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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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:


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.


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?

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},
    \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.

    Correlation functions and power spectrum

    Let (X, f, \mu) be a mixing system, \phi\in L^2(\mu) with \mu(\phi)=0.
    The auto-correlation function is defined by \rho(\phi,f,k)=\mu(\phi\circ f^k\cdot f).
    In the following we assume \sigma^2:=\sum_{\mathbb{Z}}\rho(\phi, f,k) converges. Under some extra condition, we have the central limit theorem \frac{S_n \phi}{\sqrt{n}} converges to a normal distribution.

    The power spectrum of (f,\mu,\phi) is defined by (when the limit exist)
    \displaystyle S:\omega\in [0,1]\mapsto \lim_{n\to\infty} \frac{1}{n}\int |\sum_{k=0}^{n-1} e^{2\pi \i k\omega} \phi\circ f^k|^2 d\mu.
    Note that S(0)=S(1)=\sigma^2 whenever \sum_{\mathbb{Z}}\rho(\phi, f,k) converges.
    Proof. Let T_k=\sum_{n=-k}^k \rho(n). Then T_k\to \sigma^2. So
    \mu|\sum_{k=0}^{n-1}\phi\circ f^k|^2 =\sum_{k,l=0}^{n-1} \mu(\phi\cdot \phi\circ f^{k-l})=\sum_{k,l=0}^{n-1}\rho(k-l)
    =n\cdot \rho(0)+(n-1)\cdot (\rho(1)+\rho(-1))+\cdots +2\cdot (\rho(n-1)+\rho(1-n))
    =T_0+ T_1+\cdots + T_{n-1}. Then we have
    S(0)=S(1)=\lim \frac{1}{n}(T_0+ T_1+\cdots + T_{n-1})=\sigma^2 since T_k\to \sigma^2.

    More generally, we have \mu|\sum_{k=0}^{n-1} e^{2\pi \i k\omega} \phi\circ f^k|^2  =\sum_{k,l=0}^{n-1} e^{2\pi \i (k-l)\omega} \mu(\phi\cdot \phi\circ f^{k-l})=T_0^\omega + T_1^\omega +\cdots + T_{n-1}^\omega,
    where T_k^\omega=\sum_{n=-k}^k e^{2\pi \i n\omega}\rho(n). So S(\omega) exists whenever \sum_{\mathbb{Z}} e^{2\pi \i n\omega}\rho(n) converges.

    This is the power spectrum of (X,f,\mu,\phi). Some observations:

    Proposition. Assume \sum |\rho(n)|<\infty.
    Then S(\omega) is well-defined, continuous function on 0\le \omega\le 1. Moreover,
    S(\cdot) is C^{r-2} if |\rho(k)|\le C k^{-r} for all k;
    S(\cdot) is C^{\infty} if \rho(k) decay rapidly;
    S(\cdot) is C^{\omega} if \rho(k) decay exponentially.

    Some preparations.

    Hardy: Let a_n be a sequence of real numbers, A_n=a_1+\cdots +a_n such that
    |n\cdot a_n|\le M for all n\ge 1;
    \sigma_n=\frac{A_1+\cdots +A_n}{n}\to A.
    Then \sum_{n\ge 1}a_n also converges to A.

    Proof. Let \epsilon>0 be given, N large such that |\sigma_n-A|\le \epsilon for all n\ge N.
    Then for any p\ge 1, we have
    (n+p)\sigma_{n+p}-n\sigma_n=A_{n+1}+\cdots + A_{n+p};
    R_{n,p}=(n+p)(\sigma_{n+p}-A)-n(\sigma_n-A)-p(A_n-A)=A_{n+1}+\cdots + A_{n+p}-pA
    =a_{n+1}+(a_{n+1}+a_{n+2})+\cdots +(a_{n+1}+\cdots+ a_{n+p}).
    Note that |R_{n,p}|\le \frac{M}{n+1}+(\frac{M}{n+1}+\frac{M}{n+2})+\cdots +(\frac{M}{n+1}+\cdots+ \frac{M}{n+p})   \le \frac{p(p+1)}{2}\cdot \frac{M}{n}.
    Then |A_n-A|\le \frac{n+p}{p}|\sigma_{n+p}-A|+\frac{n}{p}|\sigma_{n}-A|+\frac{|R_{n,p}|}{p}  \le \epsilon+2n\epsilon/p+\frac{p+1}{2n}M. So we can pick p\sim n\sqrt{\epsilon}, which leads to
    |A_n-A|\le \epsilon+3\sqrt{\epsilon}+M\sqrt{\epsilon} for all n\ge N.

    Let f\in C(\mathbb{T}) be differentiable, and f'\in L^1(\mathbb{T}). Let f(t)\sim \sum_n f_n e^{2\pi\text{i} nt} and f'(t)\sim \sum_n d_n e^{2\pi\text{i} nt} be the Fourier series. Then d_n=\text{i} n f_n.
    Proof: integrate by parts.

    Let f\in C(\mathbb{T}) be differentiable, and f'\in L^2(\mathbb{T}). Then \sum_n |f_n| converges.
    In particular, \sum_n |f_n| converge for all f\in C^1(\mathbb{T}).

    Proof. Note that \sum_{n\neq 0} |f_n|=\sum_{n\neq 0} |\frac{1}{n}\cdot d_n|  \le (\sum_{n\ge 1} \frac{2}{n^2})^{1/2}\cdot(\sum_n d_n^2)^{1/2}=\frac{\pi}{\sqrt{3}}\cdot\|f'\|_{L^2}.

    Some remarks about dominated splitting property

    Denote \mathcal{T} the set of transitive diffeos, \mathcal{DS} the set of diffeo’s with Global Dominated Splittings (GDS for short), \mathcal{M} the set of minimal diffeos.

    It is proved that

    \mathcal{DS}\bigcap \mathcal{M}=\emptyset: diffeo with GDS can’t be minimal (here).

    \mathcal{T}^o\subset \mathcal{DS}: robustly transitive diffeo always admits some GDS (here).

    So \mathcal{T}^o\bigcap \mathcal{M}=\emptyset, although \mathcal{T}\supset \mathcal{M}: the special property (minimality) can’t happen in the interior of the general property (transitivity).

    A minor change of the proof shows that a diffeomorphism with GDS can’t be uniquely ergodic, either. So we have the following conservative version:

    \mathcal{DS}\bigcap \mathcal{UE}=\emptyset: diffeos with GDS can’t be uniquely ergodic.

    \mathcal{E}^o\subset \mathcal{DS}: stably ergodic diffeos always admits some GDS (here).

    So \mathcal{E}^o\bigcap \mathcal{UE}=\emptyset, although \mathcal{E}\supset \mathcal{UE}.

    Remark. It is a little bit tricky to define \mathcal{E}^o. The most natural definition may lead to an emptyset. One well-accepted definition is: f\in\mathcal{E}^o if there exists a C^1 neighborhood f\in\mathcal{U}\subset\mathrm{Diff}^1_m(M), such that every g\in \mathcal{U}\cap \mathrm{Diff}^2_m(M) is ergodic. All volume-preserving Anosov satisfies the later definition, and this is the context of Pugh-Shub Stable Ergodicity Conjecture.

    Remark. There is an open dense subset \mathcal{R}\subset \mathcal{E}^o, such that every f\in \mathcal{R} is nonuniformly Anosov (here)

    Remark. Let (M,\omega) be a symplectic manifold with \dim M\ge 4, \mathrm{PH}^{2}_{\omega}(M,2) be the set of C^2 symplectic partially hyperbolic maps with \dim (E^c)=2.
    Then consider f\in\mathcal{E}^o\cap \mathrm{PH}^{2}_{\omega}(M,2) and \lambda^c_1(f,\omega)\ge \lambda^c_2(f,\omega) be the two central Lyapunov exponents. If the dominated splitting is not refined by the partially hyperbolic splitting, then it must split the central bundle, and \lambda^c_1(f,\omega)> \lambda^c_2(f,\omega): f is nonuniformly Anosov.

    Notes 3

    5. Let \Phi_t be a stochastic process on X, P_t(x,A) be the transition kernal (the probability for \Phi_t(x)\in A). This induces an action on the space of Borel measures, P_t:\mu\mapsto\mu\circ P_t: A\mapsto \int_X P_t(x,A)\cdot\mu(dx). Suppose that
    (1). there is a unique stationary measure \mu_t for the discretized process \{\Phi_{nt}\}_n;
    (2). t\mapsto \mu_t is continuous.
    Then \mu_t is independent of t, and \mu=\mu_1 is the unique stationary measure for the original process \{\Phi_t\}_t.

    Proof. Note that \mu_t is also stationary for \{\Phi_{nkt}\}_n, for all k\ge1. Then by the uniqueness, we get \mu_{rt}=\mu_t for all r=p/q and \{t:\mu_t=\mu_1\} is closed and dense, hence coincides with \mathbb{R}.

    4. There is a question about the mixing properties of the induced map. The answer is quite a surprise. Let T:I\to I be an ergodic measure-preserving isomorphism on the unit interval. Then Friedman and Ornstein proved (link) that the following two collections are dense in \mathcal{B}_I:
    (4.1) A\in \mathcal{B}_I such that T_A^k is not ergodic for all k\ge 2.
    (4.2) A\in \mathcal{B}_I such that T_A is mixing.

    3. Abramov Entropy Formula. Let (X,\mathcal{X},\mu) be a probability measure system, T:X\to X be a \mu-preserving isomorphism on X.
    (3.1). Let A\in \mathcal{X} such that \mu(\bigcup_{n\ge 0}T^nA)=1, and \mu_A be the conditional measure of \mu on A. For any point x\in A, let n(x)=\inf\{n\ge1: T^nx\in A\} be the first return to A (it is finite for \mu-a.e. x\in A by Poincare recurrence theorem). Define the first-return map T_A:A\to A, x\mapsto T^{n(x)}x, which preserves \mu_A. Then h(T_A,\mu_A)\cdot \mu(A)=h(T,\mu).

    (3.2). Let r:X\to (c,C) be a measurable roof function, X_r be the suspension space of X wrt r, \phi_t be the suspension flow on X_r, which preserves the (normalized) suspension measure \mu_r=\frac{1}{\mu(r)}\mu\times \ell. Then h(\phi_1,\mu_r)\cdot \mu(r)=h(T,\mu).

    (3.1=>3.2). We assume c\ge 2 for simplicity and then set I=[0,1). Then consider the set A=X\times I\subset X_r, and the induced map \phi_A:=(\phi_1)_A, which preserves \mu_A:=(\mu_r)_A=\mu\times \ell_{I}. Note that \phi_A(\{x\}\times I)=\{Tx\}\times I, for which it is just a rotation. So h(\phi_A,\mu\times \ell_{I})=h(T,\mu) (not that trivial). From (3.1), we see that h(\phi_A,\mu_A)\cdot \mu_r(A)=h(\phi_1,\mu_r), where \mu_r(A)=\frac{1}{\mu(r)}. Combining terms, we get (3.2).

    2. Some sharp contrast statements.
    C^1 generic map (in particular, among the expanding ones) has no ACIP (by Avila and Bochi). Every C^{1+\alpha} expanding map admits a (unique) ACIP (due to Krzyzewski and Szlenk).

    Consider an expanding map f:X\to X. Then every Holder potential \phi has a unique equilibrium state \mu_\phi. Consider the zero-temperature limit \mu^0_\phi=\lim_{\beta\to\infty}\mu_{\beta\phi}. Sometime \mu^0_\phi is called an \phi-maximizing measure. Let E(\phi) be the collection of \phi-maximizing measures. Then for a general Lipschitz continuous potential, the following dichotomy holds:
    (1) either \phi is cohomologous to a constant (then E(\phi) contains all invariant measures);
    (2) or it has a unique maximizing measure, which is supported on a periodic orbit.

    Clearly the first case consists of a meager subset, and open and densely in the Lipschitz continuous potential, the ground state is supported on a periodic orbit.

    An open question in ergodic optimization is: consider the doubling map \tau:\mathbb{T}\mapsto\mathbb{T}, x\mapsto 2x. Find \phi such that E(\phi)=\{m\} (the Lebesgue measure).

    Consider a hyperbolic basic set X of f. For generic (but with empty interior) potential \phi\in C(X), its has a unique ground state. Moreover, this state is fully supported.

    A useful observation made by Jenkinson: let f:X\to X be continuous, \phi be upper semi-continuous potential. Then the map \Phi: \mu\in\mathcal{M}(f)\mapsto \mu(\phi) is also upper semi-continuous.
    Proof. Since X is compact, \phi is bounded and the map \Phi is well-defined. Let \mu_n\in\mathcal{M}(f)\to \mu. We need to show that \limsup\mu_n(\phi)\le \mu(\phi). First assume \mu(\phi)\neq-\infty. Then pick a sequence of continuous functions \phi_i\ge\phi_{i+1}\to\phi pointwisely. Note that \mu(\phi-\phi_n)\to 0 by the monotone convergence theorem.

    1. Let \text{Diff}_m^r(S) be the set of C^r area-preserving diffeomorphisms on a surface S with C^r topology.
    Note that H^r=\{f\in \text{Diff}_m^r(S): h_{top}(f) > 0\} is open and dense (Pugh-Hayashi for r=1; for 2\le r\le \infty: Pixton for S^2, Oliveira for \mathbb{T}^2 and general surfaces with irreducible homology actions, Xia for Hamiltonian on general surface. still open for not that complicated action on general surface).

    What about H^r_m=\{f\in \text{Diff}_m^r(S): h_m(f) > 0\}?

    In the case r=1 and S\neq \mathbb{T}^2, Bochi-Mane Theorem states that h_m(f)=0 generically, and hence H^r_m is of first category. So C^1-generically, h_m(f)=0 < h_{top}(f).

    I don’t have a specific example with h_m(f)=0 < h_{top}(f) (even for r=1). See the following post here. Interesting cases: standard maps, convex billiards, geodesic flow on spheres with convex shape, perturbations of completely integrable ones. In particular, approximate ellipse with positive metric entropy.

    Collections again

    8. Definition. Given a family of maps T_\epsilon:X\to X with corresponding invariant densities \phi_\epsilon. Then T_0 is said to be acim-stable if lim_{\epsilon\to 0}T_\epsilon=T_0 implies lim_{\epsilon\to 0}\phi_\epsilon=\phi_0.
    The limits are taken with respect to properly chosen metrics on the space of maps and densities, respectively.

    Functions of the bounded variation are continuous except at a most countable number of points, at which they have two one-sided limits.

    7. Let \mathcal{H}=(\mathbb{R}^3,\ast) be the 3D Heisenberg group, with (a,b,c)\ast(x,y,z)=(a+x,b+y,c+z+ay). Let \Gamma=\langle\alpha,\beta,\gamma|\alpha\ast\beta=\beta\ast\alpha\ast\gamma,\alpha\ast\gamma=\gamma\ast\alpha,\beta\ast\gamma=\gamma\ast\beta\rangle be a cocompact discrete subgroup (for example \mathbb{Z}^3=\langle \mathbf{i},\mathbf{j},\mathbf{k}\rangle). Then M=\mathcal{H}/\Gamma is a 3D nilmanifold. A general non-toral
    three-dimensional nilmanifold is also of this form. Suppose we have a homomorphism h:\mathcal{H}\to\mathbb{R}, which is of the form (x,y,z)\mapsto ax+by, which induces a 2D-foliation, say \mathcal{F}_h on \mathcal{H} and on M.

    Theorem. Every Reebless foliation on M is almost aligned with some \mathcal{F}_h.
    Plante for C^2, Hammerlindl and Potrie for C^{1,0}.

    Theorem. Every partially hyperbolic system on M is accessible.
    J. Rodriguez Hertz, F. Rodriguez Hertz and R. Ures (convervative), Hammerlindl and Potrie (general)

    6. Let r\ge 1 and S:\mathbb{R}^{r+1}\to\mathbb{R} be a C^2 function. Consider the solutions x:\mathbb{Z}\to \mathbb{R} of the recurrence relation:
    (\ast) \displaystyle R(x_{i-r},\cdots,x_{i+r}):=\sum_j \partial_{x_i}S(x_j,\cdots,x_{j+r})=0, for all i\in\mathbb{Z}.
    Note that (\ast) is actually a finite sum of r+1 terms over j=i-r,\cdots,i. It is the derivative of formal series W(x)=\sum_j S(x_j,\cdots,x_{j+r}) with respect to \partial_{x_i}.
    Example. Billiards, or generally twist maps, where r=1 and S is the generating function, the solution gives the configuration of an orbit.

    There are some conditions:
    Periodicity. S(x+1)=S(x). So S descends to a map on \mathbb{R}^{\mathbb{Z}}/\mathbb{Z}.
    Monotone. \displaystyle\partial_{x_i,x_k}S(x_j,\cdot,x_{j+r})\le 0 for all j and all i\neq k, and \displaystyle\partial_{x_j,x_{j+1}}S(x_j,\cdot,x_{j+r}) < 0 for all j.
    Coercivity. S is bounded from below and there exists k such that S(x_j,\cdots,x_{j+r})\to\infty as |x_k-x_{k+1}|\to\infty.
    Under these conditions the (\ast) is called a monotone variational recurrence relation.

    A sequence x is said to be a global minimizer, if W(x)\le W(x+v) (understand as over all intervals) for all sequences v. Clearly a global minimizer solves (\ast). The collection of global minimizers is also closed under coordinately convergence.

    For a real number a, a sequence x with x_0=a is called an a-minimizer, if it is minimizes among all y‘s with y_0=a.
    Ana-minimizers in general need not be solutions to (\ast).
    Given a rational p/q, we consider the operator \tau_{p,q} (shift p and subtract q) and Birkhoff orbits of rotational number p/q prime sum W_{p,q}=S(x_0,\cdots, x_{r})+\cdots+S(x_{p-1},\cdots,x_{p-1+r}) over the periodic ones x=\tau_{p,q}(x).

    Periodic Peierls barrier. Let a be a real and p,q be coprime. Then as
    \displaystyle P_{p,q}(a):= \min_{\tau_{p,q}x=x,x_0=a} W_{p,q}(x)-\min_{\tau_{p,q}x=x}W_{p,q}(x).

    It is easy to see that
    There exists a periodic minimizer x\in M_{p,q} with x_0 =a if and only if P_{p,q}(a)=0.

    M_{p,q} gives an invariant curve if and only if P_{p,q}(\cdot)\equiv 0.

    Then the Peierls barrier at a general frequency is defined as P_{\omega}(a)=\lim_{p/q\to\omega}P_{p,q}(a) when the limit exists (see Mramor and Rink, arxiv:1308.3073).

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