Editing Collatz conjecture (section)

===Iterating on real or complex numbers{{anchor|Collatz_fractal}}===
[[File:Collatz Cobweb.svg|thumb|[[Cobweb plot]] of the orbit 10 → 5 → 8 → 4 → 2 → 1 → ... in an extension of the Collatz map to the real line.]]

The Collatz map can be extended to the [[real line]] by choosing any function which evaluates to <math>x/2</math> when <math>x</math> is an even integer, and to either <math>3x + 1</math> or <math>(3x + 1)/2</math> (for the "shortcut" version) when <math>x</math> is an odd integer. This is called an [[interpolating]] function. A simple way to do this is to pick two functions <math>g_1</math> and <math>g_2</math>, where:
:<math>g_1(n) = \begin{cases}1, &n\text{ is even,}\\ 0, &n\text{ is odd,}\end{cases}</math>
:<math>g_2(n) = \begin{cases}0, &n\text{ is even,}\\1, &n\text{ is odd,}\end{cases}</math>
and use them as switches for our desired values:
:<math>f(x) \triangleq \frac{x}{2}\cdot g_1(x) \,+\, \frac{3x + 1}{2}\cdot g_2(x)</math>.
One such choice is <math>g_1(x) \triangleq \cos^2\left(\tfrac{\pi}{2} x\right)</math> and <math>g_2(x) \triangleq \sin^2\left(\tfrac{\pi}{2} x\right)</math>. The [[iterations]] of this map lead to a [[dynamical system]], further investigated by Marc Chamberland.<ref name="Chamberland (1996)"/> He showed that the conjecture does not hold for positive real numbers since there are infinitely many [[Fixed point (mathematics)|fixed points]], as well as [[Orbit (dynamics)|orbits]] escaping [[monotonic function|monotonically]] to infinity. The function <math>f</math> has two [[attractor|attracting]] cycles of period <math>2</math>: <math>(1;\,2)</math> and <math>(1.1925...;\,2.1386...)</math>. Moreover, the set of unbounded orbits is conjectured to be of [[Lebesgue measure|measure]] <math>0</math>.

Letherman, Schleicher, and Wood extended the study to the [[complex plane]].<ref name="Letherman, Schleicher, and Wood (1999)"/> They used Chamberland's function for [[Trigonometric_functions#In_the_complex_plane|complex sine and cosine]] and added the extra term <math>\tfrac{1}{\pi}\left(\tfrac12 - \cos(\pi z)\right)\sin(\pi z)\,+</math>
<math>h(z)\sin^2(\pi z)</math>, where <math>h(z)</math> is any [[entire function]]. Since this expression evaluates to zero for real integers, the extended function
:<math>\begin{align}f(z) \triangleq \;&\frac{z}{2}\cos^2\left(\frac{\pi}{2} z\right) + \frac{3z + 1}{2}\sin^2\left(\frac{\pi}{2} z\right) \, + \\
 &\frac{1}{\pi}\left(\frac12 - \cos(\pi z)\right)\sin(\pi z) + h(z)\sin^2(\pi z)\end{align}</math>

is an interpolation of the Collatz map to the complex plane. The reason for adding the extra term is to make all integers [[Critical point (mathematics)|critical points]] of <math>f</math>. With this, they show that no integer is in a [[Classification_of_Fatou_components#Baker_domain|Baker domain]], which implies that any integer is either eventually periodic or belongs to a [[wandering set|wandering domain]]. They conjectured that the latter is not the case, which would make all integer orbits finite.

[[File:Collatz Fractal.jpg|thumb|left|A Collatz [[fractal]] centered at the origin, with real parts from –5 to 5.]]

Most of the points have orbits that diverge to infinity. Coloring these points based on how fast they diverge produces the image on the left, for <math>h(z) \triangleq 0</math>. The inner black regions and the outer region are the [[Classification of Fatou components|Fatou components]], and the boundary between them is the [[Julia set]] of <math>f</math>, which forms a [[fractal]] pattern, sometimes called a "Collatz fractal".

[[File:Exponential Collatz Fractal.jpg|thumb|right|Julia set of the exponential interpolation.]]

There are many other ways to define a complex interpolating function, such as using the [[Exponential_function#Complex_plane|complex exponential]] instead of sine and cosine:
:<math>f(z) \triangleq \frac{z}{2} + \frac14(2z + 1)\left(1 - e^{i\pi z}\right)</math>,
which exhibit different dynamics. In this case, for instance, if <math>\operatorname{Im}(z) \gg 1</math>, then <math>f(z) \approx z + \tfrac14</math>. The corresponding Julia set, shown on the right, consists of uncountably many curves, called ''hairs'', or ''rays''.

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