On the fractal structure of the rescaled evolution set of Carlitz sequences of polynomials

AbstractSelf-similarity properties of the coefficient patterns of the so-called m-Carlitz sequences of polynomials are considered. These properties are coded in an associated fractal set – the rescaled evolution set. We extend previous results on linear cellular automata with states in a finite field. Applications are given for the sequence of Legendre polynomials and sequences associated with the zero Bessel function

KMS states on Quantum Grammars

We consider quantum (unitary) continuous time evolution of spins on a lattice together with quantum evolution of the lattice itself. In physics such evolution was discussed in connection with quantum gravity. It is also related to what is called quantum circuits, one of the incarnations of a quantum computer. We consider simpler models for which one can obtain exact mathematical results. We prove existence of the dynamics in both Schroedinger and Heisenberg pictures, construct KMS states on appropriate C*-algebras. We show (for high temperatures) that for each system where the lattice undergoes quantum evolution, there is a natural scaling leading to a quantum spin system on a fixed lattice, defined by a renormalized Hamiltonian.Comment: 22 page

Combinatorics of Pisot Substitutions

Siirretty Doriast

Cellular Probabilistic Automata - A Novel Method for Uncertainty Propagation

We propose a novel density based numerical method for uncertainty propagation under certain partial differential equation dynamics. The main idea is to translate them into objects that we call cellular probabilistic automata and to evolve the latter. The translation is achieved by state discretization as in set oriented numerics and the use of the locality concept from cellular automata theory. We develop the method at the example of initial value uncertainties under deterministic dynamics and prove a consistency result. As an application we discuss arsenate transportation and adsorption in drinking water pipes and compare our results to Monte Carlo computations

Automaticity and Invariant Measures of Linear Cellular Automata

We show that spacetime diagrams of linear cellular automata with (-p)-automatic initial conditions are automatic.This extends existing results on initial conditions which are eventually constant.Each automatic spacetime diagram defines a jointly invariant subset of \F_p^\Z, and if the initial condition is not eventually periodic then this invariant set is nontrivial.We construct, for the Ledrappier cellular automaton, a family of nontrivial jointly-invariant measures on the space of configurations with entries from the finite field with 3 elements..Finally, given a linear cellular automaton, we construct a nontrivial jointly-invariant measure on the space of configurations with entries from the finite field with p for all but finitely many p

Symbolic Dynamics and its Applications

Book review of Symbolic Dynamics and its Applications, edited by Susan Williams, AMS