*Bratteli graphs.*

This package deals with Bratteli graphs. In every function of the package, the Bratteli graph is given by a function returning for a level `n`

of the graph the incidence matrix of the graph between level `n`

and level `n+1`

: the `(i,j)`

-entry of this matrix is the number of edges between the `i`

-th vertex at level `n`

and the `j`

-th vertex at level `n+1`

.

For example, the binary tree is defined by:

```
tree <- function(n) {
M <- matrix(0, nrow = 2^n, ncol = 2^(n+1))
for(i in 1:nrow(M)) {
M[i, ][c( 2*(i-1)+1, 2*(i-1)+2 )] <- 1
}
M
}
```

The function `bratteliGraph`

generates some LaTeX code which renders the graph up to a given level:

If you donâ€™t like the style, you are free to modify the LaTeX code.

Here is a binary tree with double edges:

```
tree2 <- function(n) {
M <- matrix(0, nrow = 2^n, ncol = 2^(n+1))
for(i in 1:nrow(M)) {
M[i, ][c( 2*(i-1)+1, 2*(i-1)+2 )] <- 2
}
M
}
bratteliGraph("binaryTree2.tex", tree2, 3)
```

Here is the code for the Pascal graph:

```
Pascal <- function(n) {
M <- matrix(0, nrow = n+1, ncol = n+2)
for(i in 1:(n+1)) {
M[i, ][c( i, i+1L )] <- 1
}
M
}
```

The *dimension* of a vertex of a Bratteli graph is the number of paths of the graph going from the root vertex to this vertex. The function `bratteliDimensions`

of the package computes these numbers:

```
## [[1]]
## [1] "1" "1"
##
## [[2]]
## [1] "1" "2" "1"
##
## [[3]]
## [1] "1" "3" "3" "1"
```

Bratteli graphs are of interest to ergodicians, and particularly to Vershik, who introduced the *intrinsic kernels* of a Bratteli graph and the *intrinsic distance* between the vertices of a Bratteli graph. Here is a picture of the Pascal graph showing the intrinsic kernels:

The intrinsic kernels are returned by the function `bratteliKernels`

. The intrinsic distances between two vertices at the same level are returned by the function `bratteliDistances`

.