When does iterating $z \to\mapsto z^2 + c$ have an exact solution?

If one iterates the map $z \to z^2 + c$$z \mapsto z^2 + c$ there is obviously a simple formula for the sequence one gets if $c=0$. Less obviously, there is also a simple formula when $c = -2$ (use the identity $2 \cos(2x) = (2\cos(x))^2 - 2)$. Are there any other values of $c$ for which one can solve this recurrence explicitly? (For all initial values of course: there are many trivial explicit solutions for special initial values, such as fixed points.)

Related links:
http://en.wikipedia.org/wiki/Mandelbrot_set (the points $c$ where 0 remains bounded under iteration of this map: this strongly suggests that there is no simple exact solution for general $c$).
http://en.wikipedia.org/wiki/Logistic_map (gives the explicit solutions above, after a change of variable)

Motivation: I once used the map with $c=-2$ in a lecture to show that one could prove limits exist even without a formula for the exact solution. A first year calculus student pointed out the non-obvious exact solution above, and I don't want to be caught out like this again.

When does iterating $z \to z^2 + c$ have an exact solution?

If one iterates the map $z \to z^2 + c$ there is obviously a simple formula for the sequence one gets if $c=0$. Less obviously, there is also a simple formula when $c = -2$ (use the identity $2 \cos(2x) = (2\cos(x))^2 - 2)$. Are there any other values of $c$ for which one can solve this recurrence explicitly? (For all initial values of course: there are many trivial explicit solutions for special initial values, such as fixed points.)

Related links:
http://en.wikipedia.org/wiki/Mandelbrot_set (the points $c$ where 0 remains bounded under iteration of this map: this strongly suggests that there is no simple exact solution for general $c$).
http://en.wikipedia.org/wiki/Logistic_map (gives the explicit solutions above, after a change of variable)

Motivation: I once used the map with $c=-2$ in a lecture to show that one could prove limits exist even without a formula for the exact solution. A first year calculus student pointed out the non-obvious exact solution above, and I don't want to be caught out like this again.

When does iterating $z \mapsto z^2 + c$ have an exact solution?

If one iterates the map $z \mapsto z^2 + c$ there is obviously a simple formula for the sequence one gets if $c=0$. Less obviously, there is also a simple formula when $c = -2$ (use the identity $2 \cos(2x) = (2\cos(x))^2 - 2)$. Are there any other values of $c$ for which one can solve this recurrence explicitly? (For all initial values of course: there are many trivial explicit solutions for special initial values, such as fixed points.)

Related links:
http://en.wikipedia.org/wiki/Mandelbrot_set (the points $c$ where 0 remains bounded under iteration of this map: this strongly suggests that there is no simple exact solution for general $c$).
http://en.wikipedia.org/wiki/Logistic_map (gives the explicit solutions above, after a change of variable)

Motivation: I once used the map with $c=-2$ in a lecture to show that one could prove limits exist even without a formula for the exact solution. A first year calculus student pointed out the non-obvious exact solution above, and I don't want to be caught out like this again.

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edit approved Sep 9, 2018 at 13:44

Glorfindel

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When does iterating z ->$z \to z^2 + cc$ have an exact solution?

If one iterates the map z -> z^2 + c$z \to z^2 + c$ there is obviously a simple formula for the sequence one gets if c=0$c=0$. Less obviously, there is also a simple formula when c = -2$c = -2$ (use the identity 2 cos(2x) = (2cos(x))^2 - 2)$2 \cos(2x) = (2\cos(x))^2 - 2)$. Are there any other values of c$c$ for which one can solve this recurrence explicitly? (For all initial values of course: there are many trivial explicit solutions for special initial values, such as fixed points.)

Related links:
http://en.wikipedia.org/wiki/Mandelbrot_set (the points c$c$ where 0 remains bounded under iteration of this map: this strongly suggests that there is no simple exact solution for general c$c$).
http://en.wikipedia.org/wiki/Logistic_map (gives the explicit solutions above, after a change of variable)

Motivation: I once used the map with c=-2$c=-2$ in a lecture to show that one could prove limits exist even without a formula for the exact solution. A first year calculus student pointed out the non-obvious exact solution above, and I don't want to be caught out like this again.

When does iterating z -> z^2 + c have an exact solution?

If one iterates the map z -> z^2 + c there is obviously a simple formula for the sequence one gets if c=0. Less obviously, there is also a simple formula when c = -2 (use the identity 2 cos(2x) = (2cos(x))^2 - 2). Are there any other values of c for which one can solve this recurrence explicitly? (For all initial values of course: there are many trivial explicit solutions for special initial values, such as fixed points.)

Related links:
http://en.wikipedia.org/wiki/Mandelbrot_set (the points c where 0 remains bounded under iteration of this map: this strongly suggests that there is no simple exact solution for general c).
http://en.wikipedia.org/wiki/Logistic_map (gives the explicit solutions above, after a change of variable)

Motivation: I once used the map with c=-2 in a lecture to show that one could prove limits exist even without a formula for the exact solution. A first year calculus student pointed out the non-obvious exact solution above, and I don't want to be caught out like this again.

When does iterating $z \to z^2 + c$ have an exact solution?

If one iterates the map $z \to z^2 + c$ there is obviously a simple formula for the sequence one gets if $c=0$. Less obviously, there is also a simple formula when $c = -2$ (use the identity $2 \cos(2x) = (2\cos(x))^2 - 2)$. Are there any other values of $c$ for which one can solve this recurrence explicitly? (For all initial values of course: there are many trivial explicit solutions for special initial values, such as fixed points.)

Related links:
http://en.wikipedia.org/wiki/Mandelbrot_set (the points $c$ where 0 remains bounded under iteration of this map: this strongly suggests that there is no simple exact solution for general $c$).
http://en.wikipedia.org/wiki/Logistic_map (gives the explicit solutions above, after a change of variable)

Motivation: I once used the map with $c=-2$ in a lecture to show that one could prove limits exist even without a formula for the exact solution. A first year calculus student pointed out the non-obvious exact solution above, and I don't want to be caught out like this again.