Bugs - ICERM - Home · October 7, 2012 1. Escher’s problem n ants on a M obius band Ant 1 chases...
Transcript of Bugs - ICERM - Home · October 7, 2012 1. Escher’s problem n ants on a M obius band Ant 1 chases...
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Bugs
Dmitri Gekhtman
October 7, 2012
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Escher’s problem
n ants on a Mobius band
Ant 1 chases ant 2, ant 2 chases bug 3, ant 4 chases
What happens?
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Escher’s problem
n ants on a Mobius band
Ant 1 chases ant 2, ant 2 chases bug 3, ant 4 chases
What happens?
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Lucas’s problem
Three bugs on the corners of an equilateral triangle and each onechases the next one at unit speed.
What happens (Lucas, 1877)?
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Lucas’s problem
Three bugs on the corners of an equilateral triangle and each onechases the next one at unit speed.
What happens (Lucas, 1877)?
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Solution
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Regular Hexagon
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Regular Decagon
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The general problem in Euclidean space
Nonsymmetric configurations of n bugs in Rm?
Bugs sweep out{bi : R+ → Rm}i∈Z/n.
Rule of motion
bi =bi+1 − bi‖bi+1 − bi‖
.
When bi catches bi+1, they stay together
Ends when all collide
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The general problem in Euclidean space
Nonsymmetric configurations of n bugs in Rm?
Bugs sweep out{bi : R+ → Rm}i∈Z/n.
Rule of motion
bi =bi+1 − bi‖bi+1 − bi‖
.
When bi catches bi+1, they stay together
Ends when all collide
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The general problem in Euclidean space
Nonsymmetric configurations of n bugs in Rm?
Bugs sweep out{bi : R+ → Rm}i∈Z/n.
Rule of motion
bi =bi+1 − bi‖bi+1 − bi‖
.
When bi catches bi+1, they stay together
Ends when all collide
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The general problem in Euclidean space
Nonsymmetric configurations of n bugs in Rm?
Bugs sweep out{bi : R+ → Rm}i∈Z/n.
Rule of motion
bi =bi+1 − bi‖bi+1 − bi‖
.
When bi catches bi+1, they stay together
Ends when all collide
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The general problem in Euclidean space
Nonsymmetric configurations of n bugs in Rm?
Bugs sweep out{bi : R+ → Rm}i∈Z/n.
Rule of motion
bi =bi+1 − bi‖bi+1 − bi‖
.
When bi catches bi+1, they stay together
Ends when all collide
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200 beetles
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Typical behavior
Beetles start out in a random configuration.
They form a nice knot shape
Knot shape undoes itself into a circular loop
Circular loop contracts to a point
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Typical behavior
Beetles start out in a random configuration.
They form a nice knot shape
Knot shape undoes itself into a circular loop
Circular loop contracts to a point
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Typical behavior
Beetles start out in a random configuration.
They form a nice knot shape
Knot shape undoes itself into a circular loop
Circular loop contracts to a point
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Typical behavior
Beetles start out in a random configuration.
They form a nice knot shape
Knot shape undoes itself into a circular loop
Circular loop contracts to a point
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100 beetles
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1000 beetles
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Finite time until collision
Proposition: Beetles all collide in finite time.Li (t) ≡ d(bi (t), bi+1(t)).
d
dtLi (t) = −1 + cos(θi ),
θi is i−th exterior angle of the piecewise geodesic path connectingthe bugsBorsuk (1947):
n∑i=1
|θi | > 2π,
so exists j for which |θj | ≥ 2πn .
d
dt
(n∑
i=1
Li (t)
)=
n∑i=1
cos(θi )− 1 ≤ cos(θj)− 1 ≤ cos
(2π
n
)− 1,
so process terminates in time less than or equal to∑n
i=1 Li (0)1−cos(2π/n−1) .
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Finite time until collision
Proposition: Beetles all collide in finite time.Li (t) ≡ d(bi (t), bi+1(t)).
d
dtLi (t) = −1 + cos(θi ),
θi is i−th exterior angle of the piecewise geodesic path connectingthe bugsBorsuk (1947):
n∑i=1
|θi | > 2π,
so exists j for which |θj | ≥ 2πn .
d
dt
(n∑
i=1
Li (t)
)=
n∑i=1
cos(θi )− 1 ≤ cos(θj)− 1 ≤ cos
(2π
n
)− 1,
so process terminates in time less than or equal to∑n
i=1 Li (0)1−cos(2π/n−1) .
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Finite time until collision
Proposition: Beetles all collide in finite time.Li (t) ≡ d(bi (t), bi+1(t)).
d
dtLi (t) = −1 + cos(θi ),
θi is i−th exterior angle of the piecewise geodesic path connectingthe bugsBorsuk (1947):
n∑i=1
|θi | > 2π,
so exists j for which |θj | ≥ 2πn .
d
dt
(n∑
i=1
Li (t)
)=
n∑i=1
cos(θi )− 1 ≤ cos(θj)− 1 ≤ cos
(2π
n
)− 1,
so process terminates in time less than or equal to∑n
i=1 Li (0)1−cos(2π/n−1) .
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Finite time until collision
Proposition: Beetles all collide in finite time.Li (t) ≡ d(bi (t), bi+1(t)).
d
dtLi (t) = −1 + cos(θi ),
θi is i−th exterior angle of the piecewise geodesic path connectingthe bugsBorsuk (1947):
n∑i=1
|θi | > 2π,
so exists j for which |θj | ≥ 2πn .
d
dt
(n∑
i=1
Li (t)
)=
n∑i=1
cos(θi )− 1 ≤ cos(θj)− 1 ≤ cos
(2π
n
)− 1,
so process terminates in time less than or equal to∑n
i=1 Li (0)1−cos(2π/n−1) .
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Asymptotic circularity and stable configurations
Question: When does it happen that shape of bug loop isasymptotically circular as we approach collapse time?
Partial answers from (Richardson 2001).
The only invariant configurations are regular n−gons.
The only locally attracting configurations are regular n− gons forn ≥ 7. (For n < 7, tend to get collapse to a line.)
Question: What is basin of attraction of stable configurations?
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Asymptotic circularity and stable configurations
Question: When does it happen that shape of bug loop isasymptotically circular as we approach collapse time?
Partial answers from (Richardson 2001).
The only invariant configurations are regular n−gons.
The only locally attracting configurations are regular n− gons forn ≥ 7. (For n < 7, tend to get collapse to a line.)
Question: What is basin of attraction of stable configurations?
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Asymptotic circularity and stable configurations
Question: When does it happen that shape of bug loop isasymptotically circular as we approach collapse time?
Partial answers from (Richardson 2001).
The only invariant configurations are regular n−gons.
The only locally attracting configurations are regular n− gons forn ≥ 7. (For n < 7, tend to get collapse to a line.)
Question: What is basin of attraction of stable configurations?
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Asymptotic circularity and stable configurations
Question: When does it happen that shape of bug loop isasymptotically circular as we approach collapse time?
Partial answers from (Richardson 2001).
The only invariant configurations are regular n−gons.
The only locally attracting configurations are regular n− gons forn ≥ 7. (For n < 7, tend to get collapse to a line.)
Question: What is basin of attraction of stable configurations?
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Bugs on manifolds
Manifold: Space that looks locally like Euclidean space
Riemannian manifold: Locally Euclidean space with extra structurefor measuring lengths, angles, and volumes.
Examples: Sphere, torus
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Bugs on manifolds
Manifold: Space that looks locally like Euclidean space
Riemannian manifold: Locally Euclidean space with extra structurefor measuring lengths, angles, and volumes.
Examples: Sphere, torus
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Bugs on manifolds
Manifold: Space that looks locally like Euclidean space
Riemannian manifold: Locally Euclidean space with extra structurefor measuring lengths, angles, and volumes.
Examples: Sphere, torus
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Geodesics
Straight line segments, paths of least resistance, paths followed byphysical particles.
On manifolds embedded in Rn, paths of zero tangential acceleration.
Example: On a sphere, great circle arcs.
Closed geodesics are smooth geodesic loops, periodic paths followedby physical particles.
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Geodesics
Straight line segments, paths of least resistance, paths followed byphysical particles.
On manifolds embedded in Rn, paths of zero tangential acceleration.
Example: On a sphere, great circle arcs.
Closed geodesics are smooth geodesic loops, periodic paths followedby physical particles.
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Geodesics
Straight line segments, paths of least resistance, paths followed byphysical particles.
On manifolds embedded in Rn, paths of zero tangential acceleration.
Example: On a sphere, great circle arcs.
Closed geodesics are smooth geodesic loops, periodic paths followedby physical particles.
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Geodesics
Straight line segments, paths of least resistance, paths followed byphysical particles.
On manifolds embedded in Rn, paths of zero tangential acceleration.
Example: On a sphere, great circle arcs.
Closed geodesics are smooth geodesic loops, periodic paths followedby physical particles.
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Bugs on compact manifolds
Place consecutive bugs close enough together so that there’s a uniquegeodesic connecting each one to the next
Give each velocity equal to the unit tangent to the geodesic.
Goal: understand what happens to the piecewise geodesic closed loopβt connecting consecutive beetles t →∞Problem is interesting on manifold because not all loops arecontractible
Conjecture: Bug loops which do not contract to a point converge toclosed geodesics.
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Bugs on compact manifolds
Place consecutive bugs close enough together so that there’s a uniquegeodesic connecting each one to the next
Give each velocity equal to the unit tangent to the geodesic.
Goal: understand what happens to the piecewise geodesic closed loopβt connecting consecutive beetles t →∞Problem is interesting on manifold because not all loops arecontractible
Conjecture: Bug loops which do not contract to a point converge toclosed geodesics.
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Bugs on compact manifolds
Place consecutive bugs close enough together so that there’s a uniquegeodesic connecting each one to the next
Give each velocity equal to the unit tangent to the geodesic.
Goal: understand what happens to the piecewise geodesic closed loopβt connecting consecutive beetles t →∞Problem is interesting on manifold because not all loops arecontractible
Conjecture: Bug loops which do not contract to a point converge toclosed geodesics.
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Bugs on compact manifolds
Place consecutive bugs close enough together so that there’s a uniquegeodesic connecting each one to the next
Give each velocity equal to the unit tangent to the geodesic.
Goal: understand what happens to the piecewise geodesic closed loopβt connecting consecutive beetles t →∞Problem is interesting on manifold because not all loops arecontractible
Conjecture: Bug loops which do not contract to a point converge toclosed geodesics.
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Bugs on compact manifolds
Place consecutive bugs close enough together so that there’s a uniquegeodesic connecting each one to the next
Give each velocity equal to the unit tangent to the geodesic.
Goal: understand what happens to the piecewise geodesic closed loopβt connecting consecutive beetles t →∞Problem is interesting on manifold because not all loops arecontractible
Conjecture: Bug loops which do not contract to a point converge toclosed geodesics.
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Subsequential convergence
Proposition: Exists (tj) going to ∞, and geodesic α so that βtj → α.
Li (t) ≡ d(bi (t), bi+1(t)),
L(t) =∑i
Li (t).
d
dtL(t) =
∑i
(−1 + cos(θi )) ,
θi is i−th exterior angle of the piecewise geodesic path connectingthe bugsSince Li (t) decreasing, bounded from below, exists subsequence(tj)∞j=1, tj →∞ s.t. d
dtL(tj)→ 0.Now, θi (tj)→ 0 for all i .Passing to subsequence, assume bi (tj) converges, say to ai . Let α bep.g. loop connecting ai .By continuity, θi = 0, so αi is a geodesic.
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Subsequential convergence
Proposition: Exists (tj) going to ∞, and geodesic α so that βtj → α.
Li (t) ≡ d(bi (t), bi+1(t)),
L(t) =∑i
Li (t).
d
dtL(t) =
∑i
(−1 + cos(θi )) ,
θi is i−th exterior angle of the piecewise geodesic path connectingthe bugsSince Li (t) decreasing, bounded from below, exists subsequence(tj)∞j=1, tj →∞ s.t. d
dtL(tj)→ 0.Now, θi (tj)→ 0 for all i .Passing to subsequence, assume bi (tj) converges, say to ai . Let α bep.g. loop connecting ai .By continuity, θi = 0, so αi is a geodesic.
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Subsequential convergence
Proposition: Exists (tj) going to ∞, and geodesic α so that βtj → α.
Li (t) ≡ d(bi (t), bi+1(t)),
L(t) =∑i
Li (t).
d
dtL(t) =
∑i
(−1 + cos(θi )) ,
θi is i−th exterior angle of the piecewise geodesic path connectingthe bugsSince Li (t) decreasing, bounded from below, exists subsequence(tj)∞j=1, tj →∞ s.t. d
dtL(tj)→ 0.Now, θi (tj)→ 0 for all i .Passing to subsequence, assume bi (tj) converges, say to ai . Let α bep.g. loop connecting ai .By continuity, θi = 0, so αi is a geodesic.
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Subsequential convergence
Proposition: Exists (tj) going to ∞, and geodesic α so that βtj → α.
Li (t) ≡ d(bi (t), bi+1(t)),
L(t) =∑i
Li (t).
d
dtL(t) =
∑i
(−1 + cos(θi )) ,
θi is i−th exterior angle of the piecewise geodesic path connectingthe bugsSince Li (t) decreasing, bounded from below, exists subsequence(tj)∞j=1, tj →∞ s.t. d
dtL(tj)→ 0.Now, θi (tj)→ 0 for all i .Passing to subsequence, assume bi (tj) converges, say to ai . Let α bep.g. loop connecting ai .By continuity, θi = 0, so αi is a geodesic.
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Subsequential convergence
Proposition: Exists (tj) going to ∞, and geodesic α so that βtj → α.
Li (t) ≡ d(bi (t), bi+1(t)),
L(t) =∑i
Li (t).
d
dtL(t) =
∑i
(−1 + cos(θi )) ,
θi is i−th exterior angle of the piecewise geodesic path connectingthe bugsSince Li (t) decreasing, bounded from below, exists subsequence(tj)∞j=1, tj →∞ s.t. d
dtL(tj)→ 0.Now, θi (tj)→ 0 for all i .Passing to subsequence, assume bi (tj) converges, say to ai . Let α bep.g. loop connecting ai .By continuity, θi = 0, so αi is a geodesic.
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Subsequential convergence
Proposition: Exists (tj) going to ∞, and geodesic α so that βtj → α.
Li (t) ≡ d(bi (t), bi+1(t)),
L(t) =∑i
Li (t).
d
dtL(t) =
∑i
(−1 + cos(θi )) ,
θi is i−th exterior angle of the piecewise geodesic path connectingthe bugsSince Li (t) decreasing, bounded from below, exists subsequence(tj)∞j=1, tj →∞ s.t. d
dtL(tj)→ 0.Now, θi (tj)→ 0 for all i .Passing to subsequence, assume bi (tj) converges, say to ai . Let α bep.g. loop connecting ai .By continuity, θi = 0, so αi is a geodesic.
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Convergence in a special case
If there is a unique closed geodesic α that has a subsequence (βtj )converging to, then βt converges to α.
Assume for the sake of contradiction that βt doesn’t converge to α.
Then there’s an ε and a sequence (tj)→∞ s.t. d(βtj , α) > ε for alltj .
Pass to subsequence so that βtj converges to a geodesic α′.
d(α′, α) > ε, so α 6= α′. Contradiction.
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Convergence in a special case
If there is a unique closed geodesic α that has a subsequence (βtj )converging to, then βt converges to α.
Assume for the sake of contradiction that βt doesn’t converge to α.
Then there’s an ε and a sequence (tj)→∞ s.t. d(βtj , α) > ε for alltj .
Pass to subsequence so that βtj converges to a geodesic α′.
d(α′, α) > ε, so α 6= α′. Contradiction.
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Convergence in a special case
If there is a unique closed geodesic α that has a subsequence (βtj )converging to, then βt converges to α.
Assume for the sake of contradiction that βt doesn’t converge to α.
Then there’s an ε and a sequence (tj)→∞ s.t. d(βtj , α) > ε for alltj .
Pass to subsequence so that βtj converges to a geodesic α′.
d(α′, α) > ε, so α 6= α′. Contradiction.
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Convergence in a special case
If there is a unique closed geodesic α that has a subsequence (βtj )converging to, then βt converges to α.
Assume for the sake of contradiction that βt doesn’t converge to α.
Then there’s an ε and a sequence (tj)→∞ s.t. d(βtj , α) > ε for alltj .
Pass to subsequence so that βtj converges to a geodesic α′.
d(α′, α) > ε, so α 6= α′. Contradiction.
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Convergence for nonpositively curved manifolds
Def: Tubular ε-nhd Nε(α) of geodesic α is {p ∈ M| infs d(p, α) < ε}.Let α be a geodesic to which subsequence βtj converges.
Fact: For ε small enough, Nε(α) is geodesically convex: ifp, q ∈ Nε(α), so are shortest geodesics connecting p, q.
Take any ε > 0 small enough that Nε(α) is convex. Find T so thatβT ⊂ Nε(α). By convexity, βt ⊂ Nε(α) for all t > T .
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Convergence for nonpositively curved manifolds
Def: Tubular ε-nhd Nε(α) of geodesic α is {p ∈ M| infs d(p, α) < ε}.Let α be a geodesic to which subsequence βtj converges.
Fact: For ε small enough, Nε(α) is geodesically convex: ifp, q ∈ Nε(α), so are shortest geodesics connecting p, q.
Take any ε > 0 small enough that Nε(α) is convex. Find T so thatβT ⊂ Nε(α). By convexity, βt ⊂ Nε(α) for all t > T .
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Convergence for nonpositively curved manifolds
Def: Tubular ε-nhd Nε(α) of geodesic α is {p ∈ M| infs d(p, α) < ε}.Let α be a geodesic to which subsequence βtj converges.
Fact: For ε small enough, Nε(α) is geodesically convex: ifp, q ∈ Nε(α), so are shortest geodesics connecting p, q.
Take any ε > 0 small enough that Nε(α) is convex. Find T so thatβT ⊂ Nε(α). By convexity, βt ⊂ Nε(α) for all t > T .
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On a torus
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Convergence for unique local minimizers
Closed geodesic α is a unique local minimizer of length if all closedloops in an nhd of α are longer.
Proposition: If α is unique local minimizer and subsequence βtj → α,then βt → α.
Suffices to show that α is unique geodesic with subsequence of bugloops converging to it.
Assume fsoc α′ also has subsequence of bug loops converging to it.Since, L(βt) decreasing, L(α) = L(α′) ≡ l .
Since L(βt) decreasing, and there is a subsequence converging toeach of α, α′, there is for each ε > 0 a path of loops of lengthbetween l and l + ε connecting α to α′. Call path of loopsbε : [0, 1]→ Loops(M).
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Convergence for unique local minimizers
Closed geodesic α is a unique local minimizer of length if all closedloops in an nhd of α are longer.
Proposition: If α is unique local minimizer and subsequence βtj → α,then βt → α.
Suffices to show that α is unique geodesic with subsequence of bugloops converging to it.
Assume fsoc α′ also has subsequence of bug loops converging to it.Since, L(βt) decreasing, L(α) = L(α′) ≡ l .
Since L(βt) decreasing, and there is a subsequence converging toeach of α, α′, there is for each ε > 0 a path of loops of lengthbetween l and l + ε connecting α to α′. Call path of loopsbε : [0, 1]→ Loops(M).
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Convergence for unique local minimizers
Closed geodesic α is a unique local minimizer of length if all closedloops in an nhd of α are longer.
Proposition: If α is unique local minimizer and subsequence βtj → α,then βt → α.
Suffices to show that α is unique geodesic with subsequence of bugloops converging to it.
Assume fsoc α′ also has subsequence of bug loops converging to it.Since, L(βt) decreasing, L(α) = L(α′) ≡ l .
Since L(βt) decreasing, and there is a subsequence converging toeach of α, α′, there is for each ε > 0 a path of loops of lengthbetween l and l + ε connecting α to α′. Call path of loopsbε : [0, 1]→ Loops(M).
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Convergence for unique local minimizers
Closed geodesic α is a unique local minimizer of length if all closedloops in an nhd of α are longer.
Proposition: If α is unique local minimizer and subsequence βtj → α,then βt → α.
Suffices to show that α is unique geodesic with subsequence of bugloops converging to it.
Assume fsoc α′ also has subsequence of bug loops converging to it.Since, L(βt) decreasing, L(α) = L(α′) ≡ l .
Since L(βt) decreasing, and there is a subsequence converging toeach of α, α′, there is for each ε > 0 a path of loops of lengthbetween l and l + ε connecting α to α′. Call path of loopsbε : [0, 1]→ Loops(M).
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Convergence for unique local minimizers
Closed geodesic α is a unique local minimizer of length if all closedloops in an nhd of α are longer.
Proposition: If α is unique local minimizer and subsequence βtj → α,then βt → α.
Suffices to show that α is unique geodesic with subsequence of bugloops converging to it.
Assume fsoc α′ also has subsequence of bug loops converging to it.Since, L(βt) decreasing, L(α) = L(α′) ≡ l .
Since L(βt) decreasing, and there is a subsequence converging toeach of α, α′, there is for each ε > 0 a path of loops of lengthbetween l and l + ε connecting α to α′. Call path of loopsbε : [0, 1]→ Loops(M).
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Convergence for unique local minimizers
Choose δ so that d(γ, α) ≤ δ and L(γ) = L(α) implies γ is areparameterization of α.
For each ε, pick a time sε ∈ [0, 1] so that d(βε(sε), α) = δ.
By compactness, there is a sequence (εj), j → 0, so that βεj (sεj )converges to some closed geodesic γ.
By continuity of distance, d(γ, α) = δ > 0, and by sub-continuity oflength d(γ) < l . Contradiction.
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Convergence for unique local minimizers
Choose δ so that d(γ, α) ≤ δ and L(γ) = L(α) implies γ is areparameterization of α.
For each ε, pick a time sε ∈ [0, 1] so that d(βε(sε), α) = δ.
By compactness, there is a sequence (εj), j → 0, so that βεj (sεj )converges to some closed geodesic γ.
By continuity of distance, d(γ, α) = δ > 0, and by sub-continuity oflength d(γ) < l . Contradiction.
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Convergence for unique local minimizers
Choose δ so that d(γ, α) ≤ δ and L(γ) = L(α) implies γ is areparameterization of α.
For each ε, pick a time sε ∈ [0, 1] so that d(βε(sε), α) = δ.
By compactness, there is a sequence (εj), j → 0, so that βεj (sεj )converges to some closed geodesic γ.
By continuity of distance, d(γ, α) = δ > 0, and by sub-continuity oflength d(γ) < l . Contradiction.
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Convergence for unique local minimizers
Choose δ so that d(γ, α) ≤ δ and L(γ) = L(α) implies γ is areparameterization of α.
For each ε, pick a time sε ∈ [0, 1] so that d(βε(sε), α) = δ.
By compactness, there is a sequence (εj), j → 0, so that βεj (sεj )converges to some closed geodesic γ.
By continuity of distance, d(γ, α) = δ > 0, and by sub-continuity oflength d(γ) < l . Contradiction.
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On the projective plane
Projective plane is upper half of sphere with opposite points on theequator glued together.
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Connection to curve shortening flow?
Beetle dynamics looks a lot like curve shortening flow
Curve-shortening pushes each point on smooth curve in the directionof the curvature vector
Curvature vector is direction in which curve turns
Knot shape undoes itself into a circular loop
Circular loop contracts to a point
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Connection to curve shortening flow?
Beetle dynamics looks a lot like curve shortening flow
Curve-shortening pushes each point on smooth curve in the directionof the curvature vector
Curvature vector is direction in which curve turns
Knot shape undoes itself into a circular loop
Circular loop contracts to a point
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Connection to curve shortening flow?
Beetle dynamics looks a lot like curve shortening flow
Curve-shortening pushes each point on smooth curve in the directionof the curvature vector
Curvature vector is direction in which curve turns
Knot shape undoes itself into a circular loop
Circular loop contracts to a point
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Connection to curve shortening flow?
Beetle dynamics looks a lot like curve shortening flow
Curve-shortening pushes each point on smooth curve in the directionof the curvature vector
Curvature vector is direction in which curve turns
Knot shape undoes itself into a circular loop
Circular loop contracts to a point
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Connection to curve shortening flow?
Beetle dynamics looks a lot like curve shortening flow
Curve-shortening pushes each point on smooth curve in the directionof the curvature vector
Curvature vector is direction in which curve turns
Knot shape undoes itself into a circular loop
Circular loop contracts to a point
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Connection to curve shortening flow?
From Curtis McMullen’s site
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Something else to think about: Billiard bugs.
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