Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco...

Sample slidesFranco Vivaldi

Hamiltonian stabilityover discrete spaces

Franco VivaldiQueen Mary, University of London

Smooth area-preserving maps

integrable


integrable

foliation by invariant curves


near-integrable: stableintegrable



KAM curves

near-integrable: stableintegrable



KAM curves

near-integrable: stableintegrable strong perturbation: unstable



KAM curves



What happens if the space is discrete?


KAM curves



What happens if the space is discrete?

?


Some methods for discretizing space


Truncation (computer arithmetic).



Restricting coordinates to a discrete field (ring, module).




Geometric discretization.




Geometric discretization.

Reduction to a finite field.

Early investigations:

F. Rannou (1974)


First study of an invertible lattice map on the torus

F. Rannou (1974)


x

t+1

= x

t

+ y

t

+m

2p

✓1� cos

2pm

y

t

◆(mod m)

y

t+1

= y

t

� lm

2p

✓sin

2pm

x

t+1

+1� cos

2pm

x

t+1

◆(mod m)

1


F. Rannou (1974)


x

t+1

= x

t

+ y

t

+m

2p

✓1� cos

2pm

y

t

◆(mod m)

y

t+1

= y

t

� lm

2p

✓sin

2pm

x

t+1

+1� cos

2pm

x

t+1

◆(mod m)

1


F. Rannou (1974)

lattice size


x

t+1

= x

t

+ y

t

+m

2p

✓1� cos

2pm

y

t

◆(mod m)

y

t+1

= y

t

� lm

2p

✓sin

2pm

x

t+1

+1� cos

2pm

x

t+1

◆(mod m)

1


rounding to nearest integer

F. Rannou (1974)

lattice size


All orbits are periodic.

x

t+1

= x

t

+ y

t

+m

2p

✓1� cos

2pm

y

t

◆(mod m)

y

t+1

= y

t

� lm

2p

✓sin

2pm

x

t+1

+1� cos

2pm

x

t+1

◆(mod m)

1



F. Rannou (1974)

lattice size



Orbit representing curves develop some“thickness”, but remain stable.

x

t+1

= x

t

+ y

t

+m

2p

✓1� cos

2pm

y

t

◆(mod m)

y

t+1

= y

t

� lm

2p

✓sin

2pm

x

t+1

+1� cos

2pm

x

t+1

◆(mod m)

1



F. Rannou (1974)

lattice size




In the chaotic regions the map behaves like a random permutation.

x

t+1

= x

t

+ y

t

+m

2p

✓1� cos

2pm

y

t

◆(mod m)

y

t+1

= y

t

� lm

2p

✓sin

2pm

x

t+1

+1� cos

2pm

x

t+1

◆(mod m)

1



F. Rannou (1974)

lattice size




In the chaotic regions the map behaves like a random permutation.

x

t+1

= x

t

+ y

t

+m

2p

✓1� cos

2pm

y

t

◆(mod m)

y

t+1

= y

t

� lm

2p

✓sin

2pm

x

t+1

+1� cos

2pm

x

t+1

◆(mod m)

1



F. Rannou (1974)

lattice size

why?

Renormalizationin parametrised families

of polygon-exchange transformations

Franco VivaldiQueen Mary, University of London

with J H Lowenstein (New York)

Piecewise isometries

a finite collection of pairwise disjoint open polytopes (intersection of open half-spaces),called the atoms.

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

the space:

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1



�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

is an isometrythe dynamics:

the space:

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1



�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1


the space:

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

If F is invertible, then F is volume-preserving.



�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1


the space:

Theorem (Gutkin & Haydin 1997, Buzzi 2001)The topological entropy of a piecewise isometry is zero.

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

�� 11 0

⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2

� ⇥ Rn � =⇤

�i F : �⇤ � F |�i

1

If F is invertible, then F is volume-preserving.

Higher dimensions: topology

type I

Higher dimensions: topologyIterate the boundary of the atoms:

type I

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[

t2ZFt(∂W) P = W\D E = D\D

l = 2cos(2p p/q)

1


type I

discontinuity set

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1


type I

discontinuity set

not dense, typically

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1


periodic set

type I

discontinuity set


Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1


periodic set

type I

discontinuity set

(union of cells of positive measure)


Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1


periodic set

exceptional set

type I

discontinuity set



Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[

t2ZFt(∂W) P = W\D E = D \D

l = 2cos(2p p/q)

1


periodic set

exceptional set

type I

discontinuity set


(asymptotic phenomena)


Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[


l = 2cos(2p p/q)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1

W2

W3

W4

|W1

|, |W2

|, |W3

|, |W4

| K =Q(|W1

|, . . . , |Wn|)∂W =

[∂Wi D =

[

t2ZFt(∂W) P = W\D E = D \D

l = 2cos(2p p/q)

1

One-parameter families of polygon-exchange transformations

type I

Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)


type I


O

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)

1


type I


O

parameter: position of O along diagonal

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)

1


type I


O O

rotate about O


Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)

1


type I


O OO

rotate about O translate back to (parameter-dependent)


Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)

1


type I


quadratic rotation fields:

O OO



Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)

1


type I


quadratic rotation fields:

translation module:

O OO



s: parameter

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)

1

Wi W ⇢ Rn W =[

Wi F : W ! W F |Wi(1234) 7! (2431)

W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)

1

Geometric discretization: strip maps


A flow with a piecewise-constant vector field is diffracted by a line



replace the flow by a map, using

the same vectors


strip



the same vectors

flow and map differ within a strip


strip



the same vectors

flow and map differ within a strip

The lattice generated by the

vectors is invariant under the map

Linked strip maps: outer billiards of polygons

Linked strip maps: outer billiards of polygonspiecewise-constant

vector field

strip


vector field

strip

closed polygon (integrable orbit)


vector field

strip

closed polygon (integrable orbit)

Linked strip maps: outer billiards of polygons

perturbed orbit

piecewise-constant vector field

The arithmetic of chaos

Franco Vivaldi Queen Mary, University of London

‘‘God gave us the integers, the rest is the work of man”L Kronecker

‘‘God gave us the integers, the rest is the work of man”L Kronecker

2

0 1 2 3 4

the work of man

computer programs to build numbers

print(123123123123123123123123123123123123123123123)

print(123) 15 times


print(123123123123123123123123123123123123123123123)

print(123) 15 times

smart

dumb


print(123123123123123123123123123123123123123123123)

print(123) 15 times

smart

dumb

print(3846264338327950288419716939937510582097494)

dumb


print(123123123123123123123123123123123123123123123)

print(123) 15 times

smart

dumb

print(3846264338327950288419716939937510582097494)

dumb

smart ?

random numbers

A number is random if the shortest program that can build its digits is the dumb program

Kolmogorov

random numbers


Kolmogorov

randomdumb program

output

random numbers


Kolmogorov

non-random

output

smart program

randomdumb program

output

do numbers have mass?

0 1


0 1Lebesgue


1 Kg

0 1Lebesgue


1 Kg

0 1

145 g

Lebesgue


1 Kg

0 1

145 g

Lebesgue

The total mass of all fractions is zero

DARK MATTER

DARK MATTER

00

25

50

75

100

universe

visible dark

DARK MATTER

00

25

50

75

100

universe

visible dark

00

25

50

75

100

real numbers

visible dark

Theorem:

Theorem: the random numbers account for thetotal mass of the real number system.


Random numbers are :dark


Random numbers are :

they can’t be defined individually, or talked about

dark




they can’t be computed, or stored in computers

dark





they can’t be proved to be random

dark





they can’t be proved to be random

Random numbers are not meant for humans.

dark

Thank you for your attention

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