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 remarks about dominated splitting property

I talked to Prof Sandro Vaienti about my current and past research after lunch today. 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)

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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Notes about Anosov diffeomorphisms

Taken from J. Franks, Anosov diffeomorphisms, in the book ‘Global Analysis’, (1968) 61–93.

Question: given f:M\to M, for what g:N\to N does there exist a nontrivial h:N\to M such that h\circ g=f\circ h? Franks proved that for some diffeo, it reduces to a homotopy problem. So the definition:

A diffeo f:M\to M is a \pi_1-diffeo, if given any homeo g:K\to K on a CW-complex K with a continuous map h:K\to M such that h_\ast\circ g_\ast=f_\ast\circ h_ast from \pi_1(K)\to \pi_1(M), there exists a unique base-point preserving map \hat h:K\to M homotopic to h such that \hat h\circ g=f\circ\hat h.

(covering version)

Examples of Anosov: hyperbolic toral automorphisms, hyperbolic nil-manifold automorphisms, hyperbolic infra-nilmanifold automorphisms (and their endomorphisms)

Theorem 2.2. Every hyperbolic infra-nilmanifold automorphism is a \pi_1-diffeo.

Problem: are all Anosov diffeo \pi_1?

Two diffeos f:M\to M and g:N\to N are \pi_1-conjugate if there exists an isomorphism \phi:\pi_1(N)\to \pi_1(M), such that \phi\circ g_\ast=f_\ast\circ \phi from \pi_1(N)\to \pi_1(M).

So two \pi_1 diffeos are topological conjugate iff they are \pi_1-conjugate.

Theorem 3.6. Suppose \pi_1(M) is torsion-free and f is \pi_1 on M.
a. if \pi_1(M) is virtually nilpotent, then f is topologically conjugate to a hyperbolic infra-nilmanifold automorphism.
b. if \pi_1(M) is nilpotent, then f is topologically conjugate to a hyperbolic nilmanifold automorphism.
a. if \pi_1(M) is abelian, then f is topologically conjugate to a hyperbolic toral automorphism.

Theorem 6.3. Every transitive codim=1 Anosov is a hyperbolic toral automorphism. Two such diffeo are topological conjugate iff they are \pi_1-conjugate.

Theorem 8.2. If \pi_1(M) is virtually nilpotent and f is expanding, then f is topologically conjugate to a hyperbolic infra-nilmanifold endomorphism.
A key step is that if f:M\to M is expanding, then \pi_1(M) has polynomial growth.

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Billiards

9. Victor Ivrii conjecture. Let Q be a strictly convex domain, F be the billiard map on the phase space \Omega=\partial Q\times(0,\pi). Let \omega be the Lebesgue measure of \Omega, and \ell be the Lebesgue measure on Q.

Conjecture 1. \omega(\text{Per}(F))=0 for all Q with C^\infty boundaries.

Remark. This is about a general domain Q, not a generic domain.

Definition. A point q\in\partial Q is said to be an absolute looping point, if \omega_q(\bigcup_{n\neq0}F^n\Omega_q)>0. Let L(Q) be the set of absolute looping points.

Conjecture 2. \ell(L(Q))=0 for all Q.

Question: When L(Q)=\emptyset?

8. In Boltzmann gas model, the identical round molecules are confined by a box. Sinai has replaced the box by periodic boundary conditions so that the molecules move on a flat torus.

On circular and elliptic billiard tables, for all p\ge 3, the (p,q)-periodic orbits forms a continuous family and hence all the trajectories have the same length.

An invariant noncontractible topological annulus, A\subset\Omega, whose interior contains no invariant circles, is a Birkhoff instability region. The dynamics in an instability region
has positive topological entropy. Hence Birkhoff conjecture implies
that any non-elliptical billiard has positive topological entropy.

How to construct a strictly convex C^1-smooth billiard table with metric positive entropy? b) How to construct a convex C^2-smooth billiard table with positive metric entropy?

Recall Bunimovich stadium is not C^2, and not strictly convex.

A periodic orbit of period q corresponds to an (oriented) closed polygon with q sides, inscribed in Q, and satisfying the condition on the angles it makes with the boundary. Birkhoff called these the harmonic polygons.

Then the maximal circumference of 2-orbit yields the diameter of Q. The minimax circumference of 2-orbit corresponds
to the width of Q.

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Physical entropy production

A simple fact: let T:X\to X be a homeomorphism preserving the measure-class of \omega, J(T^k,x) be the Jacobian of T^k at x. Then for any sequence a(k)\to\infty with \sum_k \frac{1}{a(k)}<+\infty we have \limsup_k\frac{J(T^k,x)}{a(k)}=0 for \omega-a.e. x\in X. For example a(k)=e^{t k} with t>0 or a(k)=k^{\alpha} with \alpha>1. This is a direct corollary of Borel-Cantelli property:

Consider the set E_{k,\delta}=\{x\in X: \frac{J(T^k,x)}{a(k)}\ge\delta\}. It is easy to see \omega(E_{k,\delta})\le \frac{1}{\delta\cdot a(k)} and hence \sum\omega(E_{k,\delta})\le \sum\frac{1}{\delta\cdot a(k)}<+\infty. So \omega(x\in E_{k,\delta} \text{infinitely often})=0 for all \delta>0

Notes from papers by Jaksic, Pillet, Rey-Bellet, Ruelle and Young.

4. Let \nu\ll \mu be two probability measures on X and \phi=\frac{d\nu}{d\mu} be its density. Then 1=\nu(X)=\mu(\phi). Moreover 1/\phi is well-defined with respect to \nu and \nu(1/\phi)=\mu(1)=1, too.

The relative entropy can be defined as E(\nu|\mu)=\nu(\log\phi) when \nu\ll\mu, +\infty otherwise.
Note that E(\nu|\mu)=\nu(-\log\frac{1}{\phi})\ge-\log\nu(1/\phi)=0. So the relative entropy is nonnegative.

Convexity: assume \nu_1,\nu_2\ll\mu and p+q=1, then \phi=p\phi_1+q\phi_2 and E(p\nu_1+q\nu_2|\mu)\le p\nu_1(\log\phi_1)+q\nu_2(\log\phi_2).

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Some notations

7. Let f be an Anosov diffeomorphism and g\in\mathcal{U}(f) be close enough, which leads to a Holder continuous conjugate h_g:M\to M with g\circ h_g=h_g\circ f. Ruelle found an explicit formula of h_g.

Let f,g:M\to M be two homeomorphisms, d(f,g)=\sup_M d(fx,gx), and \mathcal{U}(f,\epsilon)=\{g \text{ homeo and }d(f,g)<\epsilon\}. Let g\in \mathcal{U}(f,\epsilon). Then the map X_g:x\in M \mapsto \exp^{-1}_{fx}(gx)\in T_{fx}M gives a shifted-vector field on M, which induces a diffeomorhism \mathcal{U}(f,\epsilon)\to \mathcal{X}(0_f,\epsilon), g\mapsto X_g.
Let f be a C^r diffeomprhism. Then \mathcal{X}^r(0_f,\epsilon)\to \mathcal{U}^r(f,\epsilon), g\mapsto X_g induces the local Banach structure and turns \mathrm{Diff}^r(M) into a Banach manifold.

Let X_g\circ f^{-1}=X_g^s+X_g^u be the decomposition of the correction X_g\circ f^{-1} with respect to the hyperbolic splitting TM= E_g^s\oplus E_g^u. Then the derivative of g\mapsto h_g in the direction of X_g is given by the vector field \displaystyle \sum_{n\ge 0}Dg^n X^s_g-\sum_{n\ge1}Dg^{-n}X^u_g.

6. Let M be a compact orientable surface of genus g\ge1, s\ge1 and let \Sigma=\{p_1,\cdots,p_s\} be a subset of M. Let \kappa= (\kappa_1,\cdots,\kappa_s) be a s-tuple of positive integers with \sum (\kappa_i-1) =2g-2.

A translation structure on (M,\Sigma) of type \kappa is an atlas on M\backslash\Sigma
for which the coordinate changes are translations, and such that each singularity p_i
has a neighborhood which is isomorphic to the \kappa_i-fold covering of a neighborhood
of 0 in \mathbb{R}^2\backslash\{0\}.

The Teichmüller space Q_{g,\kappa}= Q(M,\Sigma,\kappa) is the set of such structures modulo isotopy relative to \Sigma. It has a canonical structure of manifold.

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Area of the symmetric difference of two disks

This post goes back to high school: the area \delta_d of the symmetric difference of two d-dimensional disks when one center is shifted a little bit. Let’s start with d=1. So we have two intervals [-r,r] and [x-r,x+r]. It is easy to see the symmetric difference is of length \delta_1(x)=2x.

Then we move to d=2: two disks L and R of radius r and center distance x=2a<r. So the angle \theta(x) satisfies \cos\theta=\frac{a}{r}.

difference

The symmetric difference is the union of R\backslash L and L\backslash R, which have the same area: \displaystyle (\pi-\theta)r^2+2x\sqrt{r^2-x^2}-\theta r^2=2(\frac{\pi}{2}-\arccos\frac{x}{r})r^2+2x\sqrt{r^2-x^2}. Note that the limit
\displaystyle \lim_{x\to0}\frac{\text{area}(\triangle)}{2a}=\lim_{a\to0}2\left(\frac{r^2}{\sqrt{1-\frac{a^2}{r^2}}}\cdot\frac{1}{r}+\sqrt{r^2-a^2}\right)=4r.
So \delta_2(x)\sim 4rx.

I didn’t try for d\ge3. Looks like it will start with a linear term 2d r^{d-1}x.

—————–

Now let {\bf r}(t)=(a\cos t,\sin t) be an ellipse with a>1, and {\bf r}'(t)=(-a\sin t,\cos t) be the tangent vector at {\bf r}(t). Let \omega be the angle from {\bf j}=(0,1) to {\bf r}'(t).
Let s(t)=\int_0^t |{\bf r}'(u)|du be the arc-length parameter and K(s)=|{\bf l}''(s)| be the curvature at {\bf l}(s)={\bf r}(t(s)). Alternatively we have \displaystyle K(t)=\frac{a}{|{\bf r}'(t)|^{3}}.

ellipse

The following explains the geometric meaning of curvature:

\displaystyle K(s)=\frac{d\omega}{ds}, or equivalently, K(s)\cdot ds=d\omega. (\star).

Proof. Viewed as functions of t, it is easy to see that (\star) is equivalent to K(t)\cdot \frac{ds}{dt}=\frac{d\omega}{dt}.

Note that \displaystyle \cos\omega=\frac{{\bf r}'(t)\cdot {\bf j}}{|{\bf r}'(t)|}=\frac{\cos t}{|{\bf r}'(t)|}. Taking derivatives with respect to t, we get
\displaystyle -\sin\omega\cdot\frac{d\omega}{dt}=-\frac{a^2\sin t}{|{\bf r}'(t)|^3}. Then (\star) is equivalent to

\displaystyle \frac{a}{|{\bf r}'(t)|^{3}}\cdot |{\bf r}'(t)|=\frac{a^2\sin t}{\sin\omega\cdot |{\bf r}'(t)|^3}, or
\displaystyle \sin\omega\cdot |{\bf r}'(t)|=a\sin t. Note that \displaystyle \sin^2\omega=1-\cos^2\omega=1-\frac{\cos^2 t}{|{\bf r}'(t)|^2}. Therefore \displaystyle \sin^2\omega\cdot |{\bf r}'(t)|^2= |{\bf r}'(t)|^2-\cos^2 t=a^2\sin^2 t, which completes the verification.

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