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ebsd2021:tema10 [2021/09/22 10:06] – created escolaebsd2021:tema10 [2021/09/22 10:23] (current) – escola
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 +**Tema 10: Poincaré series**
 +
 A fairly good reference for principle ideas in this section (see results stated in [Section 2.5, Knieper '02] is also [Nicholls '89]. Let $X$ be a metric space (we have in mind a Hadamard manifold).  A fairly good reference for principle ideas in this section (see results stated in [Section 2.5, Knieper '02] is also [Nicholls '89]. Let $X$ be a metric space (we have in mind a Hadamard manifold). 
 Let $\Gamma$ be a discrete infinite subgroup of the group of isometries acting on $X$.  Let $\Gamma$ be a discrete infinite subgroup of the group of isometries acting on $X$. 
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  < a_k(p,q)e^{-sk}.   < a_k(p,q)e^{-sk}. 
 \]  \]
-Applying Remark 1 to $A_k\eqdef \log a_k$ implies the claim. +Applying Remark 1 to $A_k:= \log a_k$ implies the claim. 
  
 The group $\Gamma$ is of //divergence type// if the Poincaré series diverges at $s=\delta(\Gamma)$. The group $\Gamma$ is of //divergence type// if the Poincaré series diverges at $s=\delta(\Gamma)$.
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 **Lemma 2.** Assume that $X/\Gamma$ is a compact manifold of nonpositive curvature. The Poincaré series is of divergence type if and only if **Lemma 2.** Assume that $X/\Gamma$ is a compact manifold of nonpositive curvature. The Poincaré series is of divergence type if and only if
 \[ \[
- \int_0^\infty e^{-\delta(\Gamma)r}\vol B_r(p)\,dr+ \int_0^\infty e^{-\delta(\Gamma)r}{\rm vol} B_r(p)\,dr
 \]  \]
 diverges for one, and hence for every, $x\in X$. diverges for one, and hence for every, $x\in X$.
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 **Proof.** Let $D\subset X$ be a fundamental domain for $\Gamma$. Then, by definition, $D$ is the set which contains from any orbit $\{\gamma p\colon\gamma\in\Gamma\}$ exactly one point.  Hence **Proof.** Let $D\subset X$ be a fundamental domain for $\Gamma$. Then, by definition, $D$ is the set which contains from any orbit $\{\gamma p\colon\gamma\in\Gamma\}$ exactly one point.  Hence
 \[ \[
- \int_D\sum_{\gamma\in\Gamma}e^{-sd(p,\gamma x)}\,d\vol(x) + \int_D\sum_{\gamma\in\Gamma}e^{-sd(p,\gamma x)}\,d{\rm vol}(x) 
- = \int_Xe^{-sd(p,x)}\,d\vol(x) + = \int_Xe^{-sd(p,x)}\,d{\rm vol}(x) 
- = \int_0^\infty e^{-sr}\vol B_r(p)\,dr.+ = \int_0^\infty e^{-sr}{\rm vol} B_r(p)\,dr.
 \] \]
 This implies the claim. This implies the claim.
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 \[ \[
  h(g)  h(g)
- := \lim_{r\to\infty}\frac1r\log\vol B_r(p)+ := \lim_{r\to\infty}\frac1r\log{\rm vol} B_r(p)
 \]  \] 
 exists and is independent of $p\in X$, and called //volume entropy// of $(M,g)$. It is intimately related to the //topological entropy// $h_{\rm top}(\phi)$ of the geodesic flow $\phi=\{\phi^t\}_t$ of $SM$ by exists and is independent of $p\in X$, and called //volume entropy// of $(M,g)$. It is intimately related to the //topological entropy// $h_{\rm top}(\phi)$ of the geodesic flow $\phi=\{\phi^t\}_t$ of $SM$ by
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  = h_{\rm top}(\phi).  = h_{\rm top}(\phi).
 \]  \] 
-[Freire, Ma\~n\'e '82] extended this result to manifolds without conjugate points. +[Freire, Mañé '82] extended this result to manifolds without conjugate points. 
ebsd2021/tema10.1632315978.txt.gz · Last modified: 2021/09/22 10:06 by escola