \times 5 matrix with aij=min⁡{i,j}a_{ij}=\min\{i,j\}. Suppose f:R5→R5f:\mathbb{R}^5 \to \mathbb{R}^5 is a smooth map such that f(Σ)⊂Σf(\Sigma) \subset \Sigma, where Σ={x∈R5:xAxT=1}\Sigma=\{x \in \mathbb{R}^5: xAx^T=1\}. Denote by f(n)f^{(n)} te nn-th iterate of ff. Prove that there does not exist N≥1N \ge 1 such that inf⁡x∈Σ∥f(n)(x)−x∥>0,∀n≥N.\inf_{x \in \Sigma} \| f^{(n)}(x)-x\|>0, \forall n \ge N.
Back to ProblemsView on AoPS