Introduction
Probability in Dynamics
Going Further
Decay of Correlations
Probabilistic Methods in Dynamics
M. Smith
Department of Mathematics
University of Utah
22 March 2016 / GSAC
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Outline
1
Introduction
Definitions and Examples
Tracking Orbits
2
Probability in Dynamics
Comparisons
Central Limit Theorem
Spectral Gap Method
3
Going Further
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Outline
1
Introduction
Definitions and Examples
Tracking Orbits
2
Probability in Dynamics
Comparisons
Central Limit Theorem
Spectral Gap Method
3
Going Further
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
What is Dynamics?
A dynamical system is a deterministic rule for "time evolution".
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Time Evolution by Maps
Suppose we have a map T : X → X .
Applying T to X moves time forward by one step.
Notation:
T n = T ◦ T ◦ · · · ◦ T (n times).
For x ∈ X , T n (x) is the position of x at time n.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Other Types of Time Evolution
If T is invertible, T −n = (T −1 )n , n ≥ 0.
Z acts on X by n 7→ T n .
A group G acting on X gives "time evolution".
R-actions arise from differential equations.
Z-actions are similar to R-actions.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Examples
Discrete time:
X = your favorite space, T = idX .
X = a circle, T = rotation by some angle.
X = a compact manifold, T = a diffeomorphism.
Continuous time:
Flows on manifolds.
Frictionless motion with constraints.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Main Example: Doubling
X = S 1 = R/Z = [0, 1).
Define the doubling map:
D(x) = 2x
(mod 1)
Isomorphic to z 7→ z 2 for the unit circle in C.
D n (x) = 2n x (mod 1).
Exponential "complexity".
This example is an extreme case!
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Doubling Graph
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Repeated Doubling
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Repeated Doubling
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Outline
1
Introduction
Definitions and Examples
Tracking Orbits
2
Probability in Dynamics
Comparisons
Central Limit Theorem
Spectral Gap Method
3
Going Further
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Tracking Orbits
Suppose x ∈ X , and consider its forward orbit:
O+ (x) = {T n (x)}∞
n=0
Question: What does the orbit of x look like?
What about orbits of subsets of X ?
This is very difficult in general.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Orbits Under Doubling
Any
j
2k
is sent to 0 by D k .
Similarly
j
3k
has a periodic orbit.
Both kinds are dense, but only countable.
Refined Question: What is the orbit of a "generic point" x?
Generic means we should have a notion of "size".
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Measurable Dynamics
Let µ be a probability measure on X so that T is µ-measurable.
T is µ-measure preserving if:
µ(T −1 A) = µ(A)
for all µ-measurable sets A.
Invariant measures are the easiest to understand.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Doubling Invariant Measures
D preserves λ, Lebesgue measure on S 1 .
Warning: D has many preserved measures!
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Ergodicity
Let (X , µ, T ) be a probability measure preserving system.
If T −1 A = A implies µ(A) = 0 or µ(A) = 1, T is ergodic
with respect to µ.
Ergodicity means indecomposibility.
D is λ-ergodic (by general representation theory).
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Birkhoff Ergodic Theorem
Let (X , µ, T ) be an ergodic probability measure preserving
system.
If F ∈ L1 (µ), then for µ-almost every x ∈ X :
Z
N−1
1 X
n
lim
F (T x) =
F dµ
N→∞ N
X
n=0
Interpretation: time average = space average.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Equidistribution
Take F = χA for A ⊆ X .
χA (T n x) = 1 if T n x is in A, otherwise 0.
The ergodic theorem tells us, for µ-almost every x:
N−1
1 X
χA (T n x) = µ(A).
N→∞ N
lim
n=0
The µ-typical point "spends µ(A) of its time in A" (after a
long time).
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Definitions and Examples
Tracking Orbits
Chaos
The doubling map is "chaotic".
Almost every point equidistributes (for Lebesgue measure).
Periodic points are dense.
Points which are eventually fixed at 0 are dense.
Very sensitive to initial conditions!
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Outline
1
Introduction
Definitions and Examples
Tracking Orbits
2
Probability in Dynamics
Comparisons
Central Limit Theorem
Spectral Gap Method
3
Going Further
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Identical Distribution
Let (X , µ, T ) be a probability measure preserving system.
For F ∈ L1 (µ) consider the sequence Fn = F ◦ T n .
T is measure preserving, so Fn are identically distributed.
Unlikely to be independent, T is deterministic.
Question: Can we approximate Fn by IID?
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Strong Law of Large Numbers
Let (Ω, F, p) be a probability space.
Let {Xn }∞
n=0 be a sequence of IID real random variables
such that E(|X0 |) exists and is finite.
Then almost surely:
N−1
1 X
lim
Xn = E(X0 ).
N→∞ N
n=0
This is a special case of the Birkhoff ergodic theorem!
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Probabilistic Method
Ergodic systems have SLLN.
Question: can we get any other theorems from
probability?
More theorems implies a system is "essentially random"
(over a long time).
Dynamical systems are deterministic, so this is hard!
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Mixing
Let (X , µ, T ) be a probability measure preserving system.
T is µ-mixing if for any measurable A, B we have:
lim µ(A ∩ T −n (B)) = µ(A)µ(B).
n→∞
After a long time, B is essentially independent of A.
Diffusion-like behavior.
Mixing implies ergodicity.
"Most" systems are not mixing.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Doubling is Mixing
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Decay of Correlations
T is µ-mixing if and only if for any F , G ∈ L2 (µ) we have:
Z
Z
Z
n
lim
F · G ◦ T dµ −
F dµ
G dµ = 0.
n→∞
X
X
X
Consider the correlation between F and G ◦ T n (up to
scaling).
Mixing is equivalent to observables being uncorrelated
(after a long time).
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Outline
1
Introduction
Definitions and Examples
Tracking Orbits
2
Probability in Dynamics
Comparisons
Central Limit Theorem
Spectral Gap Method
3
Going Further
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
CLT in Probability
Let {Xm }∞
m=0 be a sequence of IID real random variables.
Suppose E[X0 ] = 0 and 0 < Var[X0 ] = σ 2 < ∞.
Then the sequence:
YM = √
1
Mσ 2
M−1
X
Xm
m=0
converges in distribution to N(0, 1).
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
CLT in Dynamics
Let (X , µ, T ) be a probability measure preserving system.
A function F ∈ L20 (µ) has the CLT under T if:
lim µ x : √
M→∞
1
Mσ 2
M−1
X
!
m
F (T x) ≤ z
m=0
1
=√
2π
Z
z
−∞
for some 0 < σ 2 < ∞.
Here L20 (µ) means integral zero.
Tells us certain orbits occur with positive measure.
M. Smith
Decay of Correlations
t2
e− 2 dt
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Proving CLT
F ◦ T m is not an IID sequence in general.
Probability: Stationary processes with fast mixing of sets
still have CLT.
F ◦ T m is a stationary process if T preserves µ.
If D has fast mixing of sets, then D will have CLT!
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
The Proof Fails
Suppose γn → 0 is a sequence of positive numbers. Then there
exist A, B ⊆ [0, 1) so that:
|λ(A ∩ D −n (B)) − λ(A)λ(B)| ≥ γn
That is, D has sets with arbitrarily slow mixing.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Outline
1
Introduction
Definitions and Examples
Tracking Orbits
2
Probability in Dynamics
Comparisons
Central Limit Theorem
Spectral Gap Method
3
Going Further
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Changing the Problem
We could not prove CLT for D and L20 (S 1 ).
L2 functions can be arbitrarily complicated.
Idea: "nicer" functions could still have CLT.
More difficult to prove.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Step 1: Transfer Operator
Define the Ruelle transfer operator PT by:
Z
Z
F · G ◦ T dµ =
PT F · G dµ
X
X
PT is well defined on various Lp spaces.
For D, we have:
x
1
1
x
+F
+
PD F (x) =
F
2
2
2 2
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Step 2: Banach Spaces
PT acts nicely on L2 (µ).
If F is constant, then PT F = F .
If F has integral 0, then PT F also has integral zero.
L2 = R ⊕ L20 , each piece is PT -invariant.
Question: what does PT do to L20 ?
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Step 3: Spectral Gap
Suppose that B ⊆ L∞
0 is a Banach space such that:
kPT F kB ≤ r kF kB with 0 < r < 1.
kF k1 ≤ CkF kB for some C > 0.
Then we have:
Z
n
F · G ◦ T dµ ≤ r n CkF kB kGk∞
X
so functions in B have exponential decay of correlations.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Step 3: Spectral Gap
Contraction by r bounds the spectrum of PT away from 1.
A "gap" in the spectrum of PT makes it a contraction on B.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Step 4: CLT, at last
If PT has a spectral gap, then we are done.
Suppose every F ∈ B has exponential decay of
correlations
Each F ∈ B then has some appropriate 0 < σ 2 < ∞.
The sequence of functions F ◦ T m satisfy CLT with
variance σ 2 .
The proof is "similar" to the proof of the usual CLT.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Comparisons
Central Limit Theorem
Spectral Gap Method
Doubling CLT
For D, take B = Lip0 = Lipschitz functions of integral zero.
kF kB is the Lipschitz constant for F .
PD is a contraction by
1
2
on B:
y
kF k
1 x
B
+c −F
+c ≤
|x − y |
F
2
2
2
4
kF kB
kF kB
|x − y | =
|x − y |
4
2
So D has CLT for Lipschitz functions!
|PD F (x) − PD F (y )| ≤ 2
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Outline
1
Introduction
Definitions and Examples
Tracking Orbits
2
Probability in Dynamics
Comparisons
Central Limit Theorem
Spectral Gap Method
3
Going Further
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Piecewise Expanding Maps
The doubling map belongs to this class of maps on S 1 .
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Other Results
There are three ways we can go further.
Piecewise expanding maps have a "nice" invariant
measure. If this measure is mixing, we get CLT for
functions in BV (bounded variation).
Stronger results like the Almost Sure Invariance Principle
are possible.
We can try to get CLT in applied systems.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
Summary
Theorems from probability are still true for deterministic
systems, especially chaotic ones.
Functional analysis gives a lot of information in dynamics.
For continuous or smooth systems, invariant measures
interact with topological or smooth structures in nice ways.
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
References and Further Reading
General ergodic theory
Walters, Peter. An Introduction to Ergodic Theory.
Springer. (2000)
CLT for the doubling map
<https://vaughnclimenhaga.wordpress.com/
2013/01/30/spectral-methods-in-dynamics/>
Split into separate posts, there are links at the bottom.
Slowly mixing sets
<https://vaughnclimenhaga.wordpress.com/
2014/04/22/slowly-mixing-sets/>
M. Smith
Decay of Correlations
Introduction
Probability in Dynamics
Going Further
References and Further Reading
CLT for piecewise expanding maps
G. Keller and C. Liverani. A Spectral Gap for a
One-dimensional Lattice of Coupled Piecewise Expanding
Interval Maps. Lect. Notes Phys. 671, 115-151 (2005)
M. Smith
Decay of Correlations