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

(x^2-\mu^2)f_\mu'(x)=y+x^2-\mu^2=f_\mu(x)+x^2-\mu^2.

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 :)

Some simple dynamical systems

Dynamical formulation of Prisoner’s dilemma
Originally, consider the two players, each has a set of stratagies, say \mathcal{A}=\{a_{i}\} and \mathcal{B}=\{b_{j}\}. The pay-off P_k=P_k(a_{i},b_{j}) for player k depends on the choices of both players.

Now consider two dynamical systems (M_i,f_i). The set of stratagies consists of the invariant probability measures, and the pay-off functions can be

\phi_k(\mu_1,\mu_2)=\int \Phi_k(x,y)d\mu_1 d\mu_2, where \mu_i\in\mathcal{M}(f_i);

\psi_k(\mu_1,\mu_2)=\int \Phi_k(x,y)d\mu_1 d\mu_2-h(f_i,\mu_i).

The frist one is related to Ergodic optimization. The second one does sound better, since one may want to avoid a complicated (measured by its entropy) stratagy that has the same \phi pay-off.

Gambler’s Ruin Problem
A gambler starts with an initial fortune of $i,
and then either wins $1 (with p) or loses $1 (with q=1-p) on each successive gamble (independent of the past). Let S_n denote the total fortune after the n-th gamble. Given N>i, the gambler stops either when S_n=0 (broke), or S_n=N (win), whichever happens first.

Let \tau be the stopping time and P_i(N)=P(S_\tau=N) be the probability that the gambler wins. It is easy to see that P_0(N)=0 and P_N(N)=1. We need to figure out P_i(N) for all i=1,\cdots,N-1.

Let S_0=i, and S_n=S_{n-1}+X_n. There are two cases according to X_1:

X_1=1 (prob p): win eventually with prob P_{i+1}(N);

X_1=-1 (prob q): win eventually with prob P_{i-1}(N).

So P_i(N)=p\cdot P_{i+1}(N)+q\cdot P_{i-1}(N), or equivalently,
p\cdot (P_{i+1}(N)-P_i(N))=q\cdot (P_i(N)-P_{i-1}(N)) (since p+q=1), i=1,\cdots,N-1.

Recall that P_0(N)=0 and P_N(N)=1. Therefore P_{i+1}(N)-P_i(N)=\frac{q^i}{p^i}(P_1(N)-P_{0}(N))=\frac{q^i}{p^i}P_1(N), i=1,\cdots,N-1. Summing over i, we get 1-P_1(N)=P_1(N)\cdot\sum_{1}^{N-1}\frac{q^i}{p^i}, P_1(N)=\frac{1}{\sum_{0}^{N-1}\frac{q^i}{p^i}}=\frac{1-q/p}{1-q^N/p^N} (if p\neq .5) and P_1(N)=\frac{1}{N} (if p= .5). Generally P_i(N)=P_1(N)\cdots\sum_{0}^{i-1}\frac{q^j}{p^j}=\frac{1-q^i/p^i}{1-q^N/p^N} (if p\neq .5) and P_1(N)=\frac{i}{N} (if p= .5).

Observe that for fixed i, the limit P_i(\infty)=1-q^i/p^i>0 only when p>.5, and P_i(\infty)=0 whenever p\le .5.

Finite Blaschke products
Let f be an analytic function on the unit disc \mathbb{D}=\{z\in\mathbb{C}: |z|<1 \} with a continuous extension to \overline{\mathbb{D}} with f(S^1)\subset S^1. Then f is of the form

\displaystyle f(z)=\zeta\cdot\prod_{i=1}^n\left({{z-a_i}\over {1-\bar{a_i}z}}\right)^{m_i},

where \zeta\in S^1, and m_i is the multiplicity of the zero a_i\in \mathbb{D} of f. Such f is called a finite Blaschke product.

Proposition. Let f be a finite Blaschke product. Then the restriction f:S^1\to S^1 is measure-preserving if and only if f(0)=0. That is, a_i=0 for some i.

Proof. Let \phi be an analytic function on \overline{\mathbb{D}}. Then \int_{S_1}\phi d(\theta)=\phi(0) and \int_{S_1}\phi\circ f d(\theta)=\phi\circ f(0).

Significance: there are a lot of measure-preserving covering maps on S^1.

Kalikov’s Random Walk Random Scenery
Let X=\{1,-1\}^{\mathbb{Z}}, and \sigma:X\to X to the shift \sigma((x_n))=(x_{n+1}). More generally, let A be a finite alphabet and p be probability vector on A, and Y=A^{\mathbb{Z}}, \nu=p^{\times\mathbb{Z}}. Consider the skew-product T:X\times Y\to X\times Y, (x,y)\mapsto (\sigma x, \sigma^{x_0}y). It is clear that T preserves any \mu\times \nu, where \mu is \sigma-invariant.

Proposition. Let \mu=(.5,.5)^{\times\mathbb{Z}}. Then h(T,\mu\times \nu)=h(\sigma,\mu)=\log 2 for all (A,p).

Proof. Note that T^n(x,y)\mapsto (\sigma^n x, \sigma^{x_0+\cdot+x_{n-1}}y). CLT tells that \mu(x:|x_0+\cdot+x_{n-1}|\ge\kappa\cdot \sqrt{n})< \delta(\kappa) as n\to\infty, where \delta(\kappa)\to 0 as \kappa\to\infty. There are only 2^{n+\kappa \sqrt{n}} different n-strings (up to an error).

Significance: this gives a natural family of examples that are K, but not isomorphic to Bernoulli.

 Creation of one sink. 1D case. Consider the family f_t:x\mapsto x^2+2-t, where 0\le t\le 2. Let t_\ast the first parameter such that the graph is tangent to the diagonal at x_\ast=f_{t_\ast}(x_\ast). Note that x_\ast is parabolic. Then for t\in(t_\ast,t_\ast+\epsilon), f_t(x)=x has two solutions x_1(t)<x_2(t), where x_1(t) is a sink, and x_2(t) is a source.

2D case. Let B=[-1,1]\times[-\epsilon,\epsilon] be a rectangle, f be a diffeomorphism such that f(B) is a horseshoe lying  above B of shape ‘V’. Moreover we assume |\det Df|<1. Let f_t(x,y)=f(x,y)-(0,t) such that f_1(B) is the regular horseshoe intersection:  V . Clearly there exists a fixed point p_1 of f_1 in B. We assume \lambda_1(1)<-1<\lambda_2(1)<0. Then Robinson proved that f_t admits a fixed point in B which is a sink.

First note that for any t, and any fixed point of f_t (if exists), it is not on the boundary of B. Since p_1 is a nondegenerate fixed point of f_1, the fixed point continues to exist for some open interval (t_1,1) (assume it is maximal, and denote the fixed point by p_t). Clearly t_1>0. Note that p_{t_1} is also fixed by f_{t_1}, since it is a closed property. If there is some moment with \lambda_1(t)=\lambda_2(t) for the fixed point p_t of f_t, then it is already a sink, since \det Df=\lambda_1\cdot\lambda_2<1. So in the following we consider the case \lambda_1\neq\lambda_2 for all p_t, t\in[t_1,1]. Then the continuous dependence of parameters implies that both are continuous functions of t. The fixed point p_{t_1} must be degenerate, since the fixed point ceases to exist beyond t_1, which means: \lambda_i(t_1)=1 for some i\in\{1,2\}.

Case 1. \lambda_1(t_1)=1. Note that \lambda_1(1)<-1. So \text{Re}\lambda_1(t_\ast)=0 for some t_\ast\in(t_1,1), which implies that \lambda_1(t_\ast)=ai for some a\neq 0. In particular, \lambda_2(t_\ast)=-ai, and a^2=|\det Df|<1. So p_{t_\ast} is a (complex) sink.

Case 2. \lambda_2(t_1)=1. Note that \lambda_2(1)<0. Similarly \text{Re}\lambda_2(t_\ast)=0 for some t_\ast\in(t_1,1).

So in the orientation-preserving case there always exists a complex sink. In the orientation-reversing case (\lambda_2(1)\in(0,1)), we need modify the argument for case 2:

Case 2′. \lambda_2(t_1)=1. Note that \lambda_2(1)\in(0,1). So |\lambda_1(t_\ast)|<1 for some t_\ast\in(t_1,1). We pick t_\ast close to t_1 in the sense that |\lambda_1(t_\ast)|>|\det Df|, which implies |\lambda_1(t_\ast)|\ast<1, too. So p_{t_\ast} is also a sink.

Playing pool with pi

This is a short note based on the paper 

Playing pool with π (the number π from a billiard point of view) by G. Galperin in 2003.

Let’s start with two hard balls,  denoted by B_1 and B_2, of masses 0<m\le M on the positive real axis with position 0<x< y, and a rigid wall at the origin. Without loss of generality we assume m=1. Then push the ball B_2 towards B_1, and count the total number N(M) of collisions (ball-ball and ball-wall) till the B_2 escapes to \infty faster than B_1.

Case. M=1: first collision at y(t)=x, then B_2 rests, and B_1 move towards the wall; second collision at x(t)=0, then B_1 gains the opposite velocity and moves back to B_2; third collision at x(t)=x, then B_1 rests, and B_2 move towards \infty.

Total counts N(1)=3, which happens to be first integral part of \pi. Well, this must be coincidence, one might wonder.

However, Galperin proved that, if we set M=10^{2k}, then N(M) gives the integral part of 10^k\pi. For example, N(10^2)=31; and  N(10^4)=314.

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Notes-09-14

4. Borel–Cantelli Lemma(s). Let (X,\mathcal{X},\mu) be a probability space. Then

If \sum_n \mu(A_n)<\infty, then \mu(x\in A_n \text{ infinitely often})=0.

If A_n are independent and \sum_n \mu(A_n)=\infty, then for \mu-a.e. x, \frac{1}{\mu(A_1)+\cdots+\mu(A_n)}\cdot|\{1\le k\le n:x\in A_k\}|\to 1.

The dynamical version often involves the orbits of points, instead of the static points. In particular, let T be a measure-preserving map on (X,\mathcal{X},\mu). Then

\{A_n\} is said to be a Borel–Cantelli sequence with respect to (T,\mu) if \mu(T^n x\in A_n \text{ infinitely often})=1;

\{A_n\} is said to be a strong Borel–Cantelli sequence if \frac{1}{\mu(A_1)+\cdots+\mu(A_n)}\cdot|\{1\le k\le n:T^k x\in A_k\}|\to 1 for \mu-a.e. x.

3. Let H(q,p,t) be a Hamiltonian function, S(q,t) be the generating function in the sense that \frac{\partial S}{\partial q_i}=p_i. Then the Hamilton–Jacobi equation is a first-order, non-linear partial differential equation

H + \frac{\partial S}{\partial t}=0.

Note that the total derivative \frac{dS}{dt}=\sum_i\frac{\partial S}{\partial q_i}\dot q_i+\frac{\partial S}{\partial t}=\sum_i p_i\dot q_i-H=L. Therefore, S=\int L is the classical action function (up to an undetermined constant).

2. Let \gamma_s(t) be a family of geodesic on a Riemannian manifold M. Then J(t)=\frac{\partial }{\partial s}|_{s=0} \gamma_s(t) defines a vector field along \gamma(t)=\gamma_0(t), which is called a Jacobi field. J(t) describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic \gamma.

Alternatively, A vector field J(t) along a geodesic \gamma is said to be a Jacobi field, if it satisfies the Jacobi equation:

\frac{D^2}{dt^2}J(t)+R(J(t),\dot\gamma(t))\dot\gamma(t)=0,

where D denotes the covariant derivative with respect to the Levi-Civita connection, and R the Riemann curvature tensor on M.

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Exponential map on the complex plane

Let f(z)=e^z=e^x(\cos y+\i\sin y) (for z=x+\i y) be the exponential map. Note that f(x)>0 for all real numbers and f^{n+1}(x):=f(f^nx) goes to \infty really fast: the dynamics of f on \mathbb{R} is trivial. But the dynamics of f on \mathbb{C} is completely different. First note that e^{2k\pi\i}=1: the map is not a diffeomorphism, but a covering map branching at the origin. The following theorem was conjectured by Fatou (1926) and proved by Misiurewicz (1981).

Theorem (Orbits of the complex exponential map).
Let \mathcal{O}_e(z) be the orbit of a point z\in\mathbb{C} under the iterates of f(z)=e^z. Then each of the following sets is dense in the complex plane:
1. the basin of \infty, B_e(\infty)=\{z\in\mathbb{C}: f^n(z)\to\infty\};
2. the set of transitive points, \text{Tran}(e)=\{z\in\mathbb{C}: \mathcal{O}_e(z)\text{ is dense}\};
3. the set of periodic points, \text{Per}(e)=\{z\in\mathbb{C}: \mathcal{O}_e(z)\text{ is finite}\}.

So the exponential map is chaotic on the complex plane.

Reference:

The exponential map is chaotic: An invitation to transcendental dynamics,
Zhaiming Shen and Lasse Rempe-Gillen arXiv

Hegselmann-Krause system

There is an interesting preprint today on arxiv by Sascha Kurz:
How long does it take to consensus in the Hegselmann-Krause model? http://arxiv.org/abs/1405.5757

Consider k-particle system with total energy E, and let \delta=\delta(E) be the range of interactions. The evolution of energy distribution x=(x_i):\mathbb{N}\to \mathbb{R}^k is defined by the following algorithm:

  1. set the initial energies x_i(0)\ge 0 with E=\sum_{i} x_i(0).
  2. as time develops, x_i interacts (i.e. exchanges energy) only with the particles with similar energies: let N_i(0)=\{1\le j\le k: |x_j(0)-x_i(0)|\le \delta\}\ni i and n_i(0)=|N_i(0)|\ge 1. Then set x_i(1)=\frac{1}{n_i(0)}\sum_{j\in N_i(0)}x_j(0).
  3. Suppose we have defined x_i(n). Then let N_i(n)=\{1\le j\le k: |x_j(n)-x_i(n)|\le 1\}\ni i and n_i(n)=|N_i(n)|\ge 1. Then set x_i(n+1)=\frac{1}{n_i(n)}\sum_{j\in N_i(n)}x_j(n).
  • Note that the total energy E is preserved, and the system is scaling invariant. So we can assume \delta=1. Then the system can be viewed as a map on the simplex \Delta_k=\{x=(x_i)\in\mathbb{R}^k:x_i\ge0\text{ and }E=\sum_i x_i\}.
  • Note that the order is preserved during the process: x_i(0)\le x_{i+1}(0) implies x_i(n)\le x_{i+1}(n) for all n. In particular, if x_i(n)= x_{i+1}(n) for some n, then these two particles are identical in the system. So we can replace them by a single particle but with weight 2. In general we can consider the weighted system \Omega_k=\{(x_i,k_i):x_i\ge0, k=\sum_i k_i\text{ and }E=\sum_i k_ix_i\} (mass conservation and energy conservation). How to characterize the dynamics on \Omega_k?
  • A simple test example is x_i(0)=i for i=1,\cdots k.

    1. k=1: trivial system.
    2. k=2: x_1(0)=1 and x_2(0)=2. Then x_i(n)=1.5 for all i=1,2 and all n\ge 1. All x_i(n) reach the same energy at time T(2)=1.
    3. k=3: x_i(0)=i. Then x_1(1)=1.5, x_2(1)=2 and x_3(1)=2.5. Now note that N_i(1)=\{1,2,3\} for all i. So x_1(n)=2 for all i=1,2 and all n\ge 2. Then reach the same energy at time T(3)=2.

    The set N_i(n) plays an important role. One can check that T(4)=5 and T(5)=6: the small groups can always reach the same energy. But the case k=6 is different:

  • X(0)=\langle 1,2,3,4,5,6 \rangle
  • X(1)=\langle \frac{3}{2},2,3,4,5,\frac{11}{2} \rangle
  • X(2)=\langle 1\frac{3}{4},2\frac{1}{6},3,4,4\frac{5}{6},5\frac{1}{4} \rangle
  • X(3)=\langle 1\frac{23}{24},2\frac{11}{36},3\frac{1}{18},3\frac{17}{18},4\frac{25}{36},5\frac{1}{24} \rangle
  • Eventually this leads to two different clusters x_i(6)=A for i=1,2,3 and x_i(6)=B for i=4,5,6 with large energy difference B-A > 1. The two groups of particles never intersect with the other group and will stay in this status forever. So we set T(6)=6.

    There is no general formula for T(k). It is proved that T(k)=O(k^3), and conjectured that T(k)=O(k), and the worse scenario is given the initial data x_i(0)=i for i=1,\cdots, k.

    Continuous time Markov process

    Let \Sigma=\{1,\cdots, d\}^{\mathbb{N}}, B:\Sigma\to\mathbb{R} be a Lipschitz potential, and L_B(f):x\mapsto \sum_{\sigma y=x} e^{B(y)}f(y). The potential B is said to be normalized, if \sum_{\sigma y=x} e^{B(y)}=1 for all x\in\Sigma.

    If B is normalized, then its topological pressure P(\sigma, B)=0, and its equilibrium state \mu_B is an L_B^\ast-invariant Gibbe measure. This induces a Markov process (X_t) with values on the state space \Sigma. That is, suppose X_t=x\in\Sigma. Then it stays at this state for a while, waits for T\sim \text{Exp}(1) and jumps to a point X_{t+T}=y\in\sigma^{-1}x with probability e^{B(y)}. Then \mu_B is a stationary measure for this Markov process.

    More generally, we can assign different jump rates (exponential clocks) at different states. That is, let r:\Sigma\to[c,C]. Let (X_t^r) be the modified Markov process with clock r. That is, suppose X_t^r=x\in\Sigma. Then it waits a time T\sim \text{Exp}(r(x)) and jumps to a point X_{t+T}^r=y\in\sigma^{-1}x with probability e^{B(y)}. Then the naturally related measure is \mu^r_B:E\mapsto \frac{1}{\mu_B(1/r)}\cdot\int_E \frac{1}{d}d\mu_B.

    Another setting is consider a system of N sites, each carrying an energy x_i. Assume the neighboring sites s_i and s_{i+1} exchange energy when an exponential clock \text{Exp}(\lambda_i(x_i+x_{i+1})) rings: (\hat x_i,\hat x_{i+1})=(a,1-a)(x_i+x_{i+1}), where $\alpha\sim U([0,1])$.
    Now consider a function f:\Sigma\to\mathbb{R}, and its evolution f(X_t) with X_0=x. Then the generator L is defined by
    Lf:x\mapsto \mathbb{E}_x\lim_{t\to 0+}\frac{f(X_t)}{t}  =\sum_{i}\lim_{t\to 0+}\frac{1}{t}\mathbb{P}(T_t(i))\cdot \mathbb{E}_x(f(X_t)-f(x)|T_t(i)),
    where T_t(i) describes the event that only the i-th clock rings during the time (0,t). For independent exponential clocks, we have
    \mathbb{P}(T_t(i))=e^{-\lambda_i t}\lambda_i t\cdot\prod_{j\neq i}e^{-\lambda_j t},
    and
    \mathbb{E}_x(f(X_t)-f(x)|T_t(i))=\int_I [f(T_{ia}x)-f(x)]\cdot U(da). So
    Lf: x\mapsto\sum_i \lambda_i(x_i+x_{i+1})\cdot\int_I [f(T_{ia}x)-f(x)]\cdot U(da)

    Asymmetry of Bowen’s dimensional entropy

    1. Bowen and Dinaburg gave a alternative definition of topological entropy h_{\text{top}}(f) by calculating the exponential growth rate of the (n,\epsilon)-covers. This definition resembles the box dimension of Euclidean subset E\subset\mathbb{R}^k, and gives the same value while using the definition given by Adler, Konheim, and McAndrew. In particular, the entropy is time-reversal invariant: h_{\text{top}}(f^{-1})=h_{\text{top}}(f).

    2. Later Bowen introduced another definition of topological entropy for noncompact subset in 1973, which resembles the Hausdorff dimension.
    Let f:X\to X be a homeomorphism on a compact metric space, E\subset X and h_B(f,E) be Bowen’s topological entropy of E (may not be compact).

    Bowen proved that, for any ergodic measure \mu, h_B(f,G_{\mu})=h(f,\mu), where G_{\mu} is the set of \mu-generic points. This identity has been generalized to general invariant measures of transitive Anosov systems:

    Theorem 1. (Pfister–Sullivan link) Let f:M\to M be a transitive Anosov diffeomorphism. Then h_B(f,G_{\mu})=h(f,\mu) for any invariant measure \mu.

    Note that \mu(G_\mu)=0 whenever \mu is invariant but non-ergodic.

    3. An interesting fact is that h_B(f,E) may not be time-reversal invariant.

    Example 2. Let f:M\to M be a transitive Anosov diffeomorphism, p be a periodic point, D=W^u(x,\epsilon). Then h_B(f,D) > 0, but h_B(f^{-1},D)=0.

    Now let \mu,\nu be two different invariant measures of f, W^s(\mu,f)=G_\mu be the set of \mu-generic points with respect to f, and W^u(\nu,f)=W^s(\nu,f^{-1}) be the set of \mu-generic points with respect to f^{-1}. Let H_f(\mu,\nu)=B^s(\mu,f)\cap B^u(\nu,f) (resemble the heteroclinic intersection of different saddles). Then it is proved (Proposition D in here) that

    Proposition 3. Let f:M\to M be a transitive Anosov diffeomorphism. Then h_B(f,H_f(\mu,\nu))=h_\mu(f) and h_B(f^{-1},H_f(\mu,\nu))=h_\nu(f).

    A well known fact is that, for any 0\le t\le h_{\text{top}}(f), there exists some invariant measure \mu with h_\mu(f)=t. So a direct corollary of Proposition 3 is:

    Corollary. Let f:M\to M be a transitive Anosov diffeomorphism. Then for any a, b\in [0, h_{\text{top}}(f)], there exists an invariant subset E such that h_B(f,E)=a and h_B(f^{-1},E)=b.

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