# American Institute of Mathematical Sciences

June  2009, 2(2): 301-314. doi: 10.3934/dcdss.2009.2.301

## A note on universality in multidimensional symbolic dynamics

 1 Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, United States

Received  February 2008 Revised  September 2008 Published  April 2009

We show that in the category of effective $\mathbb{Z}$-dynamical systems there is a universal system, i.e. one that factors onto every other effective system. In particular, for $d\geq3$ there exist $d$-dimensional shifts of finite type which are universal for $1$-dimensional subactions of SFTs. On the other hand, we show that there is no universal effective $\mathbb{Z}^{d}$-system for $d\geq2$, and in particular SFTs cannot be universal for subactions of rank $\geq2$. As a consequence, a decrease in entropy and Medvedev degree and periodic data are not sufficient for a factor map to exists between SFTs.
We also discuss dynamics of cellular automata on their limit sets and show that (except for the unavoidable presence of a periodic point) they can model a large class of physical systems.
Citation: Michael Hochman. A note on universality in multidimensional symbolic dynamics. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 301-314. doi: 10.3934/dcdss.2009.2.301
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