Discrete Mathematics & Theoretical Computer Science, Vol 4, No 2 (2001)

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The topological entropy of iterated piecewise affine maps is uncomputable

Pascal Koiran


We show that it is impossible to compute (or even to approximate) the topological entropy of a continuous piecewise affine function in dimension four. The same result holds for saturated linear functions in unbounded dimension. We ask whether the topological entropy of a piecewise affine function is always a computable real number, and conversely whether every non-negative computable real number can be obtained as the topological entropy of a piecewise affine function. It seems that these two questions are also open for cellular automata.

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