## 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).

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.

## 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 $Q$.

## 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 $E(p\nu_1+q\nu_2|\mu)\le p\nu_1(\log\phi_1)+q\nu_2(\log\phi_2)$.

## 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.

## 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. So the angle $\theta(x)$ satisfies $\cos\theta=\frac{a}{r}$.

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 $\displaystyle K(t)=\frac{a}{|{\bf r}'(t)|^{3}}$.

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.