On a class of transformations which have unique absolutely continuous invariant measures

Authors:
Abraham Boyarsky and Manny Scarowsky

Journal:
Trans. Amer. Math. Soc. **255** (1979), 243-262

MSC:
Primary 28D05; Secondary 60F05

DOI:
https://doi.org/10.1090/S0002-9947-1979-0542879-2

MathSciNet review:
542879

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A class of piecewise transformations from an interval into itself with slopes greater than 1 in absolute value, and having the property that it takes partition points into partition points is shown to have unique absolutely continuous invariant measures. For this class of functions, a central limit theorem holds for all real measurable functions. For the subclass of piecewise linear transformations having a fixed point, it is shown that the unique absolutely continuous invariant measures are piecewise constant.

**[1]**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)****[2]**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)****[3]**J. Guckenheimer, G. Oster and A. Ipaktchi,*The dynamics of density dependent population models*, J. Math. Biol.**4**(1977), 101-147. MR**0441401 (55:14264)****[4]**M. Scarowsky, A. Boyarsky and H. Proppe,*Properties of piecewise linear chaotic functions*, J. Nonlinear Analysis (to appear). MR**557617 (80m:26006)****[5]**S. Wong,*Some metric properties of piecewise monotonic mappings of the unit interval*, Trans. Amer. Math. Soc.**246**(1978), 493-500. MR**515555 (80c:28014)****[6]**T. Y. Li,*Finite approximation for the Frobenius-Perron operator. A solution of Ulam's conjecture*, J. Approximation Theory**17**(1976), 177-186. MR**0412689 (54:811)****[7]**J. B. Moore and S. S. Sengupta,*Existence and control of Markov chains in systems of deterministic motion*, SIAM J. Control**13**(1975), 1103-1114. MR**0423538 (54:11514)****[8]**I. A. Ibragimov and Yu. V. Linnik,*Independent and stationary sequences of random variables*, Wolters-Noordhoff, Groningen, 1971. MR**0322926 (48:1287)****[9]**S. Wong,*A central limit theorem for piecewise monotonic mappings of the unit interval*, (preprint). MR**528327 (81b:60025)****[10]**M. Marcus and H. Minc,*A survey of matrix theory and matrix inequalities*, Allyn and Bacon, Boston, Mass., 1964. MR**0162808 (29:112)****[11]**W. Parry,*Symbolic dynamics and transformations of the unit interval*, Trans. Amer. Math. Soc.**122**(1966), 368-378. MR**0197683 (33:5846)****[12]**R. Bowen,*Bernoulli maps of the interval*, Israel J. Math.**28**(1977), 161-168. MR**0453974 (56:12225)****[13]**R. May,*Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos*, Science**186**(1974), 645-647.**[14]**C. Bowen,*On axiom A diffeomorphisms*, CBMS Regional Conf. Ser. Math., no. 35, Amer. Math. Soc., Providence, R. I., 1978. MR**0482842 (58:2888)****[15]**P. Billingsley,*Convergence of probability measures*, Wiley, New York, 1968. MR**0233396 (38:1718)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
28D05,
60F05

Retrieve articles in all journals with MSC: 28D05, 60F05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0542879-2

Article copyright:
© Copyright 1979
American Mathematical Society