--- title: "The Chacon map in R" author: "Stéphane Laurent" date: "2016-05-25" output: html_document --- ```{r setup, include=FALSE} knitr::opts_chunk\$set(echo = TRUE, fig.path="./assets/fig/ChaconMap-") fscale <- 1 ``` The `Chacon` function below is an implementation of the Chacon transformation of the unit interval \$(0,1)\$. The definition of the Chacon transformation can be found in [this thesis](https://cdr.lib.unc.edu/indexablecontent/uuid:bfc41b0c-b048-440f-9a57-533e02ea4f76). ```{r} Chacon <- function(x){ n <- 1L while(3L^(n+1L)*x >= 3L^(n+1L) - 1L) n <- n+1L while(3L^(n+1L)*x >= 2L*3L^n - 2L && 3L*x < 2L) n <- n+1L i <- 1L; A <- c(0L, 2L, 6L, 4L)/9L if((3^(n+1)-1)/2 > 2^31-1) stop("there's a too long vector") while(i < n){ A <- c(A, A + 2L/3L^(i+2L), 1L - 1L/3L^(i+1L), A + 4L/3L^(i+2L)) i <- i+1L } dists <- x-A j <- which(dists >= 0 & 3L^(n+1L)*dists < 2) return(A[j+1L]+dists[j]) } ``` Below is a plot of the Chacon transformation. ```{r ChaconPlot, fig.width=fscale*5, fig.height=fscale*5} par(mar=c(4,4,0,0)) u <- seq(0, 0.995, by=.005) Cu <- sapply(u, Chacon) plot(u, Cu, pch=19, cex=0.2, asp=1, xlim=c(0,1), pty="s") ``` This is probably rather useless.