  Let $\Gamma$ be a higher rank lattice acting on a nondegenerate dendrite $X$
with no infinite order points. We show that there exists a nondegenerate
subdendrite $Y$ which is $\Gamma$-invariant and satisfies the following items:
  (1) There is an inverse system of finite actions $\{(Y_i,
\Gamma):i=1,2,3,\cdots\}$ with monotone bonding maps $\phi_i:
Y_{i+1}\rightarrow Y_i$ and with each $Y_i$ being a dendrite, such that $(Y,
\Gamma|Y)$ is topologically conjugate to the inverse limit
$(\underset{\longleftarrow}{\lim}(Y_i, \Gamma), \Gamma)$.
  (2) The first point map $r:X\rightarrow Y$ is a factor map from $(X, \Gamma)$
to $(Y, \Gamma|Y)$; if $x\in X\setminus Y$, then $r(x)$ is an end point of $Y$
with infinite orbit; for each $y\in Y$, $r^{-1}(y)$ is contractible, that is
there is a sequence $g_i\in \Gamma$ with ${\rm diam}(g_ir^{-1}(y))\rightarrow
0$.
