## Tag Archives: measure

### Simple proporties

Let $X$ be a compact Hausdorff space, $\mathcal{B}(X)$ be Borel $\sigma$-algebra, and $\mathcal{M}(X)$ be the collection of Borel Probability measures on $X$. This again is a compact Hausdorff space.

Fo r each $B\in\mathcal{B}(X)$ there exists a $G_\delta$ set $\tilde{B}\supset B$ such that $\mu(\tilde{B}\backslash B)=0$ for every $\mu\in\mathcal{M}(X)$.

If $\mu_k,\mu\in\mathcal{M}(X)$, then $\mu_k\rightarrow\mu$ in the weak$*$-topolgy is equivalent to either one of the following:

For each closed subset $F\subseteq X$, $\limsup\mu_k(F)\le\mu(F)$.

For each open subset $U\subseteq X$, $\liminf\mu_k(U)\ge\mu(U)$.

For every $A\subseteq E$ with $\mu(\partial A)=0$, $\mu_k(A)\rightarrow\mu(A)$.

If $\{x\in X:f(x)>c\}$ is an open set in $X$ for all $c\in\mathbb{R}$, then $f$ is said to be lower semicontinuous. If $\{x\in X:f(x) is an open set in $X$ for all $c\in\mathbb{R}$, then $f$ is said to be upper semicontinuous. If $f_\lambda$ are a family of l.s.c. (or $g_\lambda$ u.s.c.), then so is $\sup_{\lambda}f_\lambda$ (or $\inf_{\lambda}g_\lambda$) since

$\{x\in X:\sup_{\lambda}f_\lambda(x)>c\}=\bigcup_{i\ge1}\bigcup_{\lambda}\{x:f_\lambda(x)>c-1/i\}.$
$\{x\in X:\inf_{\lambda}g_\lambda(x)

The Lyapunov exponent of a smooth diffeo $f:M\rightarrow M$ is defined as

$\chi(x,v)=\limsup\frac{1}{n}\log\|D_xf^n(v)\|$ for each $v\in T_x M$ and $x\in M$. It behaves quite good in the measure sense. But in general there is no more information about its topological properties. For examle it may not be semicontinuous.

Let $f:M\to M$ be a diffeo and $\Lambda$ be a compact $f$-invariant set. Denote
$D^s(x) =\{v\in T_xM|\,\|Df^nv\|\to0\text{ as }n\to+\infty\},$
$D^u(x) =\{v\in T_xM|\,\|Df^{-n}v\|\to0\text{ as }n\to+\infty\}.$
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These are $Df$-invariant linear subspaces of $T_xM$, and vectors of $D^s$ and $D^u$ are asymptotic to zero, but not necessarily exponentially fast. However, if the two subspaces form a direct sum at every point of $\Lambda$, exponential
convergence will follow:
Proposition (Mane, Liao). $\Lambda$ is hyperbolic if and only if $D^s(x) \oplus D^u(x) = T_xM$ for all $x\in\Lambda$.