10. Let f_a:S^1\to S^1, a\in[0,1] be a strictly increasing family of homeomorphisms on the unit circle, \rho(a) be the rotation number of f_a. Poincare observed that \rho(a)=p/q if and only if f_a admits some periodic points of period q. In this case f_a^q admits fixed points.

Note that a\mapsto \rho(a) is continuous, and non-decreasing. However, \rho may not be strictly increasing. In fact, if \rho(a_0)=p/q and f^q\neq Id, then \rho is locked at p/q for a closed interval I_{p/q}\ni a_0. More precisely, if f^q(x) > x for some x, then \rho(a)=p/q on [a_0-\epsilon,a_0] for some \epsilon > 0; if f^q(x) < x for some x, then \rho(a)=p/q on [a_0,a_0+\epsilon] for some \epsilon > 0; while a_0\in \text{Int}(I_{p/q}) if both happen.

Also oberve that if r=\rho(a)\notin \mathbb{Q}, then I_r is a singelton. So assuming f_a is not unipotent for each a\in[0,1], the function a\mapsto \rho(a) is a Devil’s staircase: it is constant on closed intervals I_{p/q}, whose union \bigcup I_{p/q} is dense in I.

9. Let X:M\to TM be a vector field on M, \phi_t:M\to M be the flow induced by X on M. That is, \frac{d}{dt}\phi_t(x)=X(\phi_t(x)). Then we take a curve s\mapsto x_s\in M, and consider the solutions \phi_t(x_s). There are two ways to take derivative:

(1) \displaystyle \frac{d}{dt}\phi_t(x_s)=X(\phi_t(x_s)).

(2) \displaystyle \frac{d}{ds}\phi_t(x_s)=D\phi_t(\frac{d}{ds}x_s)), which induces the tangent flow D\phi_t:TM\to TM of \phi_t:M\to M.

Combine these two derivatives together:

\displaystyle \frac{d}{dt}D_x\phi_t(x_s')=\frac{d}{dt}\frac{d}{ds}\phi_t(x_s) =\frac{d}{ds}\frac{d}{dt}\phi_t(x_s)=\frac{d}{ds}X(\phi_t(x_s)) =D_{\phi_t(x)}X\circ D_x\phi_t(x_s').

This gives rise to an equation \displaystyle \frac{d}{dt}D_x\phi_t=D_{\phi_t(x)}X\circ D_x\phi_t.


Formally, one can consider the differential equation along a solution x(t):
\displaystyle \frac{d}{dt}D(t)=D_{\phi_t(x)}X\circ D(t), D(0)=Id. Then D(t) is called the linear Poincare map along x(t). Suppose x(T)=x(0). Then D(T) determines if the periodic orbit is hyperbolic or elliptic. Note that the path D(t), 0\le t\le T contains more information than the above characterization.

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The real decay rate

Let f be a C^2 uniform expanding map on the 1-torus \mathbb{T}, \mu be the unique ACIP of f, which is exponentially mixing. That is, there exists \lambda\in(0,1) such that |C(\phi,\psi,f^n)|\le C\lambda^n for any two Lipschitz functions \phi,\psi on \mathbb{T}, where
C(\phi,\psi,f^n,\mu)=\int \phi\circ f^n\cdot \psi d\mu -\mu(\phi)\cdot\mu(\psi) be the correlation function.

Let h be a C^2 diffeomorphism on \mathbb{T}, g=h^{-1}fh be the induced map, and h_\ast \nu=\mu. The new correlation function

C(\phi,\psi,g^n,\nu)=\int \phi\circ g^n\cdot \psi d\nu-\nu(\phi)\cdot\nu(\psi)
=\int \hat\phi\circ f^n(hx)\cdot \hat\psi(hx) d\nu-\nu(\phi)\cdot\nu(\psi)
=\int \hat\phi\circ f^n \cdot \hat\psi d\mu-\mu(\hat\phi)\cdot\mu(\hat\psi),

where \hat\phi=\phi\circ h^{-1}. Therefore, the two smoothly conjugate systems (g,\nu) and (f,\mu) have the same mixing rate.

Assuming h is close to identity, we see that g=h^{-1}fh is also C^2 uniformly expanding, and one may derive the mixing rate of (g,\nu) independently. However, this new rate may be different (better or worse) from \lambda. For example, f could be the linear expanding ones and archive the best possible rate among its conjugate class. Could one detect this rate from (g,\nu) itself?

In the general case, two expanding maps on \mathbb{T} are only topologically conjugate (via full shifts). So it is possible that the decay rate varies in the topologically conjugate classes.


Dynamics of the Weil-Petersson flow: basic geometry of the Weil-Petersson metric II

Disquisitiones Mathematicae

In the first post of this series, we planned to discuss in the third and fourth posts the proof of the following ergodicity criterion for geodesic flows in incomplete negatively curved manifolds of Burns-Masur-Wilkinson:

Theorem 1 (Burns-Masur-Wilkinson) Let $latex {N}&fg=000000$ be the quotient $latex {N=M/Gamma}&fg=000000$ of a contractible, negatively curved, possibly incomplete, Riemannian manifold $latex {M}&fg=000000$ by a subgroup $latex {Gamma}&fg=000000$ of isometries of $latex {M}&fg=000000$ acting freely and properly discontinuously. Denote by $latex {overline{N}}&fg=000000$ the metric completion of $latex {N}&fg=000000$ and $latex {partial N:=overline{N}-N}&fg=000000$ the boundary of $latex {N}&fg=000000$.Suppose that:

  • (I) the universal cover $latex {M}&fg=000000$ of $latex {N}&fg=000000$ is geodesically convex, i.e., for every $latex {p,qin M}&fg=000000$, there exists an unique geodesic segment in $latex {M}&fg=000000$ connecting $latex {p}&fg=000000$ and $latex {q}&fg=000000$.
  • (II) the metric completion $latex {overline{N}}&fg=000000$ of $latex {(N,d)}&fg=000000$ is compact.
  • (III) the boundary $latex {partial N}&fg=000000$ is volumetrically cusplike, i.e., for…

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Dynamics of the Weil-Petersson flow: basic geometry of the Weil-Petersson metric I

Disquisitiones Mathematicae

Today we will define the Weil-Petersson (WP) metric on the cotangent bundle of the moduli spaces of curves and, after that, we will see that the WP metric satisfies the first three items of the ergodicity criterion of Burns-Masur-Wilkinson (stated as Theorem 3 in the previous post).

In particular, this will “reduce” the proof of the Burns-Masur-Wilkinson theorem (of ergodicity of WP flow) to the verification of the last three items of Burns-Masur-Wilkinson ergodicity criterion for the WP metric and the proof of the Burns-Masur-Wilkinson ergodicity criterion itself.

We organize this post as follows. In next section we will quickly review some basic features of the moduli spaces of curves. Then, in the subsequent section, we will start by recalling the relationship between quadratic differentials on Riemann surfaces and the cotangent bundle of the moduli spaces of curves. After that, we will introduce the Weil-Petersson and the Teichmüller metrics…

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Dynamics of the Weil-Petersson flow: Introduction

Disquisitiones Mathematicae

Boris Hasselblatt and Françoise Dal’bo are organizing the event “Young mathematicians in Dynamical Systems” at CIRM (Luminy/Marseille, France) from November 25 to 29, 2013.

This event is part of the activities around the chaire Jean-Morlet of Boris Hasselblatt. Among the topics scheduled in this event, there is a mini-course by Keith Burns and myself around the dynamics of the Weil-Petersson (WP) geodesic flow.

In our mini-course, Keith and I plan to cover some aspects of Burns-Masur-Wilkinson theorem on the ergodicity of WP flow and, maybe, some points of our joint work with Masur and Wilkinson on the rates of mixing of WP flow.

In order to help me prepare my talks, I thought it could be a good idea to make my notes available on this blog.

So, this post starts a series of 6 posts (vaguely corresponding the 6 lectures of the mini-course) on the dynamics of…

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Martingale and its application to dynamical systems

In the last week of May I attended two lectures given by Professor Matthew Nicol.

Let (\Omega,\mu) be a prob space with a \sigma-algebra \mathcal{B}. Let \mathcal{F}\prec \mathcal{B} be a sub \sigma-algebra.

Example. f(x)=2x (\text{mod} 1) on \mathbb{T}, and \mathcal{B} be the Borel \sigma-algebra. Let \mathcal{F}=f^{-1}\mathcal{B}. Note that (0.2,0.3)\notin\mathcal{F}.

Let Y be a \mathcal{B}-measurable r.v. and Y\in L^1(\mu). The conditional expectation E(Y|\mathcal{F}) is the unique \mathcal{F}-measurable r.v. Z satisfying Z^{-1}(a,b)\in \mathcal{F} for all (a,b), and \int_F Z d\mu=\int _A Y d\mu for all A\in \mathcal{F}.

Note that E(Y|\mathcal{F})=Y if and only if Y is \mathcal{F}-measurable; and E(Y|\mathcal{F})=E(Y) if Y is independent of \mathcal{F}.

Let (X_n)_{n\ge 0} be a stationary ergodic process with stationary initial distribution \mu. A basic problem is to find sufficient conditions on (X_n)_{n\ge 0} and on functions \phi\in L^2_0(\mu) such that \displaystyle S_n(\phi)=\sum_{k=1}^n \phi(X_k) satisfies the central limit theorem (CLT) \displaystyle \frac{1}{\sqrt{n}}S_n(\phi) \to N(0,\sigma^2), where the limit variance is given by \displaystyle \sigma^2(\phi)=\lim_{n\to\infty}\frac{1}{n}E(S^2_n(\phi)).

Let f be a conservative diffeomorphism on (M,m). There are two operators: \phi\mapsto U\phi=\phi\circ f, and \phi\mapsto P\phi via \int P\phi\cdot \psi=\int \phi\cdot \psi\circ f for all test function \psi.

Property. PU(\phi)=\phi (vol-preserving) and UP(\phi)=E(\phi|f^{-1}\mathcal{B}).

Let \mathcal{F}_n be an increasing sequence of \sigma-algebras. Then a sequence of r.v. S_n is called a martingale w.r.t. \mathcal{F}_n, if S_n is \mathcal{F}_n-measurable, E(S_{n+1}|\mathcal{F}_n)=S_n.

Let \mathcal{F}_n be a decreasing sequence of \sigma-algebras. Then a sequence of r.v. S_n is called a reverse martingale w.r.t. \mathcal{F}_n, if S_n is \mathcal{F}_n-measurable, E(S_{n}|\mathcal{F}_m)=S_m for any n\le m.

Theorem. Let \{X_n:n\ge 1\} be a stationary ergodic sequence of (reverse) martingale differences w.r.t. \{\mathcal{F}_n\}. Suppose E(X_n)=0, and \sigma^2=\text{Var}(X_i)>0. Then \displaystyle \frac{1}{\sigma\sqrt{n}}\sum_{i=1}^n X_i \to N(0,1) in distribution.

Gordin: Suppose (f,m) is ergodic. Consider the Birkhoff sum \displaystyle \sum_{i=1}^n \phi\circ f^i for some \phi with \int \phi=0. The time series \phi\circ f^i can be approximated by martingale differences provided the correlations decay quickly enough.

Suppose there exists p(n) with \sum  p(n) < \infty, such that \|P^n\phi\|\le C\cdot p(n)\|\phi\|. Then define \displaystyle g=\sum_{n\ge 1}P^n\phi, and let X=\phi+g-g\circ f.

Property. Let f:M\to M be such that f^{-n}\mathcal{B} is decreasing. \displaystyle S_n=\sum_{i=1}^n X\circ f^i is a reverse martingale with respect to f^{-n}\mathcal{B}.

Proof. Note that PX=P\phi+Pg-PUg=0. Then E(X|f^{-1}\mathcal{B})=UP(X)=U0=0.
Let k < n. It remains to show E(X\circ f^k|f^{-n}\mathcal{B})=0. To this end, we pick an element A\in f^{-n}\mathcal{B} and write it as A=f^{-k-1}C for some C\in f^{k+1-n}\mathcal{B}. Then \displaystyle \int_A X\circ f^k dm=\int_{f^{-1}C}X dm =\int_{f^{-1}C} E(X|f^{-1}\mathcal{B}) dm=\int_{f^{-1}C}0 dm=0. This completes the proof.

Three theorems of Gordin. Let (\Omega,\mu,T) be an invertible \mu-preserving ergodic system, X\in L^1(\mu) and X_k(x)=X(T^kx) be a strictly stationary ergodic sequence.

(*) \displaystyle \limsup_{n\to\infty}\frac{1}{\sqrt{n}}E|S_n| < \infty

Theorem 1. Suppose there exists \displaystyle \mathcal{F}_k\subset T^{-1}\mathcal{F}_k=\mathcal{F}_{k+1} such that \displaystyle \sum_{k\ge 0} E|E(X_0|\mathcal{F}_{-k})|<\infty, \displaystyle \sum_{k\ge 0} E|X_0-E(X_0|\mathcal{F}_{k})| < \infty. Then (*) implies \displaystyle \lambda:=\lim_{n\to\infty}\frac{1}{\sqrt{n}}E|S_n| exists, and \displaystyle \frac{1}{\sqrt{n}}S_n\to N(0,\lambda^2\pi/2) in distribution (degenerate if \lambda=0).

–Mixing condition. Let \displaystyle \alpha(n):=\sup\{P(A\cap B)-P(A)P(B):A\in\mathcal{F}^0_{-\infty}, B\in\mathcal{F}^{\infty}_n\}.

Theorem 2. Suppose for some 1/p+1/q=1, X\in L^p(\mu) and \displaystyle \sum_{n\ge 1}\alpha(n)^{1/q} < \infty. Then (*) implies the conclusion of Theorem 1.

–uniform mixing condition. Let \displaystyle \phi(n):=\sup\{P(B|A)-P(B):A\in\mathcal{F}^0_{-\infty}, \mu(A) > 0, B\in\mathcal{F}^{\infty}_n\}.

Theorem 3. Suppose X\in L^1(\mu) and \displaystyle \sum_{n\ge 1}\phi(n) < \infty. Then (*) implies the conclusion of Theorem 1.

Cuny–Merlevede: not only the CLT, but also the ASIP holds under the above conditions.

Note that we started with an invariant measure m. The operator U and P can be defined for all non-conservative maps. To emphasize the difference, we use \hat P. Suppose \hat P h=h for some h\in L^1(m). Then \mu=hm is an absolutely continuous invariant prob. measure:

\displaystyle \int \phi\circ f d\mu=\int \phi\circ f h dm=\int \phi\cdot \hat P h dm=\int \phi hdm=\int\phi d\mu.

Then we can rewrite \displaystyle P\phi=\frac{1}{h}\hat P(h\phi), in the sense that \displaystyle \int P(\phi)\cdot \psi d\mu=\int \phi\cdot \psi\circ f d\mu  =\int \phi h\cdot \psi\circ f dm
\displaystyle =\int\hat P(\phi h)\cdot \psi dm  \int \frac{1}{h}\hat P(\phi h)\cdot \psi d\mu.

Perron–Frobenius theorem

Today I attended a lecture given by Vaughn Climenhaga. He presented a proof of the following version of Perron–Frobenius theorem:

Let \Delta\subset \mathbb{R}^d be the set of probability vectors, P be a stochastic matrix with positive entries. Then
–there is a positive probability \pi\in \Delta fixed by P
–the eigenspace E_1=\mathbb{R}\pi
–the spectra \Sigma(P)\subset B(0,r)\cup\{1\} for some r<1
–for all v\in\Delta, P^n v\to \pi exponentially as n\to \infty.

Proof. (1) Let v\in\Delta. Then \sum_i v_i=1, and
\sum_i (Pv)_i=\sum_i \sum_j p_{ij}v_j=\sum_j v_j=1. So Pv\in \Delta. Moreover, Pv is positive and P(\Delta)\subset \text{Int}(\Delta). Therefore there exists some point \pi\in\text{Int}(\Delta) fixed by P.

(2). Suppose on the contrary that there exists v\notin \mathbb{R}\pi that is also fixed by P. Then P fixes every vector in the plane \Pi:=\mathbb{R}v\oplus\mathbb{R}\pi, in particular the points on \Pi\cap \partial \Delta. This contradicts (1).

(3). We use the norm |v|=\sum|v_i|. Note that |Pv|=\sum_i |(Pv)_i|\le \sum_{ij}p_{ij}|v_j|=|v|. So \Sigma(P)\subset D(0,1). It suffices to show \Sigma(P)\backslash\{1\}\cap S^1=\emptyset. If not, pick one ,say \lambda, and n\ge 1 such that \text{Re}(\lambda^n)1 for any \epsilon>0.

Consider the matrix A=P^n-\epsilon I, which is positive if \epsilon is small enough. Then we have |A|\le |P_n| and hence \Sigma(A)\subset D(0,1). This contradicts the fact \lambda^n-\epsilon is an eigenvalue of A.

(4). Let W\subset \mathbb{R}^d be the subset of vectors with zero mean: \sum v_i=0, and consider the decomposition \mathbb{R}^d=\mathbb{R}\pi\oplus W. Note that PW\subset W and hence \Sigma(P|_{W})\subset D(0,r). For any v\in\Delta, we have v=\pi+w for some w\in W. Then |P^nv-\pi|=|P^n(v-\pi)|=|P^nw|\le Cr^n|w|.

Only light calculations are used in his lecture. As pointed by Vaughn, this approach does not give precise information of the r.

Some notes

Let M be a complete manifold, \mathcal{K}_M be the set of compact/closed subsets of $M$. Let X be a complete metric space.

A map \phi: X\to \mathcal{K}_M is said to be upper-semicontinuous at x, if
for any open neighbourhood U\supset \phi(x), there exists a neighbourhood V\ni x, such that \phi(x')\subset U for all x' \in V.
or equally,
for any x_n\to x, and any sequence y_n\in \phi(x_n), the limit set \omega(y_n:n\ge 1)\subset \phi(x).
Viewed as a multivalued function, let G(\phi)=\{(x,y)\subset X\times M: y\in\phi(x)\} be the graph of \phi. Then \phi is u.s.c. if and only if G(\phi) is a closed graph.

And \phi is said to be lower-semicontinuous at x, if
for any open set U intersecting \phi(x), there exists neighbourhood V\ni x such that \phi(x')\cap U\neq\emptyset for all x'\in V.
or equally, for any y\in \phi(x), and any sequence x_n\to x, there exists y_n\in \phi(x_n) such that y\in \omega(y_n:n\ge 1).

Let \mathrm{Diff}^r(M) be the set of C^r diffeomorphisms, and H(f) be the closure of transverse homoclinic intersections of stable and unstable manifolds of some hyperbolic periodic points of f. Then H is lower semicontinuous.

Given f_n\to f. Note that it suffices to consider those points x\in W^s(p,f)\pitchfork W^u(q,f). Let p_n and q_n be the continuations of p and q for f_n. Pick \rho large enough such that x\in W^s_\rho(p,f)\pitchfork W^u_\rho(q,f). Then for f_n sufficiently close to f, W^s_\rho(p_n,f_n) and W^u_\rho(q_n,f_n) are C^1 close to W^s_\rho(p,f) and W^u_\rho(q,f). In particular x_n\in W^s_\rho(p_n,f_n)\pitchfork W^u_\rho(q_n,f_n) is close to x.

Admissible perturbations of the tangent map

Franks’s Lemma is a major tool in the study of differentiable dynamical systems. It says that along a simple orbit segment E=\{x,fx,\cdots,f^nx\}, the perturbation of A\sim D_xf^n can be realized via a perturbation of the map g\sim f (which preserves the orbit segment). Moreover, such a perturbation is localized in a neighborhood of E, and it can be made arbitrarily C^1-close to f.

There have been various generalizations of Franks’ Lemma. Some constraints have been noticed when generalizing to geodesic flows and billiard dynamics, since one can’t perturb the dynamics directly, but have to make geometric deformations. See D. Visscher’s thesis for more details.

Let Q be a strictly convex domain, x be the orbit along the/a diameter of Q. Clearly x is 2-period. Let r\le R be the radius of curvatures at x, fx, respectively. Then
D_xf^2=\frac{1}{rR}\begin{pmatrix}2d(d-r-R)+rR & 2d(d-R)\\ 2(d-r)(d-r-R) & 2d(d-r-R)+rR\end{pmatrix}, where d stands for the diameter of Q.
Note that the two entries on the diagonal are always the same. Therefore any linearization with different entries on the diagonal can’t be realized as the tangent map along a periodic billiard orbit of period 2. In other words, even through there are three parameters that one can change: the distance d, the radii of curvature at both ends r,R, the effects lie in a 2D-subspace \{\begin{pmatrix}a & b \\ c & d\end{pmatrix}:ad-bc=1, a=d\} of the 3D \{\begin{pmatrix}a & b \\ c & d\end{pmatrix}:ad-bc=1\}.

Visscher was able to prove that generically, for each periodic orbit of period at least 3, every small perturbation of D_xF^3 is actually realizable by deforming the boundary of billiard table. For more details, see Visscher’s paper:

A Franks’ lemma for convex planar billiards.

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Regularity of center manifold

Let X:\mathbb{R}^d\to \mathbb{R}^d be a C^\infty vector field with X(o)=0. Then the origin o is a fixed point of the generated flow on \mathbb{R}^d. Let T_o\mathbb{R}^d=\mathbb{R}^s\oplus\mathbb{R}^c\oplus\mathbb{R}^u be the splitting into stable, center and unstable directions. Moreover, there are three invariant manifolds (at least locally) passing through o and tangent to the corresponding subspaces at o.

Theorem (Pliss). For any n\ge 1, there exists a C^n center manifold C^n(o)=W^{c,n}(o).

Generally speaking, the size of the center manifold given above depends on the pre-fixed regularity requirement. Theoretically, there may not be a C^\infty center manifold, since C^n(o) could shrink to o as n\to\infty. An explicit example was given by van Strien (here). He started with a family of vector fields X_\mu(x,y)=(x^2-\mu^2, y+x^2-\mu^2). It is easy to see that (\mu,0) is a fixed point, with \lambda_1=2\mu<\lambda_2=1. The center manifold can be represented (locally) as the graph of y=f_\mu(x).

Lemma. For n\ge 3, \mu=\frac{1}{2n}, f_\mu is at most C^{n-1} at (\frac{1}{2n},0).

Proof. Suppose f_\mu is C^{k} at (\frac{1}{2n},0), and let \displaystyle f_\mu(x)=\sum_{i=1}^{k}a_i(x-\mu)^i+o(|x-\mu|^{k}) be the finite Taylor expansion. The vector field direction (x^2-\mu^2, y+x^2-\mu^2) always coincides with the tangent direction (1,f'_\mu(x)) along the graph (x,f_\mu(x)), which leads to


Note that x^2-\mu^2=(x-\mu)^2+2\mu(x-\mu). Then up to an error term o(|x-\mu|^{k}), the right-hand side in terms of (x-\mu): (a_1+2\mu)(x-\mu)+(a_2+1)(x-\mu)^2+\sum_{i=3}^{k}a_i(x-\mu)^i; while the left-hand side in terms of (x-\mu):

(x-\mu)^2f_\mu'(x)+2\mu(x-\mu)f_\mu'(x)=\sum_{i=1}^{k}ia_i(x-\mu)^{i+1}+\sum_{i=1}^{k}2\mu i a_i(x-\mu)^i

=\sum_{i=2}^{k}(i-1)a_{i-1}(x-\mu)^{i}+\sum_{i=1}^{k}2\mu i a_i(x-\mu)^i.

So for i=1: 2\mu a_1=a_1+2\mu, a_1=\frac{-2\mu}{1-2\mu}\sim 0;

i=2: a_2+1=a_1+4\mu a_2, a_2=\frac{a_1-1}{1-4\mu}\sim -1;

i=3,\cdots,k: a_i=(i-1)a_{i-1}+2i\mu a_i, (1-2i\mu)a_i=(i-1)a_{i-1}.

Note that if k\ge n, we evaluate the last equation at i=n to conclude that a_{n-1}=0. This will force a_i=0 for all i=n-2,\cdots,2, which contradicts the second estimate that a_2\sim -1. Q.E.D.

Consider the 3D vector field X(x,y,z)=(x^2-z^2, y+x^2-z^2,0). Note that the singular set S are two lines x=\pm z, y=0 (in particular it contains the origin O=(0,0,0)). Note that D_OX=E_{22}. Hence a cener manifold W^c(O) through O is tangent to plane y=0, and can be represented as y=f(x,z). We claim that f(x,x)=0 (at least locally).

Proof of the claim. Suppose on the contrary that c_n=f(x_n,x_n)\neq0 for some x_n\to 0. Note that p_n=(x_n,c_n,x_n)\in W^c(O), and W^c(O) is flow-invariant. However, there is exactly one flow line passing through p_n: the line L_n=\{(x_n,c_nt,x_n):t>0\}. Therefore L_n\subset W^c(O), which contradicts the fact that W^c(O) is tangent to plane y=0 at O. This completes the proof of the claim.

The planes z=\mu are also invariant under the flow. Let’s take the intersection W_\mu=\{z=\mu\}\cap W^c(O)=\{(x,f(x,\mu),\mu)\}. Then we check that \{(x,f(x,\mu))\} is a (in fact the) center manifold of the restricted vector field in the plane z=\mu. We already checked that f(x,\mu) is not C^\infty, so is W^c(O).