Editing Collatz conjecture (section)

==Extensions to larger domains==

===Iterating on all integers===
An extension to the Collatz conjecture is to include all integers, not just positive integers. Leaving aside the cycle 0 → 0 which cannot be entered from outside, there are a total of four known cycles, which all nonzero integers seem to eventually fall into under iteration of {{mvar|f}}. These cycles are listed here, starting with the well-known cycle for positive&nbsp;{{mvar|n}}:

Odd values are listed in large bold. Each cycle is listed with its member of least absolute value (which is always odd) first.

{| class="wikitable" style="text-align: center;"
! Cycle !! Odd-value cycle length !! Full cycle length
|-
|style="text-align: left;"| <big>'''1'''</big> → 4 → 2 → <big>'''1'''</big> '''...''' || 1 || 3
|-
|style="text-align: left;"| <big>'''−1'''</big> → −2 → <big>'''−1'''</big> '''...''' || 1 || 2
|-
|style="text-align: left;"| <big>'''−5'''</big> → −14 → <big>'''−7'''</big> → −20 → −10 → <big>'''−5'''</big> '''...''' || 2 || 5
|-
|style="text-align: left;"| <big>'''−17'''</big> → −50 → <big>'''−25'''</big> → −74 → <big>'''−37'''</big> → −110 → <big>'''−55'''</big> → −164 → −82 → <big>'''−41'''</big> → −122 → <big>'''−61'''</big> → −182 → <big>'''−91'''</big> → −272 → −136 → −68 → −34 → <big>'''−17'''</big> '''...''' || 7 || 18
|}

The generalized Collatz conjecture is the assertion that every integer, under iteration by {{mvar|f}}, eventually falls into one of the four cycles above or the cycle 0 → 0.

=== Iterating on rationals with odd denominators ===
The Collatz map can be extended to (positive or negative) rational numbers which have odd denominators when written in lowest terms.
The number is taken to be 'odd' or 'even' according to whether its numerator is odd or even. Then the formula for the map is exactly the same as when the domain is the integers: an 'even' such rational is divided by 2; an 'odd' such rational is multiplied by 3 and then 1 is added. A closely related fact is that the Collatz map extends to the ring of [[2-adic integers]], which contains the ring of rationals with odd denominators as a subring.

When using the "shortcut" definition of the Collatz map, it is known that any periodic [[#As a parity sequence|parity sequence]] is generated by exactly one rational.<ref>{{Cite journal|last=Lagarias|first=Jeffrey|date=1990|title=The set of rational cycles for the 3x+1 problem|url=https://eudml.org/doc/206298|journal=Acta Arithmetica|volume=56|issue=1|pages=33–53|issn=0065-1036|doi=10.4064/aa-56-1-33-53|doi-access=free}}</ref> Conversely, it is conjectured that every rational with an odd denominator has an eventually cyclic parity sequence (Periodicity Conjecture<ref name="Lagarias (1985)"/>).

If a parity cycle has length {{mvar|n}} and includes odd numbers exactly {{mvar|m}} times at indices {{math|''k''<sub>0</sub> < ⋯ < ''k''<sub>''m''−1</sub>}}, then the unique rational which generates immediately and periodically this parity cycle is
{{NumBlk|:|<math>\frac{3^{m-1} 2^{k_0} + \cdots + 3^0 2^{k_{m-1}}}{2^n - 3^m}.</math>|{{EquationRef|1}}}}

For example, the parity cycle {{nowrap|(1 0 1 1 0 0 1)}} has length 7 and four odd terms at indices 0, 2, 3, and 6. It is repeatedly generated by the fraction
<math display="block">\frac{3^3 2^0 + 3^2 2^2 + 3^1 2^3 + 3^0 2^6}{2^7 - 3^4} = \frac{151}{47}</math>
as the latter leads to the rational cycle
<math display="block">\frac{151}{47} \rightarrow \frac{250}{47} \rightarrow \frac{125}{47} \rightarrow \frac{211}{47} \rightarrow \frac{340}{47} \rightarrow \frac{170}{47} \rightarrow \frac{85}{47} \rightarrow \frac{151}{47} .</math>

Any cyclic permutation of {{nowrap|(1 0 1 1 0 0 1)}} is associated to one of the above fractions. For instance, the cycle {{nowrap|(0 1 1 0 0 1 1)}} is produced by the fraction
<math display="block">\frac{3^3 2^1 + 3^2 2^2 + 3^1 2^5 + 3^0 2^6}{2^7 - 3^4} = \frac{250}{47} . </math>

For a one-to-one correspondence, a parity cycle should be ''irreducible'', that is, not partitionable into identical sub-cycles. As an illustration of this, the parity cycle {{nowrap|(1 1 0 0 1 1 0 0)}} and its sub-cycle {{nowrap|(1 1 0 0)}} are associated to the same fraction {{sfrac|5|7}} when reduced to lowest terms.

In this context, assuming the validity of the Collatz conjecture implies that {{nowrap|(1 0)}} and {{nowrap|(0 1)}} are the only parity cycles generated by positive whole numbers (1 and 2, respectively).

If the odd denominator {{mvar|d}} of a rational is not a multiple of 3, then all the iterates have the same denominator and the sequence of numerators can be obtained by applying the "{{math|3''n'' + ''d''}}&nbsp;" generalization<ref name="Belaga (1998a)"/> of the Collatz function
<math display="block"> T_d(x) = \begin{cases}
 \frac{x}{2} &\text{if } x \equiv 0 \pmod{2},\\
 \frac{3x+d}{2} & \text{if } x\equiv 1 \pmod{2}.
\end{cases}</math>

===2-adic extension===
The function
<math display="block"> T(x) = \begin{cases} \frac{x}{2} &\text{if } x \equiv 0 \pmod{2}\\ \frac{3x+1}{2} & \text{if } x\equiv 1 \pmod{2} \end{cases}</math>
is well-defined on the ring <math>\mathbb{Z}_2</math> of [[2-adic integers]], where it is continuous and [[Measure-preserving transformation|measure-preserving]] with respect to the 2-adic measure. Moreover, its dynamics is known to be [[Ergodic theory|ergodic]].<ref name="Lagarias (1985)"/>

Define the ''parity vector'' function {{mvar|Q}} acting on <math>\mathbb{Z}_2</math> as
<math display="block"> Q(x) = \sum_{k=0}^{\infty} \left( T^k (x) \mod 2 \right) 2^k .</math>

The function {{mvar|Q}} is a 2-adic [[isometry]].<ref>{{Cite journal|last1=Bernstein|first1=Daniel J.|last2=Lagarias|first2=Jeffrey C.|date=1996|title=The 3''x'' + 1 conjugacy map|journal=[[Canadian Journal of Mathematics]]|language=en|volume=48|issue=6|pages=1154–1169|doi=10.4153/CJM-1996-060-x|doi-access=free|issn=0008-414X}}</ref> Consequently, every infinite parity sequence occurs for exactly one 2-adic integer, so that [[almost all]] trajectories are acyclic in <math>\mathbb{Z}_2</math>.

An equivalent formulation of the Collatz conjecture is that
<math display="block"> Q\left(\mathbb{Z}^{+}\right) \subset \tfrac13 \mathbb{Z}.</math>

===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''.

{{clear}}