The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the...
Transcript of The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the...
![Page 1: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/1.jpg)
The Fractal Boundary for the PowerTower Function
Peter LynchSchool of Mathematics & Statistics
University College Dublin
Recreational Mathematics Colloquium VMuseu Nacional de História Natural e da Ciência
Lisbon, 28-31 January 2017
![Page 2: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/2.jpg)
Outline
Introduction
Some Sample Values
Iterative Process
The Lambert W-Function
The Imaginary Power Tower
Asymptotic Behaviour
Power Tower Fractal
Conclusion
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 3: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/3.jpg)
Outline
Introduction
Some Sample Values
Iterative Process
The Lambert W-Function
The Imaginary Power Tower
Asymptotic Behaviour
Power Tower Fractal
Conclusion
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 4: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/4.jpg)
The Power Tower FunctionWe look at the function of x defined by
y(x) = xxx...
It would seem that when x > 1 this must blow up.Amazingly, this is not so.
In fact, the function converges for values
exp(−e) < x < exp(1/e)
or approximately
0.066 < x < 1.445
We call this function the power tower function.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 5: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/5.jpg)
The Power Tower FunctionWe look at the function of x defined by
y(x) = xxx...
It would seem that when x > 1 this must blow up.Amazingly, this is not so.
In fact, the function converges for values
exp(−e) < x < exp(1/e)
or approximately
0.066 < x < 1.445
We call this function the power tower function.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 6: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/6.jpg)
The Power Tower FunctionWe look at the function of x defined by
y(x) = xxx...
It would seem that when x > 1 this must blow up.Amazingly, this is not so.
In fact, the function converges for values
exp(−e) < x < exp(1/e)
or approximately
0.066 < x < 1.445
We call this function the power tower function.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 7: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/7.jpg)
Let us consider the sequence of approximations
{y1, y2, y3, . . . } ={
x , xx , xxx, . . .
}
We note the convention
xxx ≡ x (xx ) and not xxx= (xx )
x= xx2
.
Thus, the tower is constructed downwards.
It should really be denoted as
y(x) = ...xxx
as each new x is adjoined to the bottom of the tower.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 8: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/8.jpg)
Let us consider the sequence of approximations
{y1, y2, y3, . . . } ={
x , xx , xxx, . . .
}We note the convention
xxx ≡ x (xx ) and not xxx= (xx )
x= xx2
.
Thus, the tower is constructed downwards.
It should really be denoted as
y(x) = ...xxx
as each new x is adjoined to the bottom of the tower.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 9: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/9.jpg)
Let us consider the sequence of approximations
{y1, y2, y3, . . . } ={
x , xx , xxx, . . .
}We note the convention
xxx ≡ x (xx ) and not xxx= (xx )
x= xx2
.
Thus, the tower is constructed downwards.
It should really be denoted as
y(x) = ...xxx
as each new x is adjoined to the bottom of the tower.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 10: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/10.jpg)
Up and Down Values for x = 3
Let’s evaluate an example upwards and downwards:(33)3
= 273 = 19,683 3(33) = 327 = 7.6256× 1012
IT IS ESSENTIAL TO EVALUATE DOWNWARDS
MNEMONIC: Think of ex2
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 11: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/11.jpg)
Up and Down Values for x = 3
Let’s evaluate an example upwards and downwards:(33)3
= 273 = 19,683 3(33) = 327 = 7.6256× 1012
IT IS ESSENTIAL TO EVALUATE DOWNWARDS
MNEMONIC: Think of ex2
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 12: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/12.jpg)
Up and Down Values for x = 3
Let’s evaluate an example upwards and downwards:(33)3
= 273 = 19,683 3(33) = 327 = 7.6256× 1012
IT IS ESSENTIAL TO EVALUATE DOWNWARDS
MNEMONIC: Think of ex2
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 13: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/13.jpg)
Outline
Introduction
Some Sample Values
Iterative Process
The Lambert W-Function
The Imaginary Power Tower
Asymptotic Behaviour
Power Tower Fractal
Conclusion
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 14: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/14.jpg)
Sample Values
We evaluate the sequence{x , xx , xxx
, . . .}
for several particular values of x .
We will see that we may getI Convergence to a finite value.I Divergence to infinity.I Oscillation between two or more values.I More irregular (chaotic) behaviour (?).
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 15: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/15.jpg)
Sample Values
We evaluate the sequence{x , xx , xxx
, . . .}
for several particular values of x .
We will see that we may getI Convergence to a finite value.I Divergence to infinity.I Oscillation between two or more values.I More irregular (chaotic) behaviour (?).
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 16: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/16.jpg)
Sample Values: x = 1For x = 1, every term in the sequence is equal to 1.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 17: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/17.jpg)
Sample Values: x = 114
For x = 1.25, the values in the sequence grow:
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 18: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/18.jpg)
Sample Values: x = 125
For x = 1.40, the values grow to a larger value:
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 19: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/19.jpg)
Sample Values: x =√
2For x =
√2, the values grow to y = 2.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 20: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/20.jpg)
Sample Values: x = 112
For x = 1.5, the terms appear to grow without limit.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 21: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/21.jpg)
Sample Values: x = 12
For x = 0.5, we see oscillating behaviour, converging.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 22: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/22.jpg)
Sample Values: x = 110
For x = 0.1, we again see oscillating behaviour
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 23: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/23.jpg)
Sample Values: x = 120
For x = 0.05, convergence is less obvious.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 24: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/24.jpg)
Sample Values: x = 120
In fact, there is oscillation, no convergence.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 25: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/25.jpg)
Behaviour for Large and Small x
It is clear thatlim
x→∞xx =∞
So for large x the power tower function diverges.
What is less obvious is that
limx→0
xx = 1
This accounts for the counter-intuitive behaviourof the power tower for very small x.
For small x , alternate terms are close to 0 and to 1,so the sequence oscillates and does not converge.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 26: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/26.jpg)
Behaviour for Large and Small x
It is clear thatlim
x→∞xx =∞
So for large x the power tower function diverges.
What is less obvious is that
limx→0
xx = 1
This accounts for the counter-intuitive behaviourof the power tower for very small x.
For small x , alternate terms are close to 0 and to 1,so the sequence oscillates and does not converge.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 27: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/27.jpg)
Behaviour for Large and Small x
It is clear thatlim
x→∞xx =∞
So for large x the power tower function diverges.
What is less obvious is that
limx→0
xx = 1
This accounts for the counter-intuitive behaviourof the power tower for very small x.
For small x , alternate terms are close to 0 and to 1,so the sequence oscillates and does not converge.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 28: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/28.jpg)
Behaviour for Small x
y = xx for x ∈ [0,1]. Minimum at x = 1/e ≈ 0.368
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 29: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/29.jpg)
Outline
Introduction
Some Sample Values
Iterative Process
The Lambert W-Function
The Imaginary Power Tower
Asymptotic Behaviour
Power Tower Fractal
Conclusion
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 30: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/30.jpg)
Iterative ProcessIf the power tower function is to have any meaning,we need to show that it has well-defined values.
We consider the iterative process
y1 = x yn+1 = xyn .
This generates the infinite sequence
{y1, y2, y3, . . . } ={
x , xx , xxx, . . .
}
If the sequence converges to y = y(x), it follows that
y = xy
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 31: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/31.jpg)
Iterative ProcessIf the power tower function is to have any meaning,we need to show that it has well-defined values.
We consider the iterative process
y1 = x yn+1 = xyn .
This generates the infinite sequence
{y1, y2, y3, . . . } ={
x , xx , xxx, . . .
}If the sequence converges to y = y(x), it follows that
y = xy
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 32: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/32.jpg)
But y = xy leads to an explicit expression for x:
x = y1/y
Taking the derivative of this function we get
dxdy
=
(1− log y
y2
)x
which vanishes when log y = 1 or y = e.At this point, x = exp(1/e).
Moreover, it is easily shown that
limy→0
x = 0 and limy→∞
x = 1
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 33: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/33.jpg)
But y = xy leads to an explicit expression for x:
x = y1/y
Taking the derivative of this function we get
dxdy
=
(1− log y
y2
)x
which vanishes when log y = 1 or y = e.At this point, x = exp(1/e).
Moreover, it is easily shown that
limy→0
x = 0 and limy→∞
x = 1
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 34: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/34.jpg)
Plot of x = y1/y
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 35: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/35.jpg)
Plot of x = y1/y
We plotted the function x = y1/y above.
I It is defined for all positive y .I Its derivative vanishes at y = e where it
takes its maximum value exp(1/e).I It is monotone increasing on the interval (0,e)
and has an inverse function on this interval.I This inverse is the power tower function:
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 36: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/36.jpg)
Power tower function for x < exp(1/e).
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 37: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/37.jpg)
Iterative Solution
The logarithm of y = xy gives log y = y log x .
That is
y = exp(y log x) or y = exp(ξy)
where ξ = log x .
This is suited for iterative solution:given a value of x (or ξ), we seek a value y such thatthe graph of exp(ξy) intersects the diagonal line y = y .
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 38: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/38.jpg)
Iterative Solution
The logarithm of y = xy gives log y = y log x .
That is
y = exp(y log x) or y = exp(ξy)
where ξ = log x .
This is suited for iterative solution:given a value of x (or ξ), we seek a value y such thatthe graph of exp(ξy) intersects the diagonal line y = y .
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 39: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/39.jpg)
Starting from some value y(0) we iterate:
y(1) = exp(ξy(0)) , . . . y(n+1) = exp(ξy(n))
We graph exp(ξy) for selected of values of ξ.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 40: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/40.jpg)
Starting from some value y(0) we iterate:
y(1) = exp(ξy(0)) , . . . y(n+1) = exp(ξy(n))
We graph exp(ξy) for selected of values of ξ.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 41: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/41.jpg)
x ∈ [e−e,e1/e] =⇒ ξ ∈ [−e,1/e]
I For ξ < 0, corresponding to x < 1,there is a single root (top left panel).
I For 0 < ξ < 1/e (that is, for 1 < x < e1/e),there are two roots (top right panel).
I For ξ = 1/e (x = e1/e), there isone double root (bottom left panel).
I Finally, for ξ > 1/e (x > e1/e), there areno roots (bottom right panel).
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 42: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/42.jpg)
Graphs of y & exp(ξy) for some values ξ
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 43: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/43.jpg)
Graphs of y & exp(ξy) for some values ξWe compute iterations of:
y(n+1) = exp(ξy(n))
The iterative method converges only if the derivative
ddy
exp(ξy) = ξy
of the right side has modulus less than unity.
This criterion is satisfied for −e < ξ < 0, and also forthe smaller of the two roots when 0 < ξ < 1/e.
We therefore expect to obtain a single solution for−e < ξ < 1/e or exp(−e) < x < exp(1/e).
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 44: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/44.jpg)
Outline
Introduction
Some Sample Values
Iterative Process
The Lambert W-Function
The Imaginary Power Tower
Asymptotic Behaviour
Power Tower Fractal
Conclusion
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 45: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/45.jpg)
Swiss mathematician Johann Heinrich Lambert(1728–1777) introduced a function that is of widevalue and importance.
The Lambert W-function is the inverse ofz = w exp(w):
w = W (z) ⇐⇒ z = w exp(w) .
A plot of w = W (z) is presented below.
We confine attention to real values of W (z),which means that z ≥ −1/e.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 46: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/46.jpg)
Figure : Lambert W-function w = W (z). The inverse ofz = w exp(w).
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 47: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/47.jpg)
Applications of the W-Function
MATHEMATICSI Transcendental
equations.I Solving differential
equations.I In combinatorics.I Delay differential
equations.I Iterated exponentials.I Asymptotics.
PHYSICSI Analysis of
algorithms.I Water waves.I Combustion
problems.I Population growth.I Eigenstates of H2
molecule.I Quantum gravity.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 48: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/48.jpg)
Power Tower Function and W
For the Power Tower Function, x in terms of y is:
x = y1/y
This is well defined for all positive y .
Its inverse has a branch point at (x , y) = (e1/e,e).
If ξ = log x we have y = exp(ξy). We can write
(−ξy) exp(−ξy) = (−ξ)
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 49: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/49.jpg)
Power Tower Function and W
For the Power Tower Function, x in terms of y is:
x = y1/y
This is well defined for all positive y .
Its inverse has a branch point at (x , y) = (e1/e,e).
If ξ = log x we have y = exp(ξy). We can write
(−ξy) exp(−ξy) = (−ξ)
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 50: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/50.jpg)
We now define z = −ξ and w = −ξy and havez = w exp(w). By the definition of the LambertW-function, this is
w = W (z)
Returning to variables x and y , we conclude that
y =W (− log x)
− log x
which is the expression for the power tower functionin terms of the Lambert W-function.
This enables analytical continuation of the powertower function to the complex plane.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 51: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/51.jpg)
The relationship between the power tower functionand the Lambert W-function allows us to extend thepower tower function to the complex plane.
The function has a logarithmic branch point at x = 0.
The behaviour of the different branches of theW-function are described in [Corless96].
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 52: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/52.jpg)
Outline
Introduction
Some Sample Values
Iterative Process
The Lambert W-Function
The Imaginary Power Tower
Asymptotic Behaviour
Power Tower Fractal
Conclusion
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 53: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/53.jpg)
We now examine the PTF for complex z.
Specifically, we look at the case z = i:
q = i i i...
.
The first few terms of the sequence are
q1 = i q2 = i i = e−π/2 . . . qn+1 = iqn . . .
Assuming the sequence {qn} converges to Q,
Q = iQ
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 54: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/54.jpg)
We now examine the PTF for complex z.
Specifically, we look at the case z = i:
q = i i i...
.
The first few terms of the sequence are
q1 = i q2 = i i = e−π/2 . . . qn+1 = iqn . . .
Assuming the sequence {qn} converges to Q,
Q = iQ
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 55: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/55.jpg)
We now examine the PTF for complex z.
Specifically, we look at the case z = i:
q = i i i...
.
The first few terms of the sequence are
q1 = i q2 = i i = e−π/2 . . . qn+1 = iqn . . .
Assuming the sequence {qn} converges to Q,
Q = iQ
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 56: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/56.jpg)
Again,Q = iQ
Writing Q = %exp(iϑ) it follows that
ϑ tanϑ = log[π
2cosϑϑ
]and % =
2π
ϑ
cosϑ
This is easily solved to give
Q = (0.438283,0.360592)
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 57: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/57.jpg)
Here we show thesequence {qn}.
The points spiral aroundthe limit point Q, converg-ing towards it.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 58: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/58.jpg)
The points qn fall intothree distinct sets.
Three logarithmic spiralsare superimposed on theplot.
Is this pattern accidental?
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 59: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/59.jpg)
Outline
Introduction
Some Sample Values
Iterative Process
The Lambert W-Function
The Imaginary Power Tower
Asymptotic Behaviour
Power Tower Fractal
Conclusion
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 60: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/60.jpg)
Asymptotic Behaviour
We fitted a logarithmic spiral to the sequence {zn(i)}.
The points of the sequence were close tosuch a curve but did not lie exactly upon it.
Therefore, we looked at the asymptotic behaviourof the sequence for large n.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 61: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/61.jpg)
We consider the specific case z = i and supposethat zn = (1 + ε)Z where ε is small.
Then we find that zn+1 = Z ε · Z so that(zn+1 − Zzn − Z
)=
(Z ε − 1ε
).
By L’Hôpital’s rule, the limit of the right-hand sideas ε→ 0 is log Z . Thus for small ε (large n) we have
(zn+1 − Z ) ≈ log Z · (zn − Z )
and the sequence of differences {zn+k − Z}lies approximately on a logarithmic spiral
zn+k ≈ Z + (log Z )k · (zn − Z ) .
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 62: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/62.jpg)
Logarithmic Spiral
{zn(i)} for n ≥ 30.
Points zn(i) spiralaround the limit point(0.438283,0.360592)
The logarithmic spiralgives an excellent fit.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 63: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/63.jpg)
Supernumerary Spirals
Same sequence of points.
Points zn fall intothree sets.
Three logarithmicspirals superimposed.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 64: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/64.jpg)
The Asymptotic Spiral
The three “supernumerary spirals” are no accident.
Such spirals are familiar in many contexts.
In the seeds of a sun-flower, clockwiseand anti-clockwise spirals are evident.
By changing the parameter z it is possible to tune thelimit Z (z) to have spirals of a particular shape.
Patterns like this also found in pursuit problems.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 65: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/65.jpg)
A Pursuit Problem
Three ships initially at thevertices of an equilateraltriangle.
Each bears towards itscounter-clockwise neigh-bour.
Three spiral arms aretraced out.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 66: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/66.jpg)
Outline
Introduction
Some Sample Values
Iterative Process
The Lambert W-Function
The Imaginary Power Tower
Asymptotic Behaviour
Power Tower Fractal
Conclusion
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 67: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/67.jpg)
We can construct a beautiful fractal set using thePower Tower Function with complex arguments.
Repeated exponentiation is called tetration and thefractal is sometimes called the tetration fractal.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 68: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/68.jpg)
We examine the behaviour of the (tetration) function
∞z = zzz...
I For some values of z this converges.I For other values it is periodic.I For others, it “escapes” to infinity.
The boundary of the region for which the function isfinite is fractal. Let Π be the set for which ∞z is finite.
The “escape set” is the complement of this set.
The boundary of the set Π is exquisitely complex.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 69: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/69.jpg)
We examine the behaviour of the (tetration) function
∞z = zzz...
I For some values of z this converges.I For other values it is periodic.I For others, it “escapes” to infinity.
The boundary of the region for which the function isfinite is fractal. Let Π be the set for which ∞z is finite.
The “escape set” is the complement of this set.
The boundary of the set Π is exquisitely complex.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 70: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/70.jpg)
Figure : The power tower fractal for |x | < 10, |y | < 10.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 71: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/71.jpg)
Figure : The power tower fractal for |x | < 4, |y | < 4.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 72: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/72.jpg)
Figure : PTF for −3.25 ≤ x ≤ 0.25, −1.75 ≤ y ≤ 1.75.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 73: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/73.jpg)
Figure : PTF for −3.25 ≤ x ≤ 0.25, −1.75 ≤ y ≤ 1.75.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 74: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/74.jpg)
Figure : PTF for −0.525 ≤ x ≤ 0.225, −0.375 ≤ y ≤ 0.375.
A marine creature. Let’s call it the lobster.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 75: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/75.jpg)
Figure : PTF for −0.23 ≤ x ≤ −0.13, +0.2 ≤ y ≤ 0.3.
Antenna of the lobster.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 76: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/76.jpg)
Figure : PTF for −0.193 ≤ x ≤ −0.183, +0.23 ≤ y ≤ 0.24.
Spiral structure in the antenna.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 77: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/77.jpg)
Images from Website of Paul Bourke
http://paulbourke.net/fractals/tetration/
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 78: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/78.jpg)
Figure : Center = (-0.5,0.0), range = 9.0
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 79: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/79.jpg)
Figure : Center = (-1.9,0.0), range = 3.0
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 80: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/80.jpg)
Figure : Center = (-0.25,0.0), range = 0.8
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 81: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/81.jpg)
Figure : Center = (2.2,-2.5), range = 2.0
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 82: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/82.jpg)
Figure : Center = (2.15,-0.91), range = 0.5
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 83: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/83.jpg)
Figure : Center = (-2.37,-0.38), range = 0.5
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 84: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/84.jpg)
Figure : Center = (-0.94,0.41), range = 0.2
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 85: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/85.jpg)
Figure : Center = (-0.95,2.4), range = 0.1
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 86: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/86.jpg)
Figure : Center = (0.4,2.0), range = 0.2
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 87: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/87.jpg)
Outline
Introduction
Some Sample Values
Iterative Process
The Lambert W-Function
The Imaginary Power Tower
Asymptotic Behaviour
Power Tower Fractal
Conclusion
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 88: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/88.jpg)
Conclusion
Zooming in can be continued indefinitely,revealing ever more sturcture.
The fine details at any resolution are not reliable.
Structures that appear to be disjoint may beconnected by fine filaments that are visibleonly at higher resolution.
It is necessary to set the escape radius to a very largevalue (e.g. rmax = 1048) and allow many iterations.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 89: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/89.jpg)
Much more may be said about the power tower fractal.
Fixed points, for which ∞z = z. Clearly, z = 1 andz = −1 are fixed points.
Periodic orbits (see http://www.tetration.org/)
Sarkovskii’s Theorem implies that a map containingperiod three must contain all periods from one toinfinity.
Many other interesting questions to be answered.
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin
![Page 90: The Fractal Boundary for the Power Tower Functionplynch/Talks/PowerTower.pdf · So for large x the power tower function diverges. What is less obvious is that lim x!0 xx = 1 This](https://reader035.fdocuments.us/reader035/viewer/2022062611/612df5c61ecc515869428354/html5/thumbnails/90.jpg)
Thank you
Intro Values Iterations W-function Imag-z Asymptotics Fractal Fin