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Semidefinite Characterisation of Invariant Measures for One-Dimensional Discrete Dynamical Systems

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Executive Summary

Using recent results on measure theory and algebraic geometry, the authors show how semi-definite programming can be used to construct invariant measures of one-dimensional discrete dynamical systems (iterated maps on a real interval). In particular they show that both discrete measures (corresponding to finite cycles) and continuous measures (corresponding to chaotic behavior) can be recovered using standard software, paving the way for a numerical study of probabilistic properties of dynamical systems. In this paper, they show that recent results mixing measure theory and algebraic geometry can be used to construct invariant measures numerically, with the help of semi-definite programming, as called optimization over Linear Matrix Inequalities (LMIs), a versatile generalization of linear programming to the cone of positive semi-definite matrices.

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