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ebsd2021:patterson

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ebsd2021:patterson [2021/10/17 10:31] – created escolaebsd2021:patterson [2021/10/17 10:33] (current) – escola
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 \[ \[
  \delta=\delta(\Gamma,F)  \delta=\delta(\Gamma,F)
- \eqdef \limsup_{n\to\infty}\frac1n\log\sum_{n-c\le d(\gamma x,x)\le n}e^{d^F(x,\gamma x)}.+ := \limsup_{n\to\infty}\frac1n\log\sum_{n-c\le d(\gamma x,x)\le n}e^{d^F(x,\gamma x)}.
 \] \]
 One can show, as before, that the series One can show, as before, that the series
 \[ \[
  P^F(s,x,y)  P^F(s,x,y)
- \eqdef \sum_{\gamma\in\Gamma}e^{-sd^F(x,\gamma y)}+ := \sum_{\gamma\in\Gamma}e^{-sd^F(x,\gamma y)}
 \] \]
 converges for $s>\delta(\Gamma,F)$ and diverges for $s<\delta(\Gamma,F)$, and that these properties and the value of $\delta$ do not depend on $x,y\in X$.  converges for $s>\delta(\Gamma,F)$ and diverges for $s<\delta(\Gamma,F)$, and that these properties and the value of $\delta$ do not depend on $x,y\in X$. 
  
 Generalizing the before defined Busemann densities, a family of positive finite measures $\{\mu_p^F\colon p\in X\}$ on $X\cup X(\infty)$ is a //Patterson density of dimension $\delta$// (relative to $\Gamma$ and $F$) if it satisfies: Generalizing the before defined Busemann densities, a family of positive finite measures $\{\mu_p^F\colon p\in X\}$ on $X\cup X(\infty)$ is a //Patterson density of dimension $\delta$// (relative to $\Gamma$ and $F$) if it satisfies:
-  *  $\displaystyle\frac{d\mu_x^F}{d\mu_y^F}(\xi)=e^{\displaystyle -C_\xi^{F-\delta}(x,y)}$ for almost every $\xi\in X(\infty)$,\\[0.2cm]+  * $\displaystyle\frac{d\mu_x^F}{d\mu_y^F}(\xi)=e^{\displaystyle -C_\xi^{F-\delta}(x,y)}$ for almost every $\xi\in X(\infty)$,
   * $\displaystyle\{\mu_p^F\colon p\in X\}$ is $\Gamma$-invariant, that is, for every $p\in X$ and $\gamma\in\Gamma$, it holds   * $\displaystyle\{\mu_p^F\colon p\in X\}$ is $\Gamma$-invariant, that is, for every $p\in X$ and $\gamma\in\Gamma$, it holds
 \[ \[
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 \[ \[
  \mu_{p,x,s}^F  \mu_{p,x,s}^F
- \eqdef \frac{\sum_{\gamma\in\Gamma}e^{-sd^F(p,\gamma x)}\delta_{\gamma x}}+ := \frac{\sum_{\gamma\in\Gamma}e^{-sd^F(p,\gamma x)}\delta_{\gamma x}}
  {P^F(s,x,x)}.  {P^F(s,x,x)}.
 \] \]
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 \[ \[
  \mu_p^F  \mu_p^F
- \eqdef \lim_{k\to\infty}\mu_{p,x,s_k}.+ := \lim_{k\to\infty}\mu_{p,x,s_k}.
 \] \]
  
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 \[ \[
  \frac{1}{b}\,e^{d^F(x,y)-s d(x,y)}  \frac{1}{b}\,e^{d^F(x,y)-s d(x,y)}
- \le \mu_x^F\big(\pr_y(B_\rho(x))\big)+ \le \mu_x^F\big({\rm pr}_y(B_\rho(x))\big)
  \le b\,e^{d^F(x,y)-s d(x,y)}  \le b\,e^{d^F(x,y)-s d(x,y)}
 \]  \]
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 \[ \[
  B_{e^{-d(x,y)}}\big([x,y)\cap X(\infty)\big)  B_{e^{-d(x,y)}}\big([x,y)\cap X(\infty)\big)
- \subset \pr_x(B_\rho(y))+ \subset {\rm pr}_x(B_\rho(y))
  \subset B_{Cd^{-d(x,y)}}([x,y)\cap X(\infty)\big).  \subset B_{Cd^{-d(x,y)}}([x,y)\cap X(\infty)\big).
 \] \]
  
  
ebsd2021/patterson.1634477519.txt.gz · Last modified: 2021/10/17 10:31 by escola