ArticleOriginal scientific text
Title
The asymptotics of the Perron-Frobenius operator of a class of interval maps preserving infinite measures
Authors 1
Affiliations
- Institute of Mathematics, University of Salzburg, Hellbrunnerstr. 34 , A-5020 Salzburg, Austria
Abstract
We determine the asymptotic behaviour of the iterates of the Perron-Frobenius operator for specific interval maps with an indifferent fixed point which gives rise to an infinite invariant measure.
Bibliography
- [A0] J. Aaronson, An Introduction to Infinite Ergodic Theory, Amer. Math. Soc., 1997.
- [A1] J. Aaronson, The asymptotic distributional behaviour of transformations preserving infinite measures, J. Anal. Math. 39 (1981), 203-234.
- [A2] J. Aaronson, Random f-expansions, Ann. Probab. 14 (1986), 1037-1057.
- [BG] A. Boyarsky and P. Góra, Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension, Birkhäuser, 1997.
- [Fe] W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, Wiley, 1971.
- [Fr] N. A. Friedman, Mixing transformations in an infinite measure space, in: Studies in Probability and Ergodic Theory, Adv. Math. Suppl. Stud. 2, Academic Press, 1978, 167-184.
- [HK] A. B. Hajian and S. Kakutani, Weakly wandering sets and invariant measures, Trans. Amer. Math. Soc. 110 (1964), 136-151.
- [H] E. Hopf, Ergodentheorie, Springer, 1937.
- [LY] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488.
- [LM] A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, Springer, 1994.
- [Kr] K. Krickeberg, Strong mixing properties of Markov chains with infinite invariant measure, in: Proc. Fifth Berkeley Sympos. Math. Statist. Probab. 2, Part II, Univ. of California Press, 1965, 431-445.
- [M] P. Manneville, Intermittency, self-similarity and 1/f spectrum in dissipative dynamical systems, J. Phys. 41 (1980), 1235-1243.
- [P] F. Papangelou, Strong ratio limits, R-recurrence and mixing properties of discrete parameter Markov processes, Z. Wahrsch. Verw. Gebiete 8 (1967), 259-297.
- [R] M. Rychlik, Bounded variation and invariant measures, Studia Math. 76 (1983), 69-80.
- [Sch1] F. Schweiger, Invariant measures for piecewise linear fractional maps, J. Austral. Math. Soc. Ser. A 34 (1983), 55-59.
- [Sch2] F. Schweiger, Ergodic Theory of Fibered Systems and Metric Number Theory, Clarendon Press, 1995.
- [T1] M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math. 37 (1980), 303-314.
- [T2] M. Thaler, Transformations on [0,1] with infinite invariant measures, ibid. 46 (1983), 67-96.
- [T3] M. Thaler, The iteration of the Perron-Frobenius operator when the invariant measure is infinite: An example, manuscript, Salzburg, 1986.
- [T4] M. Thaler, Arc-sine limit laws for a one-parameter family of f-expansions, manuscript, Salzburg, 1993.
- [T5] M. Thaler, A limit theorem for the Perron-Frobenius operator of transformations on [0,1] with indifferent fixed points, Israel J. Math. 91 (1995), 111-127.
- [T6] M. Thaler, The invariant densities for maps modeling intermittency, J. Statist. Phys. 79 (1995), 739-741.
- [T7] M. Thaler, The Dynkin-Lamperti arc-sine laws for measure preserving transformations, Trans. Amer. Math. Soc. 350 (1998), 4593-4607.
- [TR] M. Thaler and C. Reichsöllner, Arc sine type limit laws for interval mappings, manuscript, Salzburg, 1986.
- [Z1] R. Zweimüller, Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points, Nonlinearity 11 (1998), 1263-1276.
- [Z2] R. Zweimüller, Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points, Ergodic Theory Dynam. Systems 20 (2000), 1519-1549.