Problem 2

What are the exact relations between entropies and Lyapunov exponents for cellular automata?

Using the finite information density of cellular automaton configurations, and the finite rate of information propagation in cellular automata, a number of inequalities may be derived between entropies and Lyapunov exponents (). An example is [5]. Preliminary numerical evidence suggests that for some cellular automata these inequalities may in fact be equalities. This would imply an important connection between the static properties of cellular automata, as embodied in entropies, and their dynamic properties, as measured by Lyapunov exponents. One is hampered in these studies by the lack of an efficient method for computing entropies. The best approach so far uses a conditional entropy method [11].

Lyapunov exponents can be considered to measure the rate of divergence of trajectories in the space of configurations. In continuous dynamical systems, a geometry is defined for this space, and one can identify Lyapunov exponents for various directions.

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