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The structures of higher rank lattice actions on dendrites

Let $Γ$ 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 $Γ$-invariant and satisfies the following items: (1) There is an inverse system of finite actions $\{(Y_i, Γ):i=1,2,3,\cdots\}$ with monotone bonding maps $ϕ_i: Y_{i+1}\rightarrow Y_i$ and with each $Y_i$ being a dendrite, such that $(Y, Γ|Y)$ is topologically conjugate to the inverse limit $(\underset{\longleftarrow}{\lim}(Y_i, Γ), Γ)$. (2) The first point map $r:X\rightarrow Y$ is a factor map from $(X, Γ)$ to $(Y, Γ|Y)$; 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 $g_i\in Γ$ with ${\rm diam}(g_ir^{-1}(y))\rightarrow 0$.