Fundemental Theorem of Dynamical Systems (Part 2)

April 15, 2010April 15, 2010 Conan3 Comments

Now we begin to prove the theorem.

4.Attractor-repeller pairs

Definition: A compact set $A \subseteq X$ is an attractor for $f$ if there exists $U$ open, $f(\bar U) \subseteq U$ and $\displaystyle \bigcup_{i=0}^\infty f^i(U) = A$ . $U$ is called a basin of attraction.

For any attractor $A \subseteq X$ , $U$ be a basin for $A$ , let $U^\ast = X \backslash \bar U, \ U^\ast$ is open. By definition, $f(A) = A$ and $f(A^\ast) = A^\ast$ . We also have

$f^{-1}(\overline{U^\ast}) = f^{-1}(\bar{X \backslash \bar U}) \subseteq X \backslash f^{-1}(U) \subseteq X \backslash U \subseteq \overline{U^\ast}$

Definition: A repeller for $f$ is an attractor for $f^{-1}$ . A basin of repelling for $f$ is a basin of attracting for $f^{-1}$ .

Hence $A^\ast$ is a repeller for $f$ with basin $U^\ast$ .

It’s easy to see that $A^\ast$ is defined independent of the choice of basin for $A$ . (Exercise)

We call such a pair $A, \ A^\ast$ an attracting-repelling pair.

The following two properties of attracting-repelling pairs are going to be important for our proof of the theorem.

Proposition 1: There are at most countably many different attractors for $f$ .

Proof: Since $X$ is compact metric, there exists countable basis $\mathcal{U} = \{U_i\}_{i \in \mathbb{N}}$ that generates the topology.

For any attractor $A$ , any attracting basin $\mathcal{B}$ of $A$ is a union of sets in $\mathcal{U}$ , i.e. $\displaystyle \mathcal{B} = \bigcup_{i=1}^\infty U_{n_i}$ latex for some subsequence $(n_i)$ of $\mathbb{N}$ .

Since $A$ is compact, $U_{n_i}$ is an open cover of $A$ , we have some $\{ m_1, \ m_2, \ \cdots \ m_k \} \subseteq \{n_i\}_{i \in \mathbb{N}}$ s.t. $\{ U_{m_1}, \ U_{m_2}, \ \cdots, \ U_{m_k} \}$ covers $A$ .
Let $B' = U_{m_1} \cup U_{m_2} \cup \cdots \cup U_{m_k}$ hence $A \subseteq B' \subseteq B$ . We have

$\displaystyle A \subseteq \bigcap_{n \in \mathbb{N}} f^n(B') \subseteq \bigcap_{n \in \mathbb{N}} f^n(B)$

Since $B$ is an attracting basin for $A$ , all three sets are equal. Hence $A = \bigcap_{n \in \mathbb{N}} f^n(B')$ . i.e. any attractor is intersection of foreward interates of come finite union of sets in $\mathcal{U}$ . Since $\mathcal{U}$ is countable, the set of all finite subset of it is coubtable.

Hence there are at most countably many different attractors. This establishes the proposition.

By proposition 1, we let $(A_n)_{n \in \mathbb{N}}$ be a list of all attractors for $f$ . Now we are going to relate the arrtactor-repeller pairs to the chain recurrent set and chain transitive components.

Proposition 2:

$\mathcal{R}(f) = \displaystyle \bigcap_{n \in \mathbb{N}}(A_n \cup A^\ast_n)$

Proof: i) $\mathcal{R}(f) \subseteq \bigcap_{n \in \mathbb{N}}(A_n \cup A^\ast_n)$

This is same as saying for any attractor $A$ , $\mathcal{R}(f) \subseteq (A \cup A^\ast)$ .

For all $x \notin (A \cup A^\ast)$ , let $B$ be a basin of $A$ , then there is $N \in \mathbb{N}$ for which $x \notin (f^N(B) \cup f^{-N}(B^\ast))$ (recall that $B^\ast$ is the dual basin of $B$ for $A^\ast$ ). Since $B^\ast = X \backslash \overline{B}$ and $f(\overline{B}) \subseteq B$ we conclude

$X \backslash f^{-N}(B^\ast) = f^{-N}(\overline{B}) \subseteq f^{-N-1}(f(\overline{B})) \subseteq f^{-N-1}(B)$

Hence $x \in f^{-N-1}(B)$ . Let $M$ be the smallest integer for which $x \in f^{-M}(B)$ . Hence $x \in f^{-M}(B) \backslash f^{-M+1}(B)$ . Let $U = f^{-M}(B)$ is also a basin for $A$ .

Now we show such $x$ cannot be chain recurrent: Since $X \backslash f(U)$ and $\overline{f^2(U)}$ are compact and disjoint, we may let

$\varepsilon_1 = \frac{1}{2} \min\{ d(a, b) \ | \ a \in X \backslash f(U), \ b \in \overline{f^2(U)} \}$

Since $f(x) \in f(U)$ , there exists some $\varepsilon_2$ s.t.

$\overline{B(f(x), \varepsilon_2)} \subseteq f(U)$

$f(\overline{B(f(x), \varepsilon_2)}) \subseteq f^2(U)$ so there exists $\varepsilon_3$ s.t.

$N(f(\overline{B(f(x), \varepsilon_2)}), \varepsilon_3) \subseteq f^2(U)$

(Here $B(p, r)$ denotes the ball of radius $r$ around $p$ and $N(C, r)$ denotes the $r$ -neignbourhood of compact set $C$ )

Now set $\varepsilon = \min\{ \varepsilon_1, \ \varepsilon_2, \ \varepsilon_3\}$ , for any $\varepsilon$ -chain $x, x_1, x_2, \cdots$ , we have: Since $\varepsilon < \varepsilon_2$ and $\varepsilon_3$ , $x_1 \in B(f(x), \varepsilon_2)\subseteq f(U)$ and $x_2 \in B(f(x_1), \varepsilon_3) \subseteq N(f(\overline{B(f(x), \varepsilon_2)}), \varepsilon_3) \subseteq f^2(U)$ . Hence the third term of any such chain must be in $f^2(U)$ . Since $\varepsilon < \varepsilon_1$ , no $\varepsilon$ -chain starting at $x_2$ can reach $X \backslash f(U)$ , in particular, the chain $x, x_1, x_2, \cdots$ does not come back to $x$ . Hence we conclude that $x$ is not chain recurrent. ii) $\bigcap_{n \in \mathbb{N}}(A_n \cup A^\ast_n) \subseteq \mathcal{R}(f)$ Suppose not, there is $x \in \bigcap_{n \in \mathbb{N}}(A_n \cup A^\ast_n)$ and $x \notin \mathcal{R}(f)$ . i.e. for some $\varepsilon > 0$ there is no $\varepsilon$ -chain from $x$ to itself. Let $U$ be the open set consisting all points that can be connected from $x$ by an $\varepsilon$ -chain.

We wish to generate an attractor by $V$ , to do this all we need to check is $f(\overline{V}) \subseteq V$ :
For any $y \in \overline{V}$ there exists $y' \in V$ with $d(f(y), f(y')) < \varepsilon$ . Since $y' \in V$ , there is $\varepsilon$ -chain $x, x_1, \cdots, x_n, y'$ which gives rise to $\varepsilon$ -chain $x, x_1, \cdots, x_n, y', f(y)$ . Therefore $f(\overline{V}) \subseteq V$ .

Hence $\displaystyle A = \bigcap_{n \in \mathbb{N}} f^n(V)$ is an attractor with $V$ as a basin.

By assumption, $x \in A \cup A^\ast$ , since $A \in V$ and there is no $\varepsilon$ -chain from $x$ to itself, $x$ cannot be in $A$ . i.e. $x \in A^\ast$ Take any limit point $y$ of $(f^n(x))_{n\in \mathbb{N}}$ , since $A^\ast$ is compact $f$ -invariant we have $y \in A^\ast$ . But since we can find $N$ where $d(f^N(x), y)<\varepsilon$ , $x, f(x), \cdots, f^{N-1}(x), y$ gives an $\varepsilon$ -chain from $x$ to $y$ , hence $y \in V$ .

Recall that $V$ is a basin of $A$ hence $A^\ast \cap V$ is empty. Contradiction. Establishes proposition 2.

This proposition says that to study the chain recurrent set is the same as studying each attractor-repeller pair of the system. But the dynamics is very simple for each such pair as all points not in the pair will move towards the attractor under foreward iterate. We can see that such property is goint to be of importantce for our purpose since the dynamical for each attractor-repeller pair is like the sourse-sink map.

5. Main ingredient

Here we are going to prove a lemma that’s going to produce for us the ‘building blocks’ of our final construction. Namely a function for each attracting-repelling pair that strickly decreases along the orbits not in the pair. In light of proposition $2$ , we should expect to put those functions together to get our complete Lyapunov function.

Lemma1: For each attractor-repeller pair $A, A^\ast$ there exists continuous function $g: X \rightarrow [0,1]$ s.t. $g^{-1}(0)=A, \ g^{-1}(1)=A^\ast$ and $g(f(p)) < g(p)$ for all $p \notin (A\cup A^\ast)$ .

Proof: First we define $\varphi: X \rightarrow [0,1]$ s.t.

$\varphi(x) = \frac{d(x,A)}{d(x, A)+d(x, A^\ast)}$

Note that $\phi$ takes value $0$ only on $A$ and $1$ only on $A^\ast$ . However, $\phi$ can’t care less about orbits of $f$ .

Define $\bar\varphi(x) = \sup\{\varphi(f^n(x)) \ | \ n\in\mathbb{N} \}$ . Hence automatically for all $x$ , $\bar \varphi(f(x)) \leq \bar \varphi(x)$ . Since no points accumulates to $A^\ast$ under positive iterations, we still have the $\bar \varphi^{-1}(0)=A$ and $\bar \varphi^{-1}(1) = A^\ast$ .

We now show that $\bar\varphi$ is continuous:

For $x \in A^\ast$ and $(x_i)\rightarrow x$ , $\varphi(x_i) \leq \bar\varphi(x_i) \leq 1$ and $(\varphi(x_i)) \rightarrow 1$ hence $\bar\varphi(x_i)\rightarrow 1$ i.e. $\bar\varphi$ is continuous on $A^\ast$ .

For $x \in A$ we use the fact that $A$ is attracting. Let $B$ be a basin of $A$ . For all $(x_i) \rightarrow x$ , for any $\varepsilon>0$ , there is $N \in \mathbb{N}$ s.t. $f^N(B) \subseteq N(A, \varepsilon)$ . Therefore for some $x_i \in f^N(B)$ , all $f^n(x_i)$ are in $N(A, \varepsilon)$ i.e. $\varphi(f^n(x_i))\leq \frac{\varepsilon}{\varepsilon+C}$ hence $\bar\varphi(x_i)\leq \frac{\varepsilon}{\varepsilon+C}$ . But for some $M \in \mathbb{N}$ and all $m>M$ , $x_m \in B$ . Therefore $\bar\varphi(x_i) \rightarrow 0$ . $\bar\varphi$ is continuous on $A$ .

Let $T = \overline{B} \backslash f(B)$ , for any $\displaystyle x \in T, \ r=\inf_{x \in T} \varphi(x)$ , since $f^n(T) \subseteq f^n(\overline{B})$ , there exists $N>0$ s.t. for all $n>N$ $\varphi(f^n(T))\subseteq [0,r/2]$ . i.e. $\displaystyle \bar\varphi9x) = \max_{0\leq n\leq N} \varphi(f^n(x))$ which is countinous. Since those ‘bands’ $T$ partitions the whole $X$ (by taking $B_n$ to be $f^n(B) \backslash f^{n+1}(B)$ ), hence we have proven $\bar\varphi$ is continuous on the whole $X$ .

Finally, we define

$\displaystyle g(x) = \sum_{n=0}^\infty \frac{\bar\varphi(f^n(x))}{2^{n+1}}$

We check that $g$ is continuous since $\bar\varphi$ is. $g$ takes values $0$ and $1$ only on $A$ and $A^\ast$ , respectively. For any $x \notin (A \cup A^\ast)$ ,

$\displaystyle g(f(x))-g(x) = \sum_{n=0}^\infty \frac{\bar\varphi(f^{n+1}(x))-g(f^n(x))}{2^{n+1}}$

therefore $g(f(x))-g(x) = 0$ iff $\bar\varphi(f^{n+1}(x)) = \bar\varphi(f^n(x))$ for all $n$ i.e. $\bar\varphi$ is constant on the orbit of $x$ . But this cannot be since there is a subsequence of $(f^n(x))$ converging to some point in $A$ , continuity of $\bar\varphi$ tells us this constant has to be $0$ hence $x \in A$ .

Therefore $g$ is strictly decreasing along orbits of $f$ not in $(A \cup A^\ast)$ .

Establishes lemma 1.

6.Proof of the main theorem

The proof of the main theorem now follows easily from what we have established so far.

First we restate the fundamental theorem of dynamical systems:

Theorem: Complete Lyapunov function exists for any homeomorphisms on compact metric spaces.

Proof: First we enumerate the countably many attractors as $(A_n)_{n \in \mathbb{N}}$ . For each $A_n$ , we have function $g_n: X \rightarrow R$ where $g_n$ is $0$ on $A_n$ , $1$ on $A_n^\ast$ and is strictly decreasing on $X \backslash ( A_n \cup A_n^\ast)$ .
Define $g: X \rightarrow \mathbb{R}$ by

$\displaystyle g(x) = 2 \cdot \sum_{n=1}^\infty \frac{g_n(x)}{3^n}$

Since each $g_n$ is bounded between $0$ and $1$ , the sequence of partial sums converge uniformly. Hence the limit function $g$ is continuous.

For points $p \in \mathcal{R}(f)$ , we have $p \in (A_n \cup A_n^\ast)$ for all $n \in \mathbb{N}$ . i.e. $latex \ \forall n \in \N, \ g_n(p) = 0$ or $g_n(p) = 1$ . Hence we have

$g(p) = \sum_{n=1}^\infty \frac{2 \cdot g_n(p)}{3^n} = \sum_{n=1}^\infty \frac{a_n}{3^n}$

where each $a_n$ is in $\{0, 2 \}$ . This is same as saying the base- $3$ expansion of $g(p)$ only contains digits $0$ and $2$ . We conclude $g(\mathcal{R}(f)) \subseteq \mathcal{C}$ where $\mathcal{C}$ is the standard middle-third Cantor set in $[0, 1]$ . i.e. $g(\mathcal{R}(f))$ is compact and nowhere dense in $\mathbb{R}$ .

For $p \notin \mathcal{R}(f)$ , there exists $n \in \mathbb{N}$ such that $p \notin (A_n \cup A_n^\ast)$ , hence $g_n(f(p)) < g_n(p)$ . This implies $g(f(p)) < g(p)$ since $g_i(f(p)) \leq g_i(p)$ for all $i \in \mathbb{N}$ . i.e. $g$ is strictly decreasing along orbits that are not chain recurrent.

To show $g$ is constant only on the chain-transitive components, we need the following lemma:

Claim: $p, \ q \in \mathcal{R}(f)$ are in the same chain-transitive component iff there is no attracting-repelling pair $A, \ A^\ast$ where one of $p, \ q$ is in $A$ while the other in $A^\ast$ .

Proof (of claim): ” $\Rightarrow$ ” Suppose $x, y \in \mathcal{R}(f)$ and $x \sim y$ , for any attractor $A$ , if $x \notin A$ and $y \notin A$ , then $x, y$ are both in $A^\ast$ and we are done. Hence suppose at least one of $x, y$ is in $A$ . W.L.O.G. suppose $x \in A$ . Let $B$ be a basin of $A$ . Since $\overline{f(B)}, \ X\backslash B$ are closed and disjoint, we may choose $\varepsilon<\min\{ d(a, b) \ | \ a \in (X\backslash B), \ b \in \overline{f(B)} \}$ . By the same arguement as in proposition $2$ , there are no $\varepsilon/2$ -chain (with length $>1$ ) from any point in $f{B}$ to any point in $X \backslash B$ . Hence there is also no $\varepsilon/2$ -chain from any point in $A$ to any point in $A^\ast$ . Hence $y \notin A^\ast$ i.e. $y \in A$ .

” $\Leftarrow$ ” Suppose for any attractor $A$ , $x \in A$ iff $y \in A$ . For any $\varepsilon>0$ , let $U$ be the set of all points $y$ for which there is an $\varepsilon$ -chain from $x$ to $y$ , as defined in proposition $2$ . We have showed in proposition $2$ that $U$ is a basin of some attractor $A'$ . Since $x \in \mathcal{R}(f) \subseteq (A' \cup {A'}^\ast)$ and $x \in U$ , hence $x \in A'$ . Hence by our assumption, $y$ must be also in $A'$ . Hence $y \in U$ i.e. there is an $\varepsilon$ -chain from $x$ to $y$ . Since the construction is symetric, we may also show there is an $\varepsilon$ -chain from $y$ to $x$ . i.e. $x\sim y$ .

Establishes the claim.

Finally, for $p, q \in \mathcal{R}(f)$ , $g(p) = g(q)$ means $g(p)$ and $g(q)$ has the same base- $3$ expansion in the Cantor set. This is same as saying $\forall i \in \mathbb{N}, \ g_i(p) = g_i(q) \in \{0, 2\}$ , which is to say there is no $i \in \mathbb{N}$ for which one of $p, \ q$ is in $A_i$ while the other in $A_i^\ast$ . Hence by Lemma, we conclude that $g(p) = g(q)$ iff $p, \ q$ are in the same chain transitive component.

This establishes our theorem.

Fundemental Theorem of Dynamical Systems (Part 1)

April 15, 2010April 15, 2010 Conan1 Comment

This article was written as a homework of professor Wilkinson’s dynamical systems course. Since the content is expository and detailed presentation of the theorem is missing from many books, I decided to post it here. I have mostly followed a set of notes by John Franks, with additional discussions on the intuition and ideas behind the statement and the proof.

1.Introduction

So far we have discussed various different kinds of dynamical systems ranging from topological, smooth to hyperbolic and partially hyperbolic. One might wonder if there is a united theme to the subject as a whole. Indeed, as in many other subjects, there is a so-called fundamental theorem of dynamical systems. This theorem is first stated and proved by Conley in where he studies attractor and repellers. The theorem, loosely speaking, gives a universal decomposition of any systems on compact metric spaces into invariant compact sets wandering orbits that travels between such sets.

To state this more precisely we make an analogy with Morse theory: When looking at the gradient flow on a compact embedded manifold, we ‘decompose’ the manifold into critical points and orbits that originates and ends at critical points. In a similar spirit, given any homeomorphisms on a compact metric space, we may look at it’s ‘indecomposible’ compact invariant sets and how they are ‘connected’ by wandering orbits, we then ‘place’ those compact sets on different ‘heights’ and have all other point going between the minimal sets they originates and ends at. The theorem guarantees that we can ‘place’ the space in a way that all wandering orbits are going ‘down’ at all times.

In light of the theorem, we have descried the global structure of the system except for what happens on the ‘indecomposible’ sets. i.e. The problem of understanding general topological systems on compact manifolds is reduced to understanding ‘transitive’ homeomorphisms on compact sets. The latter, unfortunately, could still be quite complicated as we have seen in the Horseshoe example.

The theorem is proposed to be the Fundemental theorem of dynamical systems because of its nature in giving concise description of all possible behaviors of a system in the given setting. In some sense, dynamics is the study of limiting behrviors of all points under interation, the theorem breaks the system down into a recurrent part and a wandering part where the behavior of the wandering part is gradient-like. Since we have developped sets of different tools for studying systems that exhibits a lot of recurrence as well as for studying gradient-like systems, this allows us to connect combine the tool sets and treat any systems in the setting.

2.Background

In this section, we define $\varepsilon$ -chains, chain recurrent sets and chain transitive components for a homeomorphism on a compact metric space. Those concepts will come up in the statement of the fundamental theorem. In fact, those are going to be the ‘minimal compact sets’ we decompose our metric space into.

Given compact metric space $X$ and homeomorphism $f: X \rightarrow X$ ,
Definition: Given two points $p, q \in X$ , an $\varepsilon$ -chain from $p$ to $q$ is a sequence $x_1, x_2, \cdots x_n$ , $n>1$ where $x_1 = p, \ x_n = q$ and for all $1 \leq i \leq n-1$ , $d(f(x_i), x_{i+1}) < \varepsilon$ .

i.e. we take a point and start applying $f$ to it, but at each iterate, we are allowed to perturb the resulting point by $\varepsilon$ . Such ‘pseudo-orbits’ are in general much easier to obtain than true orbits.

More generally, $\varepsilon$ -chains can be taken infinite. i.e. if we have a (possibly infinite) subinterval $I \subseteq \mathbb{Z}$ , an $\varepsilon$ -chain indexed by $I$ is a set of points $(p_i)_{i \in I}$ s.t. $d(f(p_i), p_{i+1}) < \varepsilon$ whenever $i, i+1$ are both in $I$ .

Definition: $p \in X$ is chain recurrent if for all $\varepsilon > 0$ , there exists an $\varepsilon$ -chain from $p$ to itself. The set of all chain recurrent points in $X$ is called the \textbf{chain recurrent set}, denoted by $\mathcal{R}(f)$ .

Note that non-wandering points are necessarily chain recurrent: If $p \in X$ is non-wandering, we may take the neighborhood to be the $\varepsilon$ -ball around $p$ , since $p$ is non-wandering, we have some $n>1$ where $f^n(B_\varepsilon(p)) \cap B_\varepsilon(p) \neq \phi$ , we pick $q$ in the intersection and define $\varepsilon$ -chain $p, f^{-n}(q), f^{-n+1}(q), \cdots, q, p$ .

At this point, it’s perhaps worthwhile to mention our completed ordering of different notions of recurrence:

$\overline{\mbox{Per}(f)} \subseteq \mbox{Rec}(f) \subseteq \mbox{NW}(f) \subseteq \mathcal{R}(f)$

Each of the above inclusion can be made strict (see Exercises). Chain recurrence is perhaps the weakest notion I’ve seen for a point to be, in any sense, recurrent. A Friendly challenge to the reader: think of a case where you feel confortable calling a point ‘recurrent’ while it’s not in the chain recurrent set of the system.

We now define equivlence relation on $\mathcal{R}(f)$ as follows:

For $p, q$ in $\mathcal{R}(f)$ , $p \sim q$ iff for all $\varepsilon > 0$ , there are $\varepsilon$ -chains from $p$ to $q$ and from $q$ to $p$ . $\sim$ is reflexive since all points in $\mathcal{R}(f)$ are chain recurrent; symmetric by definition and transitive by the obvious composition of $\varepsilon$ -chains.

Definition: The equivalence classes in $\mathcal{R}(f)$ for $\sim$ are called chain transitive components.

It’s easy to check that chain transitive components are compact and $f$ -invariant. Those are components that’s transitive in a very weak sense. i.e. any two points are connected by a ‘pseudo-orbit’, or equivalently, there is a dense (infinite) pseudo-orbit. (see exercises)

As mentioned above, throughout the rest of the chapter, we will consider chain transitive components as ‘indecomposible parts’ of our system. Those are the parts for which all points are ‘recurrently’ and each component is ‘transitive’, both in a very weak sense. We further specify how does the points that are not in the chain-recurrent set iterates between those components.

3.Statement of the theorem

Given compact metric space $X$ and homeomorphism $f: X \rightarrow X$ ,

Definition: $g: X \rightarrow \mathbb{R}$ is a complete Lyapunov function for $f$ if:

$\forall \ p \notin \mathcal{R}(f), \ g(f(p)) < g(p)$

$\forall \ p, q \in \mathcal{R}(f), \ g(p) = g(q) \ \mbox{iff} \ p \sim q$

$g(\mathcal{R}(f)) \ \mbox{is compact and nowhere dense in} \ \mathbb{R}$

Hence this is a function that stays constant only on the chain transitive components and strictly decreases along any orbit not in $\mathcal{R}(f)$ . We also require the image of $\mathcal{R}(f)$ to be compact and nowhere dense which cooresponds to the ‘critical values’ of a gradient function being compact nowhere dense as a result of Sard’s theorem.

Fundemental theorem of dynamical systems:

Complete Lyapunov function exists for any homeomorphisms on compact metric spaces.

As a historical remark, the theorem first appeared in Charles Conley’s CBMS monograph Isolated Invariant Sets and the Morse Index in 1978 [C]. In the book he developed the theory of attractor-repeller pairs in relation to Morse decomposition and index theory. The above theorem was one of the major results. Although Conley was originally more focused on the setting where instead of a homeomorphism, we have a continuous flow on the manifold (which makes it even more similar to the gradient flow), but this discrete formulation became more popular as the theory develops. The theorem is later proposed by D. Norton as the fundamental theorem of dynamical in 1995.

The proof is going to be a specific construction: First, we define a family of partitions of the chain recurrent set, each divides the set into two pieces (i.e. a attractor-repeller pair intersected with $\mathcal{R}$ ). Then we prove the family is countable and points in the same chain transitive component are not separated by any partition in the family. Furthermore, each chain-transitive component is uniquely defined by specifying which set does it belong to in each partition. i.e. the smallest common refinement for the family exactly partitions $\mathcal{R}$ into chain-transitive components. (section 4)

Next, for each attractor-repeller pair, we prove the existence of a function that takes value $0$ on the attractor and $1$ on the repeller and strictly decreases along orbits of points that’s not in $\mathcal{R}(f)$ . We should also mention the fact that all points that are contained in one of the sets in each pair must be in chain recurrent. (section 5)

The complete Lyapunov function is then constructed by taking an appropriate infinite sum of such functions. This way we get a function that separates all chain transitive components, stays constant on each component and strictly decreases along all orbits which are not in $\mathcal{R}$ . The image of the chain recurrent set will be contained in the middle-third Cantor set. (section 6)

(see part 2 for sections 4-6)

Hausdorff dimension of projections

March 30, 2010March 31, 2010 ConanLeave a comment

A few days ago, professor Wilkinson asked me the following question on google talk (while I was in Toronto):

“Say that a set in $\mathbb{R}^n$ is a $k$ -zero set for some integer $k<n$ if for every $k$ -dimensional subspace $P$ , saturating the set $X$ by planes parallel to $P$ yields a set of $n$ -dimensional Lebesgue measure zero. How big can a $k$ -zero set be?”

On the spot my guess was that the Hausdorff dimension of the set is at most $n-k$ . In deed this is the case:

First let’s note that $n$ -dimensional Lebesgue measure of the $P$ -saturated set is $0$ iff the $n-k$ dimensional Lebesgue measure of the projection of our set to the $n-k$ subspace orthogonal to $P$ is $0$ .

Hence the question can be reformulated as: If a set $E \subseteq \mathbb{R}^n$ has all $n-k$ dimensional projection being $n-k$ zero sets, how big can the set be?

Looking this up in the book ‘The Geometry of Fractal Sets’ by Falconer, indeed it’s a theorem:

Theorem: Let $E \subseteq \mathbb{R}^n$ compact, $\dim(E) = s$ (Hausdorff dimension), let $G_{n,k}$ be the Garssmann manifold consisting of all $k$ -dimensional subspaces of $\mathbb{R}^n$ , then
a) If $s \leq k$ , $\dim(\mbox{Proj}_\Pi E) = s$ for almost all $\Pi \in G_{n,k}$

b) If $s > k$ , $\mbox{Proj}_\Pi E$ has positive $k$ -dimensional Lebesgue measure for almost all $\Pi \in G_{n,k}$ .

In our case, we have some set with all $n-k$ -dimensional projection having measure $0$ , hence the set definitely does not satisfy b), i.e. it has dimensional at most $n-k$ . Furthermore, a) also gives that if we have a uniform bound on the dimension of almost all projections, this is also a bound on the dimension of our original set.

This is strict as we can easily find sets that’s $n-k$ dimensional and have all such projections measure $0$ . For example, take an $n-k$ subspace and take a full-dimension measure $0$ Cantor set on the subspace, the set will have all projections having measure $0$ .

Also, since the Hausdorff dimension of any projection can’t exceed the Hausdorff dimension of the original set, a set with one projection having positive $n-k$ measure implies the dimension of the original set is $\geq n-k$ .

Question 2: If one saturate a $k$ -zero set by any smooth foliations with $k$ -dimensional leaves, do we still get a set of Lebesgue measure $0$ ?

We answer the question in the affective.

Given foliation $\mathcal{F}$ of $\mathbb{R}^n$ and $k$ -zero set $E$ . For any point $p \in E$ , there exists a small neighborhood in which the foliation is diffeomorphic to the subspace foliation of the Euclidean space. i.e. there exists $f$ from a neighborhood $U$ of $p$ to $(-\epsilon, \epsilon)^n$ where the leaves of $\mathcal{F}$ are sent to $\{\bar{q}\} \times (-\epsilon, \epsilon)^k$ , $\bar{q} \in (-\epsilon, \epsilon)^{n-k}$ .

By restricting $f$ to a small neighborhood (for example, by taking $\epsilon$ to be half of the origional $\epsilon$ ), we may assume that $f$ is bi-Lipschitz. Hence the measure of the $\mathcal{F}$ -saturated set inside $U$ of $U \cap E$ is the same as $f(U \cap E)$ saturated by parallel $k$ -subspaces inside $(-\epsilon, \epsilon)^n$ . Dimension of $f(E)$ is the same as dimension of $E$ which is $\leq n-k$ , if the inequality is strict, then all projections of $f(E)$ onto $n-k$ dimensional subspaces has measure $0$ i.e. the saturated set by $k$ -planes has $n$ dimensional measure $0$ .

…to be continued

Whitney’s extension theorem revisited

March 9, 2010March 20, 2010 Conan1 Comment

I (very surprisingly) bumped into Charles Fefferman at Northwestern this afternoon…Hence we talked math for a little bit. Among other things I mentioned that I’ve been trying to extend $C^1$ functions to the disc volume-preservingly. After trying on the board for a while, he laughs out loud when he saw that this may be obtained applying his favorite Whitney’s extension theorem. (I’ll discuss what he did later in the poster)

Mean while, it’s a pity that I’ve never written a post on Whitney’s extension theorem, hence here it is~

Given a compact subset $K$ in $\mathbb{R}^n$ and a function $f: K \rightarrow \mathbb{R}$ , when can we extend it to a $C^r$ function on the whole $\mathbb{R}^n$ ?

First we note that there are obvious cases for which this can’t be done: for example, if we take $E$ to be a segment in $\mathbb{R}^2$ and $f$ a one-variable function of lower regularity than $r$ , then of course there are no way to find a $C^r$ extension.

Hence it’s only reasonable to restrict our attention to those $f$ that has ‘candidate derivatives’ of all orders no larger than $r$ at all points in $E$ .

i.e. For any $k$ -fold subscript $d= (d_1, d_2, \cdots, d_k)$ with $d_1+d_2+ \cdots +d_k \leq r$ (we will denote $d_1+d_2+ \cdots +d_k = |d|$ , there is a continuous function $f_d: K \rightarrow \mathbb{R}$ with the following property:

For all $x_o \in K$ , $\displaystyle f_d(x) = \sum_{|l| \leq r-|d|} \frac{f_{l+d}(x)}{l!}(x-x_0)^l+R_d(x, x_0)$ where $R_d(x, x_0) \sim o(|x-x_0|^{r-|d|})$ as $x \rightarrow x_0$ and is uniform in $x_0$ .

i.e. The functions $f_\alpha$ are compatible as Taylor coefficients of some $C^r$ function on $\mathbb{R}^n$ , which is absolutely necessary for a $C^r$ extension to exist.

Whitney’s extension theorem: (classical version)

Suppose a set of functions $f_\alpha$ with all multi-index $| \alpha | \leq r$ satisfying the above Taylor condition at all points in $K$. Then there is a $C^r$ function $\hat{f}: \mathbb{R}^n \rightarrow \mathbb{R}$ s.t. $\hat{f}|_K = f_{\bar{0}}$ and for all $\alpha \leq r$ , $(D^\alpha \hat{f})|_K = f_\alpha$ . Furthermore, $\hat{f}$ can be taken real analytic on $\mathbb{R}^n \backslash K$ .

This is indeed the best one could hope for. i.e. there is a $C^r$ extension whenever possible, furthermore the extension is at worst $C^r$ at the points which it is given to be only $C^r$ and much better (analytic) everywhere else.

However, sometimes we would like to control the $C^r$ norm of the resulting function in terms of the $C^r$ norm of the function on $K$ .

Theorem: (Fefferman)

For any $n, \ r$ , there exists $C$ such that the extension $||\hat{f}|| \leq C \cdot ||f||$ where the norm is the $C^r$ norm.

Systolic inequality on the 2-torus

March 1, 2010March 3, 2010 Conan5 Comments

Starting last summer with professor Guth, I’ve been interested in the systolic inequality for Riemannian manifolds. As a starting point of a sequence of short posts I plan to write on little observations I had related to the subject, here I’ll talk about the baby case where we find the lower bound of the systole on the $2$ -torus in terms of the area of the torus.

Given a Riemannian manifold $(M, g)$ where $g$ is the Riemannian metric.

Definition: The systole of $M$ is the length of smallest homotopically nontrivial loop in $M$ .

We are interested in bounding the systole in terms of the $n$ -th root of the volume of the manifold ( where $n$ is the dimension of $M$ ).

Note that the systole is only defined when our manifold has non-trivial fundamental group. I wish to remark that for the case of n-torus, having an inequality of the form $(\mbox{Sys}(\mathbb{T}^n))^n \leq C \cdot \mbox{Vol}(\mathbb{T}^n)$ is intuitive as we can see in the case of an embedded $2$ -torus in $\mathbb{R}^3$ , we may deform the metric (hence the embedding) to make a non-contactable loop as small as we want while keep the volume constant, however when we attempt to make the smallest such loop large when not changing the volume, we can see that we will run into trouble. Hence it’s expected that there is an upper bound for the length of the smallest loop.

Since if only one loop in some homotopy class achieves that minimal length, we should be able to enlarge it and contract some other loops in that class to enlarge the systole and keep the volume constant. Hence it’s tempting to assume that all loops in the same class are of the same length. In the $2$ -torus case, such thing is the flat torus. Since any flat torus has systole proportional to $(\mbox{Vol}(\mathbb{T}^2))^{\frac{1}{2}}$ , we have reasons to expect the optimal case fall inside this family. i.e.

$(\mbox{Sys}(\mathbb{T}^2))^2 \leq C \cdot \mbox{Vol}(\mathbb{T}^2)$ .

This is indeed the case. The result was given in an early unpublished result by Loewner.

Let’s first optimize in the class of flat torus:

My first guess was that $C$ cannot be made less than $1$ i.e. the torus $\mathbb{R}^2/ \mathbb{Z}^2$ is the optimum case. However, this is not true. Let’s be more careful:

$\mathbb{T}^2 = \mathbb{R}^2 / (0,c)\mathbb{Z} \times (a, b)\mathbb{Z}$

Since by scaling does not change ratio between $(\mbox{Vol}(\mathbb{T}^2))$ and $(\mbox{Sys}(\mathbb{T}^2))^2$ , we may normalize and let $c=1$

Let $\alpha, \beta$ be generators of the fundamental group of $\mathbb{T}^2$ length of all geodesic loops in class $[\alpha], \ [\beta]$ are the side lengths of the parallelepiped i.e. $1$ and $||(a,b)||$ . W.L.O.G we suppose $a, b > 0$ . $\alpha \beta^{-1}$ has length $||((1-a),b)||$ and all geodesics in other classes are at least twice as long as one of the above three.

Hence the systole is maximized when those three are equal, we get $a=1/2, b=\sqrt{3}/2$ . The systole in this case is $1$ and the volume is $\sqrt{3}/2$ . Hence for any flat torus, we have

$(\mbox{Sys}(\mathbb{T}^2))^2 \leq \frac{2}{\sqrt{3}} \cdot \mbox{Vol}(\mathbb{T}^2)$ .

Theorem (Loewner): This bound holds for any metric $g$ on $\mathbb{T}^2$ .

Proof: We will show this by reducing the case to flat metric.

$g$ induced an almost complex structure on $\mathbb{T}^2$ , on surfaces, any almost complex structure is integrable. Hence there exists $f:\mathbb{T}^2 \rightarrow \mathbb{R}^+$ and $g= f \dot g_0$ where $(\mathbb{T}^2, g_0)$ is a Riemann surface.

By uniformization theorem, $(\mathbb{T}^2, g_0)$ is the quotient of $\mathbb{C}$ by a discrete lattice. i.e. $(\mathbb{T}^2, g_0)$ is a flat torus $\mathbb{R}^2 / (0,c)\mathbb{Z} \times (a, b)\mathbb{Z}$ .

By scaling of the torus, we may assume the volume of the manifold is $1$ i.e.

$\displaystyle \int_{\mathbb{T}^2} f \ dV_{g_0} = 1 = \mbox{Vol}(\mathbb{T}^2, g_0)$

Any nontrivial homotopy class of loops on $(\mathbb{T}^2, g)$ can be represented by a straight loop on the flat torus. The length of such a loop in $(\mathbb{T}^2, g)$ is merely integration of $f$ along the segment.

Here we have a family of loops in the homotopy class that is straight, by taking a segment of appropriate length orthogonal to the loops, we have the one-parameter family of parallel loops foliate the torus. Hence integrating over the segment of the length of the loops gives us the total volume of the torus. By Fubini, we have at least one loop is longer than volume of the torus over length of the segment we integrated on, which is the length of the straight loop in the flat torus.

Therefore the systole of $(\mathbb{T}^2, g)$ is smaller than the minimum length of straight loops which is smaller than that of the flat torus. While the volume are the same. Hence it suffice to optimize the ratio in the class of flat tori. Establishes the theorem.

Combining the pervious statement, we get

$(\mbox{Sys}(\mathbb{T}^2))^2 \leq \frac{2}{\sqrt{3}} \cdot \mbox{Vol}(\mathbb{T}^2)$

for any metric on the torus.

Area777

Chasing dreams from mathematics to the real world

Fundemental Theorem of Dynamical Systems (Part 2)

Fundemental Theorem of Dynamical Systems (Part 1)

Hausdorff dimension of projections

Whitney’s extension theorem revisited

Systolic inequality on the 2-torus

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