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Approximating the invariant densities of transformations with infinitely many pieces on the interval

Authors: P. Góra and A. Boyarsky
Journal: Proc. Amer. Math. Soc. 105 (1989), 922-928
MSC: Primary 58F11
MathSciNet review: 953006
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Abstract: Let $ I = [0,1]$ and $ \tau :I \to I$ be a piecewise continuous, expanding transformation with infinitely many pieces of monotonicity. We construct a sequence of transformations $ \left\{ {{\tau _n}} \right\}$, each having a finite partition, such that their invariant densities converge in $ {L_1}$ to the invariant density of $ \tau $.

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  • [1] M. Rychlik, Bounded variation and invariant measures, Studia Math., LXXVI (1983), 69-80. MR 728198 (85h:28019)
  • [2] G. Pianigiani, First return map and invariant measures, Israel J. Math. 35 (1980), 32-48. MR 576460 (82a:58031)
  • [3] G. Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math. 94 (1982), 313-333. MR 685377 (84k:58130)
  • [4] P. Góra and A. Boyarsky, Compactness and invariant densities for families of expanding, piecewise monotonic transformations, preprint.
  • [5] T-Y Li, Finite approximation for the Frobenius-Perron operator. A solution to Ulam 's conjecture, J. Approx. Theory 17 (1976), 177-186. MR 0412689 (54:811)
  • [6] A. Boyarsky, Approximating the $ \sigma $-finite measure invariant under a non-expanding map, J. Math. Anal. Appl. 78 (1980), 222-232. MR 595778 (82b:28030)
  • [7] P. Góra and A. Boyarsky, Why computers like Lebesgue measure, Applic. 16, No. 4, 321-329, 1988. Comput. Math. MR 959419 (89m:58107)
  • [8] A. Lasota and J. A. Yorke,, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488. MR 0335758 (49:538)

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