Random self-similar trees: dynamical pruning and its applications to inviscid Burgers equations
We introduce generalized dynamical pruning on rooted binary trees with edge lengths. The pruning removes parts of a tree T, starting from the leaves, according to a pruning function defined on subtrees within T. The generalized pruning encompasses a number of previously studied discrete and continuous pruning operations, including the tree erasure and Horton pruning. For example, a finite critical binary Galton-Watson tree with exponential edge lengths is invariant with respect to the generalized dynamical pruning for arbitrary admissible pruning function. We will discuss an application in which we examine a one dimensional inviscid Burgers equation with a piece-wise linear initial potential with unit slopes. The Burgers dynamics in this case is equivalent to a generalized pruning of the level set tree of the potential, with the pruning function equal to the total tree length. We give a complete description of the Burgers dynamics for the initial velocity that alternates between the values of +1 and -1 at the epochs of a stationary Poisson process.
This work was done in collaboration with Ilya Zaliapin (University of Nevada Reno) and Maxim Arnold (University of Texas at Dallas).