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Metric space

Source: IMS 2014 - Day1 - Problem4

October 2, 2014
functionlimittopologyreal analysisreal analysis unsolved

Problem Statement

Let (X,d)(X,d) be a metric space and f:X→Xf:X \to X be a function such that ∀x,y∈X:d(f(x),f(y))=d(x,y)\forall x,y\in X : d(f(x),f(y))=d(x,y). a)\text{a}) Prove that for all x∈Xx \in X, lim⁡n→+∞d(x,fn(x))n\lim_{n \rightarrow +\infty} \frac{d(x,f^n(x))}{n} exists, where fn(x)f^n(x) is f(f(⋯f(x)⏟ntimes⋯ ))\underbrace{f(f(\cdots f(x)}_{n \text{times}} \cdots )). b)\text{b}) Prove that the amount of the limit does not depend on choosing xx.
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