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.
Highlights
There is an active line of research in which dynamical systems are studied from an effective point of view
We use a reduction from the nilpotency problem, albeit of piecewise-affine maps instead of cellular automata
We show that there exists a dimension n and nilpotent saturated linear maps f : Rn → Rn such that f k is not identically zero, where k is an arbitrarily large integer. This clarifies an observation of [1]. It is still unknown whether there exists n such that the nilpotency problem for saturated linear functions in dimension n is undecidable
Summary
There is an active line of research in which dynamical systems are studied from an effective point of view. We show that there exists a dimension n and nilpotent saturated linear maps f : Rn → Rn such that f k (the k-th iterate of f ) is not identically zero, where k is an arbitrarily large integer This clarifies an observation of [1]. It is still unknown whether there exists n such that the nilpotency problem for saturated linear functions in dimension n is undecidable We conclude this introduction with an open problem: is the topological entropy of a piecewise-affine map always a computable real number? One can ask whether every computable real number can be obtained as the topological entropy of an iterated piecewise-affine map These two questions make sense for cellular automata, and it seems that they have not been addressed so far
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