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corrected the range of f
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Stefan Kiefer
Stefan Kiefer
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Consider the function $f : \mathbb{R} \to [0,1]$$f : \mathbb{R} \to [-1,1]$ with $$ f(x) = \begin{cases} -1 & x \le -1 \\ +1 & x \ge +1 \\ \frac{f(\frac32 (x-\frac13)) + f(\frac32 (x+\frac13))}{2} & -1 \le x \le +1\,. \end{cases} $$ So $f$ is the average of two affine transformations of itself. The picture shows the graph of $f$ (in red) and the two affine transformations from the definition (in blue). function and two affine transformations What is known about such functions? Is $f(0.5) \approx 0.694064$ rational?

Consider the function $f : \mathbb{R} \to [0,1]$ with $$ f(x) = \begin{cases} -1 & x \le -1 \\ +1 & x \ge +1 \\ \frac{f(\frac32 (x-\frac13)) + f(\frac32 (x+\frac13))}{2} & -1 \le x \le +1\,. \end{cases} $$ So $f$ is the average of two affine transformations of itself. The picture shows the graph of $f$ (in red) and the two affine transformations from the definition (in blue). function and two affine transformations What is known about such functions? Is $f(0.5) \approx 0.694064$ rational?

Consider the function $f : \mathbb{R} \to [-1,1]$ with $$ f(x) = \begin{cases} -1 & x \le -1 \\ +1 & x \ge +1 \\ \frac{f(\frac32 (x-\frac13)) + f(\frac32 (x+\frac13))}{2} & -1 \le x \le +1\,. \end{cases} $$ So $f$ is the average of two affine transformations of itself. The picture shows the graph of $f$ (in red) and the two affine transformations from the definition (in blue). function and two affine transformations What is known about such functions? Is $f(0.5) \approx 0.694064$ rational?

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Stefan Kiefer
Stefan Kiefer
  • 165
  • 4

function that is the average of affine transformations of itself

Consider the function $f : \mathbb{R} \to [0,1]$ with $$ f(x) = \begin{cases} -1 & x \le -1 \\ +1 & x \ge +1 \\ \frac{f(\frac32 (x-\frac13)) + f(\frac32 (x+\frac13))}{2} & -1 \le x \le +1\,. \end{cases} $$ So $f$ is the average of two affine transformations of itself. The picture shows the graph of $f$ (in red) and the two affine transformations from the definition (in blue). function and two affine transformations What is known about such functions? Is $f(0.5) \approx 0.694064$ rational?