Dan Lilja (Posts about intermediate)http://danlilja.se/enContents © 2023 <a href="mailto:dan@danlilja.se">Dan Lilja</a> Sat, 15 Jul 2023 17:56:24 GMTNikola (getnikola.com)http://blogs.law.harvard.edu/tech/rss- Dynamcial systems, part 1: Introduction and definitionshttp://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/Dan Lilja<p>This post is the first in a series on dynamical systems aiming towards
explaining the contents of my PhD thesis. This first part will cover the basic
definitions, objects of interest and important results on dynamical systems in
general. The reader will be assumed to be familiar with the basics of calculus,
topology and manfolds.</p>
<h2>Introduction</h2>
<p>Dynamical systems describe how objects and their properties change with respect
to time. For example it can describe the motion of a particle under the
influence of a force such as gravity, how the concentrations of various
substances change during a chemical reaction or how the number of fish in a lake
changes from year to year.</p>
<p>Mathematically a dynamical system consists of a space \(M\) called the <em>phase
space</em> and a function \(\Phi\), commonly called the <em>evolution function</em>,
taking as its arguments a time \(t\) and a point \(x\) of phase space and
outputs another point \(y=\Phi(t,x)\) of phase space, the point to which
\(x\) has moved after time \(t\). We therefore also require that
\(\Phi(0,x)=x\) for every point \(x\), i.e. if no time passes then the
points do not move. Although in general the time parameter \(t\) can belong to
any given monoid \(\mathcal{T}\) in practice either the real numbers
\(\mathbb{R}\) or the integers \(\mathbb{Z}\) are used, possibly restricted
to their non-negative parts. Therefore one usually also talks about <em>continuous
time</em> dynamical systems and <em>discrete time</em> dynamical systems.</p>
<p>Discrete time dynamical systems are usually modeled simply as a map \(F\colon
M\to M\) taking points of phase space to other points of phase space and the
time \(t\) gives the number of iterations of this function. If the point
\(x\) is mapped to the point \(y\) after time \(t=n\) we write
\(y=F^{n}(x)\) where \(F^{n}\) denotes \(n\) iterations of \(F\). In
other words, in terms of an evolution function \(\Phi\) we have
\(\Phi(n,x)=F^{n}(x)\).</p>
<p>On the other hand, continuous time dynamical systems commonly arise as the flow
of a set of differential equations where time is modeled as a subset of the real
numbers \(\mathbb{R}\). Typically these differential equations are a set of
ordinary differential equations with a finite dimensional phase space but they
could also be partial differential equations with an infinite dimensional phase
space.</p>
<p>For simplicity we will only be concerned with phase spaces which are
finite-dimensional smooth manifolds and discrete time dynamical systems which
are diffeomorphisms of their underlying phase space in the remainder of this
post. Many concepts and results generalize to other settings but some require
technicalities that are not suited for a blog post but better left to a proper
textbook on dynamical systems. You can find some suggesitons in the "Further
Reading" section below.</p>
<h2>Fixed points and other invariant sets</h2>
<p>The goal of studying dynamical systems is often not to find a complete
description of the orbit of every point, such as finding a closed expression for
the solution of a system of ordinary differential equations, since such a
description may be impossible to find. Instead one often focuses on finding a
more qualitative description of a subset of the orbits. This can be achieved by
studying certain special subsets of phase space that limit the behavior of
orbits at or around it.</p>
<p>The simplest such subset is the <em>fixed point</em>. A fixed point of a dynamical
system \(F\) is a point \(x\in M\) such that \(F(x) = x\), i.e., a point
that doesn't move around but stays fixed in time.<sup id="fnref:1"><a class="footnote-ref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fn:1">1</a></sup> Fixed points are of
interest not only because they are the simples example of an <em>invariant set</em> but
mainly because it can often be used to limit the behavior of nearby points as
well. For example, consider a dynamical system \(F\) with a fixed point
\(x_{0}\) such that all eigenvalues of \(D_{x_{0}}F\) are contained inside
the unit circle, i.e., have absolute value less than one. There will then be
some neighborhood \(U\) of \(x_{0}\) in which the dynamics is well-described
by the linear map \(D_{x_{0}}F\) and, as a result, there is some constant \(C
< 1\) such that</p>
<p>\[
d(F(x), x_{0})\leq C d(x, x_{0}) < d(x, x_{0}).
\] </p>
<p>This means that the point \(x_{0}\) will attract all points in its vicinity.
The fixed point is therefore called an <em>attracting fixed point</em>. Conversely, if
all eigenvalues of \(D_{x_{0}}F\) ar outside the unit circle the fixed point
will instead repel all nearby points and is therefore called a <em>repellng fixed
point</em>. Assuming that \(F\) is invertible we have that any attracting fixed
point of \(F\) is also a repelling fixed point of \(F^{-1}\) and vice versa.</p>
<p>Slightly more complicated are <em>periodic points</em>. These are points \(x_{0}\)
for which \(F^{n}(x_{0}) = x_{0}\), i.e., they are fixed points of some
iteration \(F^{n}\) of \(F\). The smallest such \(n\) is called the
<em>period</em> of the periodic point. Note that if \(x_{0}\) is a periodic point
then so are all points \(F^{i}(x_{0})\) for \(i = 0,1,\ldots,n-1\). Just as
for fixed points, the <em>orbit</em></p>
<p>\[
\mathcal{O}^{F}(x_{0}) = \{F^{i}(x_{0})\colon i\geq 0\}
\]</p>
<p>is called <em>attracting</em> respectively <em>repelling</em> if it is attracting respectively
repelling as a fixed point of \(F^{n}\).<sup id="fnref:2"><a class="footnote-ref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fn:2">2</a></sup></p>
<p>In full generality an <em>invariant set</em> is a subset \(U\subset M\) such that for
every \(x\in U\) we have \(F^{i}(x)\in U\) for all \(i\geq 0\). An
<em>attractor</em> is then an invariant subset \(U\) such that there is a
neighborhood \(B(U)\) of \(U\), called the <em>basin of attraction of \(U\)</em>,
with the property that for each \(b\in B(U)\) and any neighborhood \(V\) of
\(U\) there is an \(N\geq 0\) such that \(F^{n}(b)\in V\) for all \(n\geq
N\).</p>
<p>So far so good but the eigenvalues of the differential at a fixed point are not
always all inside the unit circle or all outside the unit circle. What happens
when this condition fails? We will explore the situation when the eigenvalues
can be mixed but none of them are on the unit circle. In such cases, which
include attracting and repelling fixed points, the fixed point is called
<em>hyperbolic</em>. The same terminology also applies to periodic points: a periodic
point \(x_{0} = F^{n}(x_{0})\) is called <em>hyperbolic</em> if it is a hyperbolic
fixed point of \(F^{n}\).</p>
<p>When we have a hyperbolic fixed point the tangent space at the fixed point is
also split into two parts: a <em>stable subspace</em> \(E^{s}\) corresponding to the
eigenvalues inside the unit circle and an <em>unstable subspace</em> \(E^{u}\)
corresponding to the eigenvalues outside the unit circle. If \(x_{0}\) is a
hyperbolic fixed point we can therefore write</p>
<p>\[
T_{x_{0}}M = E_{s}\oplus E_{u}.
\]</p>
<p>Furthermore, the stable and unstable subspaces will "integrate" into the <em>stable
manifold</em>, denoted \(W^{s}(x_{0})\), and <em>unstable manifold</em>, denoted
\(W^{u}(x_{0})\), respectively.</p>
<p>The idea of the tangent space being split into a direct sum of a stable subspace
and an unstable subspace naturally generalizes to larger invariant sets. Let
\(U\subset M\) be a compact invariant set and suppose there are constants
\(\lambda\in (0,1)\) and \(C > 0\) along with a splitting of the tangent
space \(TM = E^{s}\oplus E^{u}\) over \(U\) such that for every \(x\in U\)
we have:</p>
<ol>
<li>The splitting is invariant under \(F\), i.e.,
\[
D_{x}F(E^{s}(x))\subset E^{s}(F(x))\textrm{ and } D_{x}F(E^{u}(x))\subset E^{u}(F(x)),
\]</li>
<li>\(\Vert D_{x}F^{n}v\Vert \leq C\lambda^{n}\Vert v\Vert\) for all \(v\in
E^{s}(x)\) and \(n\geq 0\),</li>
<li>\(\Vert D_{x}F^{-n}v\Vert \leq C\lambda^{n}\Vert v\Vert\) for all \(v\in
E^{u}(x)\) and \(n\geq 0\).</li>
</ol>
<p>We then call \(U\) a <em>hyperbolic set</em>. In fact, the splitting is continuous in
the sense that both \(E^{s}(x)\) and \(E^{u}(x)\) depend continuously on
\(x\in U\). Hyperbolic fixed points are the simplest example of hyperbolic
sets and just as for fixed points this invariant splitting "integrates" into
invariant manifolds. This is the content of one of the most important theorems
about hyperbolic sets and perhaps dynamical systems in general: the <em>stable
manifold theorem</em>, sometimes called the <em>stable-unstable manifold theorem</em>.<sup id="fnref:3"><a class="footnote-ref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fn:3">3</a></sup>
We state it here in terms of the <em>local</em> stable and unstable manifolds but its
proof is well beyond the scope of a simple blog post. The interested reader can
find slightly more general statements as well as their proofs in the textbooks
suggested as further reading at the end of this post.</p>
<p><strong>Stable manifold theorem</strong>: Let \(F\) be a \(C^{1}\)-diffeomorphism of
\(M\) and let \(U\) be a hyperbolic set for \(F\). Then there exists an
\(\varepsilon > 0\) such that for every \(x\in U\)</p>
<ol>
<li>
<p>The sets</p>
<p>\[W^{s}_{\varepsilon}(x) = \{y\in M\colon d(F^{n}(x), F^{n}(y)) <
\varepsilon\quad\forall n\geq 0\},\]</p>
<p>\[W^{u}_{\varepsilon}(x) = \{y\in M\colon d(F^{-n}(x), F^{-n}(y)) <
\varepsilon\quad\forall n\geq 0\},\]</p>
<p>called the <em>local stable manifold of \(x\)</em> and the <em>local unstable
manifold of \(x\)</em> respectively, are \(C^{1}\)-embedded balls,</p>
</li>
<li>
<p>\(T_{y}W_{\varepsilon}^{s}(x) = E^{s}(y)\) for all \(y\in
W_{\varepsilon}^{s}(x)\) and \(T_{y}W_{\varepsilon}^{u}(x) = E^{u}(y)\)
for all \(y\in W_{\varepsilon}^{u}(x)\),</p>
</li>
<li>\(F(W_{\varepsilon}^{s}(x))\subset W_{\varepsilon}^{s}(F(x))\) and
\(F^{-1}(W_{\varepsilon}^{u}(x))\subset W_{\varepsilon}^{u}(F^{-1}(x))\),</li>
<li>for every \(x\in U\) the map \(F\) contracts distances along
\(W_{\varepsilon}^{s}(x)\) by \(\lambda\) and \(F^{-1}\) contracts
distances along \(W_{\varepsilon}^{u}(x)\) by \(\lambda\).</li>
</ol>
<p>Furthermore, it is also possible to define the global stable and unstable
manifolds of a point \(x\) in a hyperbolic set, simply denoted \(W^{s}(x)\)
and \(W^{u}(x)\) and defined as follows:</p>
<p>\[
W^{s}(x) = \left\{y\in M\colon \lim_{n\to \infty}d(F^{n}(x), F^{n}(y)) = 0\right\},
\]</p>
<p>\[
W^{u}(x) = \left\{y\in M\colon \lim_{n\to\infty}d(F^{-n}(x), F^{-n}(y)) = 0\right\}.
\]</p>
<p>The existence of the stable and unstable manifolds on a hyperbolic set can go a
long way towards getting a description of the possible dynamics of \(F\).
Consider for example the concept of <em>homoclinic connections</em> and <em>homoclinic
intersections</em>. Let \(x_{0}\) be a hyperbolic fixed point. A <em>homoclinic
connection</em> is then a connected invariant manifold \(N\subset M\) such that
\(N\subset W^{s}(x_{0})\cap W^{u}(x_{0})\). Imagine a dynamical system \(F\)
on \(\mathbb{R}^{2}\) with a hyperbolic fixed point \(x_{0}\) having a
\(1\)-dimensional stable manifold and a \(1\)-dimensional unstable manifold,
both of which are curves in the plane intersecting transversally at \(x_{0}\).
One can then imagine one part of the "outgoing" unstable manifold turning back
in to the fixed point to become one part of the stable manifold, see the below
image.</p>
<p><a class="reference" href="http://danlilja.se/images/posts/dynamicalSystemsPart1/homoclinic_connection.png"><img alt="A hyperbolic fixed point with a homoclinic connection in red." class="image-center" title="A hyperbolic fixed point with a homoclinic connection in red." src="http://danlilja.se/images/posts/dynamicalSystemsPart1/homoclinic_connection.thumbnail.png"></a></p>
<p>A <em>homoclinic intersection</em> is a different situation where the stable and
unstable manifolds intersect but do not form a connection, for example if they
intersect transversaly. It is then clear that if \(y\in W^{s}(x_{0})\cap
W^{u}(x_{0})\) then, since both \(W^{s}(x_{0})\) and \(W^{u}(x_{0})\) are
invariant, every point \(F^{n}(y)\) must also be a point of intersection
between \(W^{s}(x_{0})\) and \(W^{u}(x_{0})\). This situation creates
something called a <em>homoclinic tangle</em>, a complicated structure with complicated
dynamics, illustrated in the below image.</p>
<p><a class="reference" href="http://danlilja.se/images/posts/dynamicalSystemsPart1/homoclinic_intersection.png"><img alt="A hyperbolic fixed point with a homoclinic intersection, the unstable manifold in red and the stable manifold in blue." class="image-center" title="A hyperbolic fixed point with a homoclinic intersection, the unstable manifold in red and the stable manifold in blue." src="http://danlilja.se/images/posts/dynamicalSystemsPart1/homoclinic_intersection.thumbnail.png"></a></p>
<p>There are also <em>heteroclinic connections</em> and <em>heteroclinic intersections</em>. Like
their homoclinic counterparts they signify situations where stable and unstable
manifolds overlap as a connected manifold or intersect but originate from
different hyperbolic fixed points. Here's an illustration of a heteroclinic
connection.</p>
<p><a class="reference" href="http://danlilja.se/images/posts/dynamicalSystemsPart1/heteroclinic_connection.png"><img alt="Two hyperbolic fixed points with a heteroclinic connection in red." class="image-center" title="Two hyperbolic fixed points with a heteroclinic connection in red." src="http://danlilja.se/images/posts/dynamicalSystemsPart1/heteroclinic_connection.thumbnail.png"></a></p>
<p>And here's an illustration of a heteroclinic intersection.</p>
<p><a class="reference" href="http://danlilja.se/images/posts/dynamicalSystemsPart1/heteroclinic_intersection.png"><img alt="Two hyperbolic fixed points with a heteroclinic intersection in red and blue." class="image-center" title="Two hyperbolic fixed points with a heteroclinic intersection in red and blue." src="http://danlilja.se/images/posts/dynamicalSystemsPart1/heteroclinic_intersection.thumbnail.png"></a></p>
<p>Note that the above situations are generic but simplified in the sense that the
homoclinic and heteroclinic intersections are transverse. There are however also
intersections which are not transverse which bring further complications.</p>
<p>We will return to the homoclinic and heteroclinic connections and intersections
in the next part of this series: <em>Chaos</em>. For now, we will briefly explore
another way in which hyperblic sets can influence not only the dynamics on the
phase space but their mere existence can sometimes restrict the possible
topology of the phase space itself.</p>
<p>Extreme examples of hyperbolic sets and their influence are given by the
so-called <em>Anosov diffeomorphisms</em>. These are diffeomorphisms for which the
entire phase space is a hyperbolic set. This poses some severe restrictions on
both the topology of the underlying phase space and also on the dynamics of the
diffeomorphism itself. For example, it is immediately clear that the two-sphere
\(S^{2}\) does not admit any Anosov diffeomorphisms since that would imply a
global splitting \(TS^{1} = E^{s}\oplus E^{u}\) where both \(E^{s}\) and
\(E^{u}\) are one-dimensional, i.e., this would give us a parallellization of
the tangent bundle of \(S^{2}\). On the other hand, the set of Anosov
diffeomorphisms is an open subset of \(\textrm{Diff}^{1}(M)\), meaning that
any sufficiently small perturbation of an Anosov diffeomorphism is also an
Anosov diffeomorphism. Additionally, when Anosov diffeomorphisms do exist they
are <em>structurally stable</em>, meaning that if \(F\colon M\to M\) is an Anosov
diffeomorphism and \(G\in \textrm{Diff}^{1}(M)\) is close enough to \(F\) in
the \(C^{1}\)-metric then we can always find a homeomorphism \(h\colon M\to
M\) conjugating \(F\) and \(G\), i.e., \(F\circ h = h\circ G\), which is
close to the identity map on \(M\) in the \(C^{0}\)-metric.</p>
<h2>Measures and ergodicity</h2>
<p>While hyperbolic sets, stable manifolds and homoclinic tangles are an important
part of the analytic aspect of dynamical systems, another very important aspect
of dynamics are the statistical properties. Where the analytic aspects use the
language of derivatives and smooth manifolds, the statistical aspect use the
lanugage of measures and integrals. As such, we need to consider phase spaces
which are measure spaces. Typically we will consider <em>probability spaces</em>, i.e.,
measure spaces \(M\) having a measure \(\mu\) such that \(\mu(M) = 1\).
Just as smooth dynamical systems behave well with respect to a differentiable
structure we also consider <em>measure-preserving dynamical systems</em> which behave
well with respect to a probability measure. Formally we say that a dynamical
system \(F\) is <em>measure-preserving</em> if with respect to the probability
measure \(\mu\) if</p>
<p>\[
\mu(F^{-1}(A)) = \mu(A)
\]</p>
<p>for every measurable subset \(A\).<sup id="fnref:4"><a class="footnote-ref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fn:4">4</a></sup> We also say that the measure \(\mu\)
is an <em>invariant measure</em> with respect to the dynamical system \(F\).</p>
<p>Given that we often start out with a dynamical system already defined on some
phase space, how often is it possible to find an invariant measure? If invariant
measures are an inredibly rare phenomenon one might not expect it to be very
importnat. This question is answered, at least in part, by the
<em>Krylov-Bogolyubov theorem</em>.</p>
<p><strong>Krylov-Bogolyubov theorem</strong>: Let \(M\) be a compact metric space and let
\(F\) be a continuous map of \(M\). Then there exists an \(F\)-invariant
Borel probability measure on \(M\).</p>
<p>According to this theorem then, invariant measures are quite common. For
instance, any continuous discrete time dynamical system defined on a compact
smooth manifold will always have an invariant measure.</p>
<p>The Krylov-Bogolyubov theorem can actually be proven without too much
difficulty, so we will go over the main points of a proof. Key to the proof is
the <em>Riesz representation theorem</em> which for us means that<sup id="fnref:5"><a class="footnote-ref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fn:5">5</a></sup> for any positive
continuous linear functional \(L\) on the space \(C(M)\), the space of
continuous functions \(f\colon M\to \mathbb{R}\), there is a finite Borel
measure \(\mu\) on \(M\) such that</p>
<p>\[
L(f) = \int_{M}fd\mu
\]</p>
<p>for all \(f\in C(M)\). The strategy will therefore be to create a positive
continuous linear functional on \(C(M)\) starting with the dynamical system
\(F\) as in the assumptions of the theorem. To start with we will select a
countable and dense subset of functions<sup id="fnref:6"><a class="footnote-ref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fn:6">6</a></sup> \(\mathcal{F}\subset C(M)\) and
consider the time average \(S_{f}^{n}(x) =
\frac{1}{n}\sum_{i=0}^{n-1}f(F^{i}(x))\) of any function \(f\in\mathcal{F}\)
under the dynamical system \(F\). Now since the phase space \(M\) is compact
the sequence of time averages \(S_{f}^{n}(x)\) is bounded and hence has a
convergent subsequence and by countability of the set \(\mathcal{F}\) we can
find a sequence \(n_{i}\to\infty\) such that the sequences of time averages
\(S_{f}^{n_{i}}\) converge as \(i\to\infty\) for every
\(f\in\mathcal{F}\). In fact, using this convergence of time averages for all
\(f\) in a dense subset we can show that the time averages will converge for
any function \(g\in C(M)\) and we use this fact to create the bounded linear
functional \(L_{x}\) that maps a function to its time average at \(x\). This
linear functional gives us what we need in order to apply the Riesz
representation theorem and get a finite Borel measure \(\mu_{x}\) and it is
not difficult to show that it will be \(F\)-invariant since the linear
functional is.</p>
<p>This settles the question of how common measure-preserving dynamical systems are
but it remains to be seen how measure-preservation can impact the dynamics. To
show this we will present a theorem called the <em>Poincaré recurrence theorem</em>.</p>
<p><strong>Poincaré recurrence theorem</strong>: Let \(F\) be a measure-preserving dynamical
system of a probability space \(M\) and let \(U\subset M\) be a measurable
subset. Then for almost every point \(x\in U\) there is an integer \(n\geq
1\) such that \(F^{n}(x)\in U\).</p>
<p>What this theorem tells us is that for any measurable subset \(U\) almost
every point will return to it at some point. In fact, almost every point will
return infinitely often to the subset \(U\). Another way of putting it is that
the set of points in any measurable subset \(U\) that never return has measure
zero.</p>
<p>The Poincaré recurrence theorem allows us to think of measure-preserving
dynamical systems as, in some sense, making sure that the points of phase space
are being very well mixed<sup id="fnref:7"><a class="footnote-ref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fn:7">7</a></sup> by the dynamics, at least within the support of
the invariant measure.</p>
<p>Let's return for a moment to the proof of the Krylov-Bogolyubov theorem. From
the way that the positive bounded linear operator is created we can see that
there is a subsequence \(n_{k}\) such that</p>
<p>\[
\lim_{k\to\infty}\frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1}f(F^{i}(x)) = \int_{M} fd\mu_{x}
\]</p>
<p>whenever \(f\) is continuous, the phase space \(M\) is compact and
metrizable and \(\mu_{x}\) is an invariant probability measure for \(F\).
This begs a few questions: Can the sequence of time averages converge without
taking a subsequence and if so for which points \(x\) does it do so? When is
the invariant measure independent of \(x\)? The <em>Birkhoff ergodic theorem</em>
answers the first of these questions.</p>
<p><strong>Birkhoff ergodic theorem</strong>: Let \(F\) be a measure-preserving dynamical
system on a probability space \((M,\mu)\) and let \(f\in L^{1}(M,\mu)\),
i.e., \(f\) is an integrable function on \(M\) with respect to the
probability measure \(\mu\). Then the limit</p>
<p>\[
\bar{f}(x) = \lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}f(F^{i}(x))
\]</p>
<p>exists for almost every \(x\in M\), it is \(\mu\)-integrable, \(F\)-invariant and</p>
<p>\[
\int_{M}\bar{f}d\mu = \int_{M}fd\mu.
\]</p>
<p>So in general the time average of an integrable function \(f\) under a
measure-preserving dynamical system does not equal the space average but rather
it equals some other function \(\bar{f}\). However, the space average of
\(\bar{f}\) equals the space average of \(f\). We therefore need some
stronger condition to ensure that the time average of a function actually equals
the space average. This is where the so-called <em>ergodic</em> dynamical systems come
in. A measure-preserving dynamical system \(F\) on a probability space
\((M,\mu)\) is called <em>ergodic with respect to \(\mu\)</em> if<sup id="fnref:8"><a class="footnote-ref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fn:8">8</a></sup> for for every
\(F\)-invariant subset \(U\) we have either \(\mu(U) = 0\) or \(\mu(U) =
1\). The first step to seeing how this can answer our question of when the time
average of a function equals the space average is to recognize that this
definition is equivalent to another characterization in terms of
\(F\)-invariant measurable functions. Namely, \(F\) is ergodic with respect
to \(\mu\) if and only if any \(F\)-invariant measurable function \(f\) is
constant outside a set of measure zero. Armed with this new knowledge and
combining it with the Birkhoff ergodic theorem we get the following answer to
the second question above.</p>
<p><strong>Theorem</strong>: A measure-preserving dynamical system \(F\) on a probability
space \((M,\mu)\) is ergodic if and only if the time average at \(x\) of any
integrable function \(f\in L^{1}(M,\mu)\) equals the space average of \(f\)
for almost every \(x\), i.e., if and only if</p>
<p>\[
\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}f(F^{i}(x)) = \int_{M}fd\mu
\]</p>
<p>for almost every \(x\).</p>
<h3>Multiplicative ergodic theorem and Lyapunov exponents</h3>
<p>Finally, we will use another of the ergodic theorems, the so-called <em>Oseledec
multiplicative ergodic theorem</em>, to reconnect with the hyperbolic dynamics from
earlier. First we will need some definitions.</p>
<p>Let \(F\) be a differentiable map of a smooth manifold \(M\). The <em>upper
Lyapunov exponent of \((x, v)\) with respect to \(F\)</em> is defined as</p>
<p>\[
\chi(x, v) = \limsup_{n\to\infty}\frac{1}{n}\log\lVert D_{x}F^{n}v\rVert
\]</p>
<p>where \(x\in M\) and \(v\in T_{x}M\). Sometimes this is also called just the
Lyapunov exponent, especially if the actual limit exists and not just the limit
superior. Thus the Lyapunov exponent measures the exponential expansion rate of
tangent vectors along orbits in the sense that</p>
<p>\[
\lVert D_{x}F^{n}v\rVert \approx e^{\chi(x, v)}\lVert v\rVert.
\]</p>
<p>It is clear that if \(F\) has uniformly bounded first derivative, for example
if \(M\) is compact, then the Lyapunov exponent always exists for every
\(x\in M\) and every \(v\in T_{x}M\). Some things that are not clear are for
example when the Lyapunov exponent exists in general and how it depends on
\(x\). One could imagine, for example, that if \(F\) was ergodic that the
Lyapunov exponent would not depend on \(x\) outside a subset of measure zero
due to the way points \(x\) seem to move around all over the support of the
measure. This intuition will lead us on the right track to answering these
questions but first we will take a detour into linear cocycles and another
ergodic theorem.</p>
<p>Suppose we have a measure-preserving dynamical system \(F\) defined on a
probability space \((M,\mu)\) which is also a Lebesgue space<sup id="fnref:9"><a class="footnote-ref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fn:9">9</a></sup>. A
<em>measurable linear cocycle over \(F\)</em>, or simply a <em>linear cocycle</em>, is a
measurable map \(A\colon M\times\mathbb{Z}\to GL(n, \mathbb{R})\) satisfying</p>
<p>\[
A(x, m+n) = A(F^{n}(x), m)A(x, n)
\]</p>
<p>for all \(m,n\in\mathbb{Z}\).</p>
<p>One of the main examples of how cocycles arise in dynamical systems, and the one
we will be the most interested in here, is if we're starting out with a
measure-preserving dynamical system that also happens to be a diffeomorphism. We
can then consider the derivative \(DF\) as a cocycle by choosing<sup id="fnref:10"><a class="footnote-ref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fn:10">10</a></sup></p>
<p>\[
A(x, m) = D_{x}F^{m} = D_{F^{m-1}(x)}F\cdot D_{F^{m-2}(x)}F\cdot\ldots\cdot D_{x}F.
\]</p>
<p>Keeping this example in mind we define the <em>upper Lyapunov exponent of A at
\((x, v)\)</em> as</p>
<p>\[
\limsup_{n\to\infty}\frac{1}{n}\log\lVert A(x, n)v \rVert.
\]</p>
<p>We can see that this definition reduces to the previous definition of the
Lyapunov exponent if the cocycle \(A\) is given as the composition of the
differential \(DF\) of the map \(F\).</p>
<p>One can show that for any real number \(C\) the set \(V_{C}(x) = \{v\in
\mathbb{R}^{n}\colon \chi(x, v)\leq C\}\) forms a linear subspace of
\(\mathbb{R}^{n}\) and these subspaces are nested as we increase the constant,
i.e., if \(\chi_{1}\leq C_{2}\) we have \(V_{C_{1}}(x)\subset
V_{C_{2}}(x)\). In fact, for each \(x\) it is possible to find a sequence of
real numbers \(\chi_{1} < \chi_{2} < \ldots < \chi_{n(x)}\) and associated
subspaces</p>
<p>\[
\{0\}\subset V_{\chi_{1}}(x)\subset V_{\chi_{2}}(x)\subset\ldots\subset
V_{\chi_{n(x)}}(x) = \mathbb{R}^{n}
\]</p>
<p>such that if \(v\in E_{\chi_{i}}(x)\setminus E_{\chi_{i-1}}(x)\) then the
upper Lyapunov exponent of \((x,v)\) equals \(\chi_{i}\). With this in mind
we call the numbers \(\chi_{i}\) the <em>upper Lyapunov exponents of \(A\) at
\(x\)</em>. We define the <em>multiplicity \(d_{i}\) of the Lyapunov exponent
\(\chi_{i}\)</em> as the number \(d_{i} = \dim E_{\chi_{i}}(x) - \dim
E_{\chi_{i-1}}\). The Lyapunov exponents of \(A\) together with their
multiplicities is called the <em>Lyapunov spectrum of \(A\) at \(x\)</em>.</p>
<p>With the preliminaries out of the way we can now state the <em>Oseledec
multiplicative ergodic theorem</em> which, like the Birkhoff ergodic theorem, will
tell us about when the Lyapunov exponents (essentially the time averages in this
situation) exist and when they are independent of the initial point \(x\).</p>
<p><strong>Oseledec multiplicative ergodic theorem</strong>: Let \(F\) be a measure-preserving
dynamical system on a probability space \((M,\mu)\) which is also a Lebesgue
space and let \(A\) be a measurable linear cocycle over \(F\) satisfying
that both the maps \(x\mapsto \log^{+} \lVert A(x, 1)\rVert\) and \(x\mapsto
\log^{+} \lVert A(x, -1)\rVert\), where \(\log^{+}(x) = \max\{0,
\log(x)\}\), are both \(L^{1}\)-integrable with respect to \(\mu\). Then
there exists a subset \(U\subset M\) of full measure such that for each
\(x\in U\) we have the following:</p>
<ol>
<li>
<p>There exists a decomposition of \(\mathbb{R}^{n}\) by subspaces
\(E_{i}(x)\) for \(i = 1,\ldots, n(x)\), i.e., </p>
<p>\[
\mathbb{R}^{n} = \bigoplus_{i = 1}^{n(x)}E_{i}(x)
\]</p>
<p>which is invariant under the induced map of \(M\times \mathbb{R}^{n}\)
given by \((x,v)\mapsto (F(x), A(x, 1)v)\).</p>
</li>
<li>
<p>The Lyapunov exponents \(\chi_{1}(x) < \chi_{2}(x) < \ldots <
\chi_{n(x)}(x)\) associated with each \(E_{i}(x)\) all exist and are
\(F\)-invariant.</p>
</li>
<li>
<p>For all non-zero vectors \(v\in E_{i}(x)\) we have
\[
\lim_{n\to\pm\infty}\frac{1}{m}\log\frac{\lVert A(x, n)v\rVert}{\lVert v\rVert} = \pm\chi_{i}(x)
\]
where convergence is uniform in \(v\).</p>
</li>
</ol>
<p>Furthermore, if \(F\) is ergodic with respect to \(\mu\) then the Lyapunov
exponents are independent of \(x\).</p>
<p>Now let's go back to the cocycle given by iteration of the differential of
\(F\) considered above. For simplicity, let's assume that \(M\) is a compact
smooth manifold and that \(F\) is a diffeomorphism of \(M\) which is
hyperbolic. Consider a vector \(v\) in the stable subspace \(E^{s}(x)\subset
T_{x}M\) where \(x\) is in some hyperbolic set \(U\) for \(F\). We then
know from hyperbolicity that there is some constant \(C > 0\) and some
constant \(0 < \lambda < 1\) such that</p>
<p>\[
\lVert D_{x}F^{n}v\rVert \leq C\lambda^{n}\lVert v\rVert
\]</p>
<p>for all \(n\geq 0\). On the other hand, let's consider the Lyapunov exponent:</p>
<p>\[
\chi(x, v) = \limsup_{n\to\infty}\frac{1}{n}\log\lVert D_{x}F^{n}v\rVert \leq \limsup_{n\to\infty}\frac{1}{n}\log\lVert C\lambda^{n}v \rVert = \log(\lambda)
\]</p>
<p>It is clear from this that the Lyapunov exponent is related to the hyperbolicity
of \(F\). Picking an invariant measure<sup id="fnref:11"><a class="footnote-ref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fn:11">11</a></sup> on \(U\) we can also use the
multiplicative ergodic theorem to guarantee us the existence of all the Lyapunov
exponents. These can then be seen as a further refinement of the constant
\(\lambda\) giving the local rate of contraction and expansion in the stable
and unstable manifolds.</p>
<p>Given that the Lyapunov exponents are related to ideas of contraction and
expansion but having a wider range of applicability than just hyperbolicity they
have found frequent use in generalizations of hyperbolicity, such as nonuniform
hyperbolicity and partial hyperbolicity. They also feature prominently in the
theory surrounding <em>chaotic dynamical systems</em>, which we will explore in the
next part of this series.</p>
<h2>Further Reading</h2>
<p>Here are some links to free online resources for those that are interested in
reading more about the topics covered in this blog post.</p>
<ul>
<li><a href="https://en.wikipedia.org/wiki/Dynamical_system">Dynamical systems</a></li>
<li><a href="https://en.wikipedia.org/wiki/Hyperbolic_set">Hyperbolic sets</a></li>
<li><a href="https://en.wikipedia.org/wiki/Invariant_manifold">Invariant manifold</a></li>
<li><a href="https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem">Poincaré recurrence theorem</a></li>
<li><a href="https://en.wikipedia.org/wiki/Ergodic_theory">Ergodic theory</a></li>
<li><a href="https://en.wikipedia.org/wiki/Lyapunov_exponent">Lyapunov exponents</a></li>
</ul>
<p>Here are also some recommended textbooks for those wishing to learn dynamical
systems.</p>
<ul>
<li>
<p><a href="https://www.amazon.com/Introduction-Chaotic-Dynamical-Systems-2nd/dp/0813340853"><em>An Introduction to Chaotic Dynamical Systems</em> by Robert L. Devaney</a></p>
<p>A fantastic introduction to dynamical systems, suitable at an undergraduate
level. Presents many interesting topics of dynamics, including chaos and
complex dynamics, while also focusing on one, two and sometimes three
dimensional systems in order to keep the presentation simple and intuitive.
Also includes several pictures of the phenomena being discussed that also
helps understanding.</p>
</li>
<li>
<p><a href="https://www.amazon.com/Introduction-Dynamical-Systems-Michael-Brin/dp/1107538947"><em>Introduction to Dynamical Systems</em> by Michael Brin and Garrett Stuck</a></p>
<p>Another good introductory textbook on dynamical systems offering a concise
presentation of a very well chosen set of topics. More advanced textbook
that doesn't shy away from presenting results in much more generality,
including using \(n\)-dimensional manifolds as phase spaces, hence more
suited for the graduate level.</p>
</li>
<li>
<p><a href="https://www.amazon.com/Introduction-Dynamical-Encyclopedia-Mathematics-Applications/dp/0521575575/"><em>Introduction to the Modern Theory of Dynamical Systems</em> by Anatole Katok and Boris Hasselblatt</a></p>
<p>Gradute level textbook covering many topics of dynamical systems, from the
basic definitions and central results, to more specialized or advanced
topics not usually appearing in a general textbook, such as Aubrey-Mather
sets and geodesic flows. Definitely requires more mathematical maturity than
the previous two. Also unlike the previous two books it also covers a good
deal continuous time dynamics. This is pretty much the endgame as far as
books on general dynamical systems go. To continue from here requires diving
into more specialized advanced textbooks.</p>
</li>
<li>
<p><a href="https://www.amazon.com/Differential-Equations-Dynamical-Systems-Introduction/dp/0123820103"><em>Differential Equations, Dynamical Systems & An Introduction to Chaos</em> by Morris W. Hirsch, Stephen Smale, and Robert L. Devaney</a></p>
<p>Undergraduate level textbook focusing on systems of ordinary differential
equations and continuous time dynamical systems. Provides a very good
introductin to the fundamentals of both ordinary differential equations and
continuous time dynamical systems. Has a large amount of very good examples,
illustrations, exercises and applications.</p>
</li>
</ul>
<div class="footnote">
<hr>
<ol>
<li id="fn:1">
<p>For a continuous time dynamical system a fixed point would be a point
\(x\) such that \(\Phi(t, x) = x\) for all \(t\). <a class="footnote-backref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fnref:1" title="Jump back to footnote 1 in the text">↩</a></p>
</li>
<li id="fn:2">
<p>Note that for a periodic point the orbit \(\{F^{i}(x_{0})\colon i\geq
0\}\) is finite since it repeats starting with \(F^{n}(x_{0}) = x_{0}\). <a class="footnote-backref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fnref:2" title="Jump back to footnote 2 in the text">↩</a></p>
</li>
<li id="fn:3">
<p>It can be considered a special case of the <em>Hadamard-Perron theorem</em> but
stating it in such generality quickly becomes unwieldy for a small blog
post. Consult a proper textbook on dynamical systems to enjoy the full
statement of this important theorem. <a class="footnote-backref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fnref:3" title="Jump back to footnote 3 in the text">↩</a></p>
</li>
<li id="fn:4">
<p>Another way of stating this is in terms of the pushforward measure:
\(f_{*}(\mu) = \mu\). <a class="footnote-backref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fnref:4" title="Jump back to footnote 4 in the text">↩</a></p>
</li>
<li id="fn:5">
<p>The full statement, which is more general, is about Hilbert spaces, see
<a href="https://en.wikipedia.org/wiki/Riesz_representation_theorem">Riesz representation
theorem</a>. <a class="footnote-backref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fnref:5" title="Jump back to footnote 5 in the text">↩</a></p>
</li>
<li id="fn:6">
<p>This can be done since by the Stone-Weierstrass theorem the space
\(C(M)\) is separable and hence it has a countable dense subset. <a class="footnote-backref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fnref:6" title="Jump back to footnote 6 in the text">↩</a></p>
</li>
<li id="fn:7">
<p>In fact, certaing measure-preserving dynamical systems also have a
stronger property called <em>mixing</em>. We will not discuss this here. <a class="footnote-backref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fnref:7" title="Jump back to footnote 7 in the text">↩</a></p>
</li>
<li id="fn:8">
<p>It is also common to say the \(\mu\) is ergodic with respect to \(F\). <a class="footnote-backref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fnref:8" title="Jump back to footnote 8 in the text">↩</a></p>
</li>
<li id="fn:9">
<p>A <em>Lebesgue space</em> is a measure space with finite measure which is
isomorphic to the a union of the Lebesgue measure on an interval \([0,a]\)
with at most countably many points of positive measure. <a class="footnote-backref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fnref:9" title="Jump back to footnote 9 in the text">↩</a></p>
</li>
<li id="fn:10">
<p>We are ignoring some technical details here, like choosing local
trivializations on \(M\), for the sake of readability. <a class="footnote-backref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fnref:10" title="Jump back to footnote 10 in the text">↩</a></p>
</li>
<li id="fn:11">
<p>It turns out that non-trivial, i.e., not just a periodic orbit,
hyperbolic sets have many invariant measures. In particular, for each
periodic orbit in the hyperbolic set we have an atomic invariant measure
supported on that periodic orbit and since there are infinitely many
periodic orbits we also have infinitely many invariant measures. Additional
invariant measures can then be created as the weak limit of these measures. <a class="footnote-backref" href="http://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/#fnref:11" title="Jump back to footnote 11 in the text">↩</a></p>
</li>
</ol>
</div>chaosdynamical systemsintermediatemathematicshttp://danlilja.se/blog/dynamical-systems-part-1-introduction-and-definitions/Wed, 19 Oct 2022 16:55:00 GMT