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)}<a.

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}.

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