sample slides - qmul mathsfvivaldi/teaching/rmms/sampleslidesrmms… · sample slides franco...
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![Page 1: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/1.jpg)
Sample slidesFranco Vivaldi
![Page 2: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/2.jpg)
Hamiltonian stabilityover discrete spaces
Franco VivaldiQueen Mary, University of London
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Smooth area-preserving maps
![Page 4: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/4.jpg)
integrable
Smooth area-preserving maps
![Page 5: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/5.jpg)
integrable
foliation by invariant curves
Smooth area-preserving maps
![Page 6: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/6.jpg)
near-integrable: stableintegrable
foliation by invariant curves
Smooth area-preserving maps
![Page 7: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/7.jpg)
KAM curves
near-integrable: stableintegrable
foliation by invariant curves
Smooth area-preserving maps
![Page 8: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/8.jpg)
KAM curves
near-integrable: stableintegrable strong perturbation: unstable
foliation by invariant curves
Smooth area-preserving maps
![Page 9: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/9.jpg)
KAM curves
near-integrable: stableintegrable strong perturbation: unstable
foliation by invariant curves
What happens if the space is discrete?
Smooth area-preserving maps
![Page 10: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/10.jpg)
KAM curves
near-integrable: stableintegrable strong perturbation: unstable
foliation by invariant curves
What happens if the space is discrete?
?
Smooth area-preserving maps
![Page 11: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/11.jpg)
Some methods for discretizing space
![Page 12: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/12.jpg)
Some methods for discretizing space
Truncation (computer arithmetic).
![Page 13: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/13.jpg)
Some methods for discretizing space
Truncation (computer arithmetic).
Restricting coordinates to a discrete field (ring, module).
![Page 14: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/14.jpg)
Some methods for discretizing space
Truncation (computer arithmetic).
Restricting coordinates to a discrete field (ring, module).
Geometric discretization.
![Page 15: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/15.jpg)
Some methods for discretizing space
Truncation (computer arithmetic).
Restricting coordinates to a discrete field (ring, module).
Geometric discretization.
Reduction to a finite field.
![Page 16: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/16.jpg)
Early investigations:
F. Rannou (1974)
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Early investigations:
F. Rannou (1974)
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Early investigations:
First study of an invertible lattice map on the torus
F. Rannou (1974)
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Early investigations:
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
First study of an invertible lattice map on the torus
F. Rannou (1974)
![Page 20: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/20.jpg)
Early investigations:
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
First study of an invertible lattice map on the torus
F. Rannou (1974)
lattice size
![Page 21: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/21.jpg)
Early investigations:
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
First study of an invertible lattice map on the torus
rounding to nearest integer
F. Rannou (1974)
lattice size
![Page 22: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/22.jpg)
Early investigations:
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
First study of an invertible lattice map on the torus
rounding to nearest integer
F. Rannou (1974)
lattice size
![Page 23: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/23.jpg)
Early investigations:
All orbits are periodic.
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
First study of an invertible lattice map on the torus
rounding to nearest integer
F. Rannou (1974)
lattice size
![Page 24: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/24.jpg)
Early investigations:
All orbits are periodic.
Orbit representing curves develop some“thickness”, but remain stable.
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
First study of an invertible lattice map on the torus
rounding to nearest integer
F. Rannou (1974)
lattice size
![Page 25: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/25.jpg)
Early investigations:
All orbits are periodic.
Orbit representing curves develop some“thickness”, but remain stable.
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
First study of an invertible lattice map on the torus
rounding to nearest integer
F. Rannou (1974)
lattice size
why?
![Page 26: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/26.jpg)
Renormalizationin parametrised families
of polygon-exchange transformations
Franco VivaldiQueen Mary, University of London
with J H Lowenstein (New York)
![Page 27: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/27.jpg)
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
![Page 28: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/28.jpg)
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
�� �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
![Page 29: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/29.jpg)
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
�� �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
If F is invertible, then F is volume-preserving.
![Page 30: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/30.jpg)
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
�� �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:
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.
![Page 31: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/31.jpg)
Higher dimensions: topology
type I
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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
![Page 33: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/33.jpg)
Higher dimensions: topologyIterate the boundary of the atoms:
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 =
[
t2ZFt(∂W) P = W\D E = D\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
![Page 34: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/34.jpg)
Higher dimensions: topologyIterate the boundary of the atoms:
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 =
[
t2ZFt(∂W) P = W\D E = D\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
![Page 35: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/35.jpg)
Higher dimensions: topologyIterate the boundary of the atoms:
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 =
[
t2ZFt(∂W) P = W\D E = D\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
![Page 36: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/36.jpg)
Higher dimensions: topologyIterate the boundary of the atoms:
periodic set
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 =
[
t2ZFt(∂W) P = W\D E = D\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
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
![Page 37: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/37.jpg)
Higher dimensions: topologyIterate the boundary of the atoms:
periodic set
type I
discontinuity set
(union of cells of positive measure)
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 =
[
t2ZFt(∂W) P = W\D E = D\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
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
![Page 38: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/38.jpg)
Higher dimensions: topologyIterate the boundary of the atoms:
periodic set
exceptional set
type I
discontinuity set
(union of cells of positive measure)
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 =
[
t2ZFt(∂W) P = W\D E = D\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
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
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
![Page 39: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/39.jpg)
Higher dimensions: topologyIterate the boundary of the atoms:
periodic set
exceptional set
type I
discontinuity set
(union of cells of positive measure)
(asymptotic phenomena)
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 =
[
t2ZFt(∂W) P = W\D E = D\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
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
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
![Page 40: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/40.jpg)
One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
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One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
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
![Page 42: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/42.jpg)
One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
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
![Page 43: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/43.jpg)
One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
O O
rotate about 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
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One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
O OO
rotate about O translate back to (parameter-dependent)
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
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
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One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
quadratic rotation fields:
O OO
rotate about O translate back to (parameter-dependent)
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
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
![Page 46: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/46.jpg)
One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
quadratic rotation fields:
translation module:
O OO
rotate about O translate back to (parameter-dependent)
parameter: position of O along diagonal
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
![Page 47: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/47.jpg)
Geometric discretization: strip maps
![Page 48: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/48.jpg)
Geometric discretization: strip maps
A flow with a piecewise-constant vector field is diffracted by a line
![Page 49: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/49.jpg)
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
![Page 50: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/50.jpg)
Geometric discretization: strip maps
strip
A flow with a piecewise-constant vector field is diffracted by a line
replace the flow by a map, using
the same vectors
flow and map differ within a strip
![Page 51: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/51.jpg)
Geometric discretization: strip maps
strip
A flow with a piecewise-constant vector field is diffracted by a line
replace the flow by a map, using
the same vectors
flow and map differ within a strip
The lattice generated by the
vectors is invariant under the map
![Page 52: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/52.jpg)
Linked strip maps: outer billiards of polygons
![Page 53: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/53.jpg)
Linked strip maps: outer billiards of polygons
![Page 54: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/54.jpg)
Linked strip maps: outer billiards of polygons
![Page 55: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/55.jpg)
Linked strip maps: outer billiards of polygons
![Page 56: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/56.jpg)
Linked strip maps: outer billiards of polygons
![Page 57: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/57.jpg)
Linked strip maps: outer billiards of polygonspiecewise-constant
vector field
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strip
Linked strip maps: outer billiards of polygonspiecewise-constant
vector field
![Page 59: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/59.jpg)
strip
closed polygon (integrable orbit)
Linked strip maps: outer billiards of polygonspiecewise-constant
vector field
![Page 60: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/60.jpg)
strip
closed polygon (integrable orbit)
Linked strip maps: outer billiards of polygons
perturbed orbit
piecewise-constant vector field
![Page 61: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/61.jpg)
The arithmetic of chaos
Franco Vivaldi Queen Mary, University of London
![Page 62: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/62.jpg)
God
![Page 63: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/63.jpg)
‘‘God gave us the integers, the rest is the work of man”L Kronecker
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‘‘God gave us the integers, the rest is the work of man”L Kronecker
2
![Page 65: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/65.jpg)
0 1 2 3 4
the work of man
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0 1 2 3 4
the work of man
![Page 67: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/67.jpg)
0 1 2 3 4
the work of man
![Page 68: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/68.jpg)
0 1 2 3 4
the work of man
![Page 69: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/69.jpg)
0 1 2 3 4
the work of man
![Page 70: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/70.jpg)
computer programs to build numbers
print(123123123123123123123123123123123123123123123)
print(123) 15 times
![Page 71: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/71.jpg)
computer programs to build numbers
print(123123123123123123123123123123123123123123123)
print(123) 15 times
smart
dumb
![Page 72: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/72.jpg)
computer programs to build numbers
print(123123123123123123123123123123123123123123123)
print(123) 15 times
smart
dumb
print(3846264338327950288419716939937510582097494)
dumb
![Page 73: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/73.jpg)
computer programs to build numbers
print(123123123123123123123123123123123123123123123)
print(123) 15 times
smart
dumb
print(3846264338327950288419716939937510582097494)
dumb
smart ?
![Page 74: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/74.jpg)
random numbers
A number is random if the shortest program that can build its digits is the dumb program
Kolmogorov
![Page 75: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/75.jpg)
random numbers
A number is random if the shortest program that can build its digits is the dumb program
Kolmogorov
randomdumb program
output
![Page 76: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/76.jpg)
random numbers
A number is random if the shortest program that can build its digits is the dumb program
Kolmogorov
non-random
output
smart program
randomdumb program
output
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do numbers have mass?
0 1
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do numbers have mass?
0 1Lebesgue
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do numbers have mass?
1 Kg
0 1Lebesgue
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do numbers have mass?
1 Kg
0 1
145 g
Lebesgue
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do numbers have mass?
1 Kg
0 1
145 g
Lebesgue
The total mass of all fractions is zero
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DARK MATTER
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DARK MATTER
00
25
50
75
100
universe
visible dark
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DARK MATTER
00
25
50
75
100
universe
visible dark
00
25
50
75
100
real numbers
visible dark
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Theorem:
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Theorem: the random numbers account for thetotal mass of the real number system.
![Page 88: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/88.jpg)
Theorem: the random numbers account for thetotal mass of the real number system.
Random numbers are :dark
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Theorem: the random numbers account for thetotal mass of the real number system.
Random numbers are :
they can’t be defined individually, or talked about
dark
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Theorem: the random numbers account for thetotal mass of the real number system.
Random numbers are :
they can’t be defined individually, or talked about
they can’t be computed, or stored in computers
dark
![Page 91: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/91.jpg)
Theorem: the random numbers account for thetotal mass of the real number system.
Random numbers are :
they can’t be defined individually, or talked about
they can’t be computed, or stored in computers
they can’t be proved to be random
dark
![Page 92: Sample slides - QMUL Mathsfvivaldi/teaching/rmms/SampleSlidesRMMS… · Sample slides Franco Vivaldi. Hamiltonian stability over discrete spaces Franco Vivaldi Queen Mary, University](https://reader036.vdocuments.us/reader036/viewer/2022063013/5fcd267af324e30abd575c3f/html5/thumbnails/92.jpg)
Theorem: the random numbers account for thetotal mass of the real number system.
Random numbers are :
they can’t be defined individually, or talked about
they can’t be computed, or stored in computers
they can’t be proved to be random
Random numbers are not meant for humans.
dark
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Thank you for your attention