Graph-theoretic bound on number of A.C.I.M. for random transformation

Author:
Paweł Góra

Journal:
Proc. Amer. Math. Soc. **116** (1992), 401-410

MSC:
Primary 28D05

DOI:
https://doi.org/10.1090/S0002-9939-1992-1094502-0

MathSciNet review:
1094502

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Abstract: Upper and lower bounds are given for the number of ergodic absolutely continuous invariant measures for a general random map. They are based on a communication graph describing the interaction of deterministic transformations used to define the random one. In particular, the case of piecewise expanding transformations is discussed.

**[1]**A. Boyarsky and W. Byers,*A graph-theoretic bound on the number of independent absolutely continuous invariant measures*, J. Math. Anal. Appl.**139**(1989), 139-151. MR**991931 (91i:58078)****[2]**P. Collet and J.-P. Eckmann,*Measures invariant under mappings of the unit interval*, Regular and Chaotic Motions in Dynamic Systems, NATO ASI Series (G. Velo and A.S. Wightman eds.), Plenum Press, New York and London, 1983, pp. 233-265.**[3]**P. Góra and A. Boyarsky,*Absolutely continuous invariant measures for piecewise expanding**transformations in*, Israel J. Math.**67**(1989), 272-286. MR**1029902 (91c:58061)****[4]**P. Góra, A. Boyarsky, and H. Proppe,*On the number of invariant measures for higher dimensional chaotic transformations*, J. Statist. Phys.**62**(1991), 1000-1017. MR**1105279 (92b:58127)****[5]**A. Lasota and J. Yorke,*On the existence of invariant measures for piecewise monotonic transformations*, Trans. Amer. Math. Soc.**186**(1973), 481-486. MR**0335758 (49:538)****[6]**T.-Y. Li and J. A. Yorke,*Ergodic transformations from an interval into itself*, Trans. Amer. Math. Soc.**235**(1978), 183-192. MR**0457679 (56:15883)****[7]**You-Shi Lou,*The existence of absolutely continuous invariant measures for**Jabloński transformations in*, Proc. Amer. Math. Soc.**109**(1990), 105-112. MR**1004420 (90h:58053)****[8]**S. Pelikan,*Invariant densities for random maps of interval*, Trans. Amer. Math. Soc.**281**(1984), 813-825. MR**722776 (85i:58070)**

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1094502-0

Article copyright:
© Copyright 1992
American Mathematical Society