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Distribution stable under iteration
x<-0.15 for (k in 1:20000) {x<-c((x[1]^2-1)/(2*abs(x[1])),x)} hist(x[abs(x)<10],breaks=100,prob=T) start <- function(x) {1+x/sqrt(1+x^2)-(2*x+sqrt(1+x^2)+x^2/sqrt(1+x^2))/sqrt(2+2*x^2+2*x*sqrt(1+x^2))} iter1 <- function(x) {(start(sqrt(1+x^2)+x)+start(-sqrt(1+x^2)-x))*(1+x/sqrt(1+x^2))} iter2 <- function(x) {(iter1(sqrt(1+x^2)+x)+iter1(-sqrt(1+x^2)-x))*(1+x/sqrt(1+x^2))} iter3 <- function(x) {(iter2(sqrt(1+x^2)+x)+iter2(-sqrt(1+x^2)-x))*(1+x/sqrt(1+x^2))} curve(iter3(x),col='red',add=T) plot(ecdf(x[abs(x)<10])) curve(1-sqrt(2+2*x^2+2*x*sqrt(1+x^2))+sqrt(1+x^2)+x,col='red',add=T)
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