Article Contents
Article Contents

# A generalization of the Babbage functional equation

• A recent refinement of Kerékjártó's Theorem has shown that in $\mathbb R$ and $\mathbb R^2$ all $\mathcal C^l$–solutions of the functional equation $f^n = \text{Id}$ are $\mathcal C^l$–linearizable, where $l\in \{0,1,\dots \infty\}$. When $l\geq 1$, in the real line we prove that the same result holds for solutions of $f^n = f$, while we can only get a local version of it in the plane. Through examples, we show that these results are no longer true when $l = 0$ or when considering the functional equation $f^n = f^k$ with $n>k\geq 2$.

Mathematics Subject Classification: Primary: 37C15, 39B12; Secondary: 30D05, 37C05, 37E05, 37E30, 39B22.

 Citation:

• Figure 1.  Graph of a generic idempotent continuous function in $\mathbb R$

Figure 2.  Graph of the function $f_\lambda$ with $\lambda = (0.11001\dots)_2$

Figure 3.  Graph of the function $f_\lambda$ with $\lambda = (0.11001\dots)_2$

Figure 4.  Concatenation of two overhand knots, one figure eight knot and one overhand knot

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