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The topological entropy of iterated piecewise affine maps is uncomputable

Abstract : 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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Pascal Koiran. The topological entropy of iterated piecewise affine maps is uncomputable. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2001, Vol. 4 no. 2 (2), pp.351-356. ⟨10.46298/dmtcs.292⟩. ⟨hal-00958966⟩

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