Category Archives: Dynamics

There is no positively expansive homeomorphism

Let f be a homeomorphism on a compact metric space (X,d). Then f is said to be \mathbb{Z}-expansive, if there exists \delta>0 such that for any two points x,y\in X, if d(f^nx,f^ny)<\delta for all n\in\mathbb{Z}, then x=y. The constant \delta is called the expansive constant of f.

Similarly one can define \mathbb{N}-expansiveness if f is not invertible. An interesting phenomenon observed by Schwartzman states that

Theorem. A homeomorphism f cannot be \mathbb{N}-expansive (unless X is finite).

This result was reported in Gottschalk–Hedlund’s book Topological Dynamics (1955), and a proof was given in King’s paper A map with topological minimal self-joinings in the sense of del Junco (1990). Below we copied the proof from King’s paper.

Proof. Suppose on the contrary that there is a homeo f on (X,d) that is \mathbb{N}-expansive. Let \delta>0 be the \mathbb{N}-expansive constant of f, and d_n(x,y)=\max\{d(f^k x, f^k y): 1\le k\le n\}.

It follows from the \mathbb{N}-expansiveness that N:=\sup\{n\ge 1: d_n(x,y)\le\delta \text{ for some } d(x,y)\ge\delta\} is a finite number. Pick \epsilon\in(0,\delta) such that d_N(x,y)<\delta whenever d(x,y)<\epsilon.

Claim. If d(x,y)<\epsilon, then d(f^{-n} x, f^{-n}y)<\delta for any n\ge 1.

Proof of Claim. If not, we can prolong the N-string since f^{k}=f^{k+n}\circ f^{-n}.

Recall that a pair (x,y) is said to be \epsilon-proximal, if d(f^{n_i}x, f^{n_i}y)<\epsilon for some n_i\to\infty. The upshot for the above claim is that any \epsilon-proximal pair is \delta-indistinguishable: d(f^{n}x, f^{n}y)<\delta for all n.

Cover X by open sets of radius < \epsilon, and pick a finite subcover, say \{B_i:1\le i\le I\}. Let E=\{x_j:1\le j\le I+1\} be a subset consisting of I+1 distinct points. Then for each n\ge 0, there are two points in f^n E share the room B_{i(n)}, say f^nx_{a(n)}, and f^nx_{b(n)}. Pick a subsequence n_i such that a(n_i)\equiv a and b(n_i)\equiv b. Clearly x_a\neq x_b, and d(f^{n_i}x_a,f^{n_i}x_b)<\epsilon. Hence the pair (x_a,x_b) is \epsilon-proximal and \delta-indistinguishable. This contradicts the \mathbb{N}-expansiveness assumption on f. QED.

An interesting lemma about the Birkhoff sum

A few days ago I attended a lecture given by Amie Wilkinson. She presented a proof of Furstenberg’s theorem on the Lyapunov exponents of random products of matrices in SL(2,\mathbb{R}).

Let \lambda be a probability measure on SL(2,\mathbb{R}), \mu=\lambda^{\mathbb{N}} be the product measure on \Omega=SL(2,\mathbb{R})^{\mathbb{N}}. Let \sigma be the shift map on \Omega, and A:\omega\in\Omega\mapsto \omega_0\in SL(2,\mathbb{R}) be the projection. We consider the induced skew product (f,A) on \Omega\times \mathbb{R}^2. The (largest) Lyapunov exponent of (f,A) is defined to be the value \chi such that \displaystyle \lim_{n\to\infty}\frac{1}{n}\log\|A_n(\omega)\|=\chi for \mu-a.e. \omega\in \Omega.

To apply the ergodic theory, we first assume \int\log\|A\| d\lambda < \infty. Then \chi(\lambda) is well defined. There are cases when \chi(\lambda)=0:

(1) the generated group \langle\text{supp}\lambda\rangle is compact;

(2) there exists a finite set \mathcal{L}=\{L_1,\dots, L_k\} of lines that is invariant for all A\in \langle\text{supp}\lambda\rangle.

Furstenberg proved that the above cover all cases with zero exponent:
\chi(\lambda) > 0 for all other \lambda.

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Symplectic and contact manifolds

Let (M,\omega) be a symplectic manifold. It said to be exact if \omega=d\lambda for some one-form \lambda on M.

(1) If \omega=d\lambda is exact, then there is a canonical isomorphism between the v.f. and 1-forms. In particular, there exists a v.f. X such that \lambda=i_X\omega. Then we have \lambda(X)=\omega(X,X)=0, and L_X\lambda=i_X d\lambda+d i_X\lambda=i_X\omega +0=\lambda, and L_X\omega=d i_X\omega=d\lambda=\omega.

(2) Suppose there exists a vector field X on M such that its Lie-derivative L_X\omega=\omega (notice the difference with L_X\omega=0). Then Cartan’s formula says that \omega=i_X d\omega+ di_X\omega=d\lambda, where \lambda=i_X\omega. So \omega=d\lambda is exact, and L_X\lambda=i_Xd\lambda+di_X\lambda=i_X\omega+0=\lambda.

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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)  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.


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).

The volume of uniform hyperbolic sets

This is a note of some well known results. The argument here may be new, and may be complete.

Proposition 1. Let f\in\mathrm{Diff}^2_m(M). Then m(\Lambda)=0 for every closed, invariant hyperbolic set \Lambda\neq M.

See Theorem 15 of Bochi–Viana’s paper. Note that Proposition 1 also applies to Anosov case, in the sense that m(\Lambda)>0 implies that \Lambda=M and f is Anosov.

Proof. Suppose m(\Lambda)>0 for some hyperbolic set. Then the stable and unstable foliations/laminations are absolutely continuous. Hopf argument shows that \Lambda is (essentially) saturated by stable and unstable manifolds. Being a closed subset, \Lambda is in fact saturated by stable and unstable manifolds, and hence open. So \Lambda=M.

Proposition 2. There exists a residual subset \mathcal{R}\subset \mathrm{Diff}_m^1(M), such that for every f\in\mathcal{R}, m(\Lambda)=0 for every closed, invariant hyperbolic set \Lambda\neq M.

Proof. Let U\subset M be an open subset such that \overline{U}\neq M, \Lambda_U(f)=\bigcap_{\mathbb{Z}}f^n\overline{U}, which is always a closed invariant set (maybe empty). Given \epsilon>0, let \mathcal{D}(U,\epsilon) be the set of maps f\in\mathrm{Diff}_m^1(M) that either \Lambda_U(f) is not a uniformly hyperbolic set, or it’s hyperbolic but  m(\Lambda_U(f))<\epsilon. It follows from Proposition 1 that \mathcal{D}(U,\epsilon) is dense. We only need to show the openness. Pick an f\in \mathcal{D}(U,\epsilon). Since m(\Lambda_U(f))<\epsilon, there exists N\ge 1 such that m(\bigcap_{-N}^N f^n\overline{U})<\epsilon. So there exists \mathcal{U}\ni f such that m(\bigcap_{-N}^N g^n\overline{U})<\epsilon. In particular, m(\Lambda_U(g))<\epsilon for every g\in \mathcal{U}. The genericity follows by the countable intersection of the open dense subsets \mathcal{D}(U_n,1/k).

The dissipative version has been obtained in Alves–Pinheiro’s paper

Proposition 3. Let f\in\mathrm{Diff}^2(M). Then m(\Lambda)=0 for every closed, transitive hyperbolic set \Lambda\neq M. In particular, m(\Lambda)>0 implies that \Lambda=M and f is Anosov.

See Theorem 4.11 in R. Bowen’s book when \Lambda is a basic set.

Doubling map on unit circle

1. Let \tau:x\mapsto 2x be the doubling map on the unit torus. We also consider the uneven doubling f_a(x)=x/a for 0\le x \le a and f(x)=(x-a)/(1-a) for a \le x \le 1. It is easy to see that the Lebesgue measure m is f_a-invariant, ergodic and the metric entropy h(f_a,m)=\lambda(m)=\int \log f_a'(x) dm(x)=-a\log a-(1-a)\log (1-a). In particular, h(f_a,m)\le h(f_{0.5},m)=\log 2 =h_{\text{top}}(f_a) and h(f_a,m)\to 0 when a\to 0.

2. Following is a theorem of Einsiedler–Fish here.

Proposition. Let \tau:x\mapsto 2x be the doubling map on the unit torus, \mu be an \tau-invariant measure with zero entropy. Then for any \epsilon>0, \beta>0, there exist \delta_0>0 and a subset E\subset \mathbb{T} with \mu(E) > 0, such that for all x \in E, and all \delta<\delta_0: \mu(B(x,\delta))\ge \delta^\beta.

A trivial observation is \text{HD}(\mu)=0, which also follows from general entropy-dimension formula.

Proof. Let \beta and \epsilon be fixed. Consider the generating partition \xi=\{I_0, I_1\}, and its refinements \xi_n=\{I_\omega: \omega\in\{0,1\}^n\} (separated by k\cdot 2^{-n})….

Furstenberg introduced the following notation in 1967

Definition. A multiplicative semigroup \Sigma\subset\mathbb{N} is lacunary, if \Sigma\subset \{a^n: n\ge1\} for some integer a. Otherwise, \Sigma is non-lacunary.

Example. Both \{2^n: n\ge1\} and \{3^n: n\ge1\} are lacunary semigroups. \{2^m\cdot 3^n: m,n\ge1\} is a non-lacunary semigroup.

Theorem. Let \Sigma\subset\mathbb{N} be a non-lacunary semigroup, and enumerated increasingly by s_i > s_{i+1}\cdot. Then \frac{s_{i+1}}{s_i}\to 1.

Example. \Sigma=\{2^m\cdot 3^n: m,n\ge1\}. It is equivalent to show \{m\log 2+ n\log 3: m,n\ge1\} has smaller and smaller steps.

Theorem. Let \Sigma\subset\mathbb{N} be a non-lacunary semigroup, and A\subset \mathbb{T} be \Sigma-invariant. If 0 is not isolated in A, then A=\mathbb{T}.

Furstenberg Theorem. Let \Sigma\subset\mathbb{N} be a non-lacunary semigroup, and \alpha\in \mathbb{T}\backslash \mathbb{Q}. Then \Sigma\alpha is dense in \mathbb{T}.

In the same paper, Furstenberg also made the following conjecture: a \Sigma-invariant ergodic measure is either supported on a finite orbit, or is the Lebesgue measure.

A countable group G is said to be amenable, if it contains at least one Følner sequence. For example, any abelian countable group is amenable. Note that for amenable group action G\ni g:X\to X, there always exists invariant measures and the decomposition into ergodic measures. More importantly, the generic point can be defined by averaging along the Følner sequences, and almost every point is a generic point for an ergodic measure. In a preprint, the author had an interesting idea: to prove Furstenberg conjecture, it suffices to show that every irrational number is a generic point of the Lebesgue measure. Then any other non-atomic ergodic measures, if exist, will be starving to death since there is no generic point for them 🙂