Exponential pathways

Simon Salamon

Dal lectio magistralis al Castello del Valentino, 4/12/2007Turin to Dulwich

Contents

Arithmetic (5 slides)

Bifurcation (5 slides)

Complex numbers (4 slides)

Dynamics (4 pictures)

Three operations and three numbers A1

m = 12

= 0.5 satisfies m + m = 1

The mean m (of 0 and 1) is rational

r =√

2 satisfies r × r = 2

The root r = 1.4142 . . . is irrational but algebraic

s = †2 satisfies ss = 2

The super-root s = 1.5596 . . . is transcendental

Roots and reciprocals A2

Let s = †2 and let u =1

s

⇒ us =(1

s

)s

=1

ss= m

⇒ usu = (us)u = mu

⇒ u = mu

How do we solve this equation for u?

Power towers A3

Consider the function f (x) = mx . Start with x = 1 and keep

applying f . For example, after 5 times, we get m(m(m(mm)))

If we apply f infinitely often we get a quantity

u = mmmmmmm↗

that should satisfy mu = u

Does it work?

Convergence A4

m↗2 = mm = 1r

= 0.707 . . .

m↗3 = m(mm) = 0.6125 . . .

m↗4 = m(m(mm)) = 0.6540 . . .

m↗5 = m(m(m(mm))) = 0.63549 . . .

m↗6 = 0.64371864172286913 . . .

m↗8 = 0.64168580704299834 . . .

m↗10 = 0.64128450906658502 . . .

m↗20 = 0.64118577420003478 . . .

. . . to a limit A5

m↗30 = 0.64118574451391446 . . .

m↗40 = 0.64118574450498867 . . .

m↗50 = 0.64118574450498599 . . .

m↗60 = 0.64118574450498598 . . .

In fact,

m↗∞ = limn→∞

m↗n = 0.64118574450498598 . . .

exists, and

s =1

u= 1.5596104694623693 . . .

satisfies ss = 2

Euler’s theorem B1

The limit

x↗∞ = limn→∞

x↗n

exists if and only if

1/ee 6 x 6 e1/e

which roughly means

0.065988 6 x 6 1.44466786

Inverse functions B2

y = x↗∞ ⇔ y = xy

⇔ x = y 1/y

Trouble near zero B3

It is well known that xx → 1 as x → 0

To see this, set y = 1/x . Then

log(xx) = x log x =− log y

y→ 0 as y →∞

It follows that x (xx ) → 01 = 0 as x → 0

By induction, x↗n → 1 as x → 0 if n is even

x↗n → 0 as x → 0 if n is odd

The bifurcation B4

Trademarks B5

Universal onset of chaos C1

A picture like this is seen in many different physical situations,representing period doubling, tripling and ultimately chaos

The Argand circle C2

To make sense of xx for x < 0 one introduces i =√−1

This is an irrational number that satisfies e it = cos t+i sin t

Complex exponents C3

One value of i i is

(e iπ/2)i = e−π/2 = 0.20788 . . .

but there are infinitely others (all transcendental)

Exponential iteration C4

If x = −t with t > 0 then

xx = (t e iπ)−t = t−te−iπt = t−t(cos πt − i sin πt)

We graph y = xx by plotting in space the points

(−t, t−tcosπt, −t−t sin πt)

One can continue by computing x (xx ) , x (x

(xx )) , . . .

Two trifurcations D1

Revealed by the graph of x↗400 for −0.11 6 x 6 0.09

But what are the legs hiding?

Wild dynamics D2

The graphs of x↗480, x↗482, x↗484 between theplane x = −0.0845 and x = −.075

A slim watch spring D3

The graph of x↗507 near x = −0.08055 forms a waveform ofthickness 0.00002 but diameter 8000

A galactic spiral D4

The graph of x↗508 near x = −0.08055 has diameter 1065