A piecewise constant method for Frobenius-Perron operators via delta function approximations. (English) Zbl 1301.37064
Summary: Let \(S:[0,1]\to[0,1]\) be a nonsingular transformation such that the corresponding Frobenius-Perron operator \(P_{S}:L^{1}(0,1)\to L^{1}(0,1)\) has a stationary density \(f^{\ast}\). We develop a piecewise constant method for the numerical computation of \(f^{\ast}\), based on the approximation of Dirac’s delta function via pulse functions. We show that the numerical scheme out of this new approach is exactly the classic Ulam’s method. Numerical results are given for several one dimensional test mappings.
MSC:
| 37M25 | Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.) |
| 37C30 | Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
Keywords:
Frobenius-Perron operator; stationary density; delta function; pulse function; Ulam’s methodReferences:
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