Finite approximations of Frobenius-Perron operators. A solution of Ulam’s conjecture to multi-dimensional transformations. (English) Zbl 0890.58035
Summary: We prove that Ulam’s piecewise constant approximation algorithm is convergent for computing an absolutely continuous invariant measure associated with a piecewise \(C^2\) expanding transformation or a Jablonski transformation \(S : [0,1]^N \subset \mathbb{R}^N\rightarrow[0,1]^N\). This solves an extension of Ulam’s conjecture to multi-dimensions and generalizes the convergence result given by T.-Y. Li for one-dimensional transformations.
MSC:
| 37A99 | Ergodic theory |
| 28D05 | Measure-preserving transformations |
| 37E99 | Low-dimensional dynamical systems |
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