## Some distinguished meausres II

A note taken from Section 20.5, Introduction to the Morden Theory of Dynamical Systems (by Katok und Hasselblatts).

Let $M$ be a closed manifold and $T^t:M\to M$ be a transitive $C^2$ Anosov flow, that is, there exists a continuous invariant splitting $TM=E^s\oplus \langle X\rangle \oplus E^u$ such that, $X$ gives the flow direction, vectors in $E^s$ get uniformly contracted and vectors in $E^s$ get uniformly expanded. Let $\mathcal{F}^{s}$ be the strong stable foliation tangent to $E^s$, $\mathcal{F}^{u}$ be the strong unstable foliation tangent to $E^u$, $\mathcal{F}^{s+1}$ be the stable foliation tangent to $E^s\oplus\langle X\rangle$, $\mathcal{F}^{u+1}$ be the unstable foliation tangent to $E^u\oplus\langle X\rangle$. It is well known all leaves in these foliations are uniformly $C^2$ submanifolds.

We denote by $E\prec \mathcal{F}$ if $E$ is contained in a single leaf, say $s(E)$, of $\mathcal{F}$. Denote by $f\prec \mathcal{F}$ if $f$ is defined on a single leaf, say $s(f)$, of $\mathcal{F}$.
All ordinary notations are extended to this leaf-version, like openness, compactness, continuity, measurability and integrability.

Denote by $\mathcal{C}^{u+1}$ the set of leaf-functions $f$ with $\text{supp}(f)\prec \mathcal{F}^{u+1}$ compact and on which $f$ is continuous.

Note that $\mathcal{C}^{u+1}$ may not be a linear space, since $f+g$ may not lie in $\mathcal{C}^{u+1}$, where $f,g\in \mathcal{C}^{u+1}$. Still a functional $F:\mathcal{C}^{u+1}\to\mathbb{R}$ is said to be linear if
(a). $F(c\cdot f)=c F(f)$ for all $f\in \mathcal{C}^{u+1}$,
(b). $F(f+g)=F(f)+F(g)$ for all $f,g\in \mathcal{C}^{u+1}$ with $f+g\in \mathcal{C}^{u+1}$.
Let $\mathcal{C}^{\ast}$ be the collection of such linear functionals.

Fix a leaf-open set $U$ and a $f_U\in \mathcal{C}^{u+1}$ with $f_U> \chi_U$. Consider a parameter family $F_t\in \mathcal{C}^{\ast}$ given by $f\mapsto \int T^tf d\mu_{u+1}$ and the convex $\mathcal{C}^{\ast}_0=\{F\in \mathcal{C}^{\ast}|F=\sum c_i F_{t_i}: c_i,t_i\ge0, F(f_U)=1\}$.

Proposition.
1. For each $f\in \mathcal{C}^{u+1}$, there exists $C_f\ge1$ such that $|F(f)|\le C_f$ for all $F\in\overline{\mathcal{C}^{\ast}_0}$.
2. If $f\in \mathcal{C}^{u+1}$ is nonnegative and not identically zero, then there exists $c_f>0$ such that $|F(f)|\ge c_f$ for all $F\in\overline{\mathcal{C}^{\ast}_0}$.
3. For each $\epsilon>0$ and $f\in \mathcal{C}^{u+1}$, there exists $\delta>0$ such that $|F(f)-F(g)|<\epsilon$ for all $g$ $\delta$-close to $f$ and all $F\in\overline{\mathcal{C}^{\ast}_0}$.

Theorem. There exists $J\in \overline{\mathcal{C}^{\ast}_0}$ and a constant $h_u>0$ such that $T^tJ= e^{t\cdot h_u}\cdot J$.

We want to use Riesz Representation Theorem to pick a measure on $M$ corresponding to $J$. But the underline space $\mathcal{C}^{u+1}$ is not linear space. So we start with a local plaque. Let $\mathrm{OC}^{u+1}$ be the collection of open $u+1$-plaques with compact closures. Pick one such plaque $U$ and consider $\mathcal{C}^{u+1}_U$ be the collection of continuous functions supported on $\overline{U}$. As a continuous linear functional on $\mathcal{C}^{u+1}_U$, the action of $J$ can be extended to $C(\overline{U})$, the usual space of continuous functions on $\overline{U}$ and hence a leaf-measure $\mu_U$ (now by Riesz Representation Theorem). After checking the compactibility of the family $\{\mu_U:U\in \mathrm{OC}^{u+1}$, we get a system of leaf-measures $\mu^{u+1}:\mathrm{OC}^{u+1}\to\mathbb{R}$ with the following properties:
1. The restriction of $\mu^{u+1}$ on $\mathrm{OC}^{u+1}(W^{u+1}(p))$ gives rise to a single, $\sigma$-finite measure on $W^{u+1}(p)$.
2. $\mu^{u+1}(T^tU)=e^{th_u}\mu^{u+1}(U)$ for all $U\in\mathrm{OC}^{u+1}$ and all $t$.
3. If $U,V\in \mathrm{OC}^{u+1}$ are $s$-equivalent, then $\mu^{u+1}(U)=\mu^{u+1}(V)$.

Further we induces a measure $\mu^u$ on $\mathrm{OC}^{u}$ by flowing forward some time. Similarly we get a measure $\mu^s$ on $\mathrm{OC}^{s}$ (by reversing the time) Denote the corresponding rate by $h_s$.

Patching these together, we will get the so called Margulis measure $\mu$ on $M$, with local product structure $\mu^s\times \ell\times \mu^u$ and $\mu(T^tA)=e^{(h_u-h_s)t}\mu(A)$. Testing $A=M$ we get that $h_u=h_s$ (from now on we denote the common value by $h$) and hence the invariance of $\mu$: $\mu(T^tA)=\mu(A)$.

Again by the local product structure we see $\mu(B(x,\epsilon,t))\sim e^{-ht}\cdot C(\epsilon)$ and hence the metric entropy $h_{\mu}(f)=h$ (for example, by Brin-Katok local entropy formula). Moreover it follows that $h=h_{top}(T)$, the topological entropy of $T$. So this Margulis measure $\mu$ coincides with Bowen measure (measure of maximal entropy).

The following part taken from Margulis’s paper [M] “Certain measures associated with U-flows on compact manifolds”.

Let’s define $T^t\mu_{u+1}:E\mapsto \mu_{u+1}(T^tE)$. In particular there exist $b>0$ and $\lambda>0$ such that $T^t\mu_{u+1}(E)\ge b e^{\lambda t} \mu_{u+1}(E)$ for every $t\ge0$. Dually we have $\int T^tf d\mu_{u+1}=\int f d(T^t\mu_{u+1})\ge b e^{\lambda t} \int f d\mu_{u+1}$ for every $f\prec \mathcal{F}$ and every $t\ge0$. Two sets $E,F\prec \mathcal{F}^{u+1}$ are said to be $\epsilon$-equivalent if the $s$-holonomy $h^s:E\to F$ between is a homeomorphism and $d(h^s,Id)\le \epsilon$.
Lemma 2.1 in [M]. Let $E$ be a leaf-open subset. Then there exist $\epsilon>0$ and $r>0$, such that for every $x\in M$, the set $\mathcal{F}^{u+1}(x,r)$ is $\epsilon$-equivalent to some subset of $E$.
It is equivalent to say that $\mathcal{F}^s$ is minimal. Note that the $\epsilon$ here may be large. The

Lemma 2.3 in [M]. For each $\delta>0$ there exists $\epsilon>0$ such that $|\frac{\mu_{u+1}(E_1)}{\mu_{u+1}(E_2)}-1|<\delta$ for every $\epsilon$-equivalent $E_1,E_2\prec \mathcal{F}^{u+1}$.

Lemma 2.4 in [M]. Let $E$ be a leaf-open subset. For $\epsilon,r$ given by Lemma 2.1, there exists $a>0$ such that for every $x\in M$ and $t\ge0$, $\frac{\mu_{u+1}(T^t\mathcal{F}^{u+1}(x,r))}{\mu_{u+1}(T^t E)}.

Let $p\in M$ and define $f_p:q\in M\mapsto (1+\mu_{u+1}[\mathcal{F}^{u+1}(p,d_{u+1}(p,q))])^{-2}$ (if $q\in \mathcal{F}^{u+1}(p)$). Then let $U_{i,p}$ be the set of points $q$ with $i\le \mu_{u+1}[\mathcal{F}^{u+1}(p,d_{u+1}(p,q))]< i+1$. Let $r_i>0$ with $\mu_{u+1}[\mathcal{F}^{u+1}(p,r_i)]=i$. Clearly $U_{i,p}=\mathcal{F}^{u+1}(p,r_{r+1})\backslash\mathcal{F}^{u+1}(p,r_i)$ and hence $\mu_{u+1}(U_{i,p})= 1$. In particular
Lemma 2.6 in [M]. For every $p\in M$, $\int f_p(q)d\mu_{u+1}(q)\le\sum_{i\ge 0}\int_{U_{i,p}}\frac{d\mu_{u+1}}{(1+i)^2}< 2$.

Let $p\in M$, $E$ be leaf-open and $r>0$ be given as above. Define a map $\hat{f}_{p,E}:q\mapsto \int_{\mathcal{F}^{u+1}(q,r)}f_p(w)d\mu_{u+1}(w)$. Clearly $\hat{f}_{p,E}$ is continuous and positive on $\mathcal{F}^{u+1}(q)$.
Lemma 2.8 in [M]. For every $p\in M$ and $t\ge0$, $\frac{\int T^t \hat{f}_{p,E} d\mu_{u+1}}{\mu_{u+1}(T^t E)}<2a$.
Proof. Applying Fubini Theorem and Lemma 2.4, we get
$\int T^t \hat{f}_{p,E} d\mu_{u+1}=\int\int_{\mathcal{F}^{u+1}(q,r)}f_p(w)d\mu_{u+1}(w) d T^t \mu_{u+1}(q)$
$=\int \int_{\mathcal{F}^{u+1}(w,r)}dT^t\mu_{u+1}(q) f_p(w)d\mu_{u+1}(w)=\int T^t\mu_{u+1}(\mathcal{F}^{u+1}(w,r)) f_p(w)d\mu_{u+1}(w)$
$\le \int a\cdot \mu_{u+1}(T^t E)\cdot f_p(w)d\mu_{u+1}(w)= a\cdot \mu_{u+1}(T^t E)\cdot\int f_p(w)d\mu_{u+1}(w)$
$< 2a\mu_{u+1}(T^t E)$.

Two functions $f,g\in \mathcal{C}^{u+1}$ are $\epsilon$-equivalent if
(a). $\text{supp}(f)$ and $\text{supp}(g)$ are $\epsilon$-equivalent (via stable holonomy $h^s$),
(b). $f=g\circ h^s$.

Lemma 2.9 in [M]. For every nonnegative and nonzero $f\in \mathcal{C}^{u+1}$, and every leaf-compact $E$, there exists $C=C(E,f)>0$ such that for every bounded, leaf-measurable $g$ vanishing outside $E$, and every $t\ge 0$,
$\frac{\int T^tg d\mu_{u+1}}{\int T^tfd\mu_{u+1}}\le C\cdot \|g\|_0$.
Proof. Pick $\epsilon(f)>0$ such that $f^{-1}(\epsilon,+\infty)$ has nonmepty leaf-interior. Then $\int T^tg d\mu_{u+1}\le T^t\mu_{u+1}(E)\cdot \|g\|_0\le \epsilon^{-1} \cdot c\cdot\int_{f^{-1}(\epsilon,+\infty)}T^tfd\mu_{u+1}\cdot \|g\|_0$. Then set $C=c\epsilon^{-1}$.

Lemma 2.11 in [M]. For any $\delta>0$ there exists $\epsilon>0$ such that for all $\epsilon$-equivalent $f,g\in \mathcal{C}^{u+1}$, $|\int f d\mu_{u+1}-\int gd\mu_{u+1}|\le \delta\cdot \int |f| d\mu_{u+1}$.