Distortion inequality for the Frobenius-Perron operator and some of its consequences in ergodic theory of Markov maps in $ℝ^d$

Piotr Bugiel

Annales Polonici Mathematici, Tome 69 (1998), p. 125-157 / Harvested from The Polish Digital Mathematics Library

Résumé

Asymptotic properties of the sequences (a) ${P^j_φ g}_{j=1}^{∞}$ and (b) ${j^{-1} ∑_{i=0}^{j-1} Pⁱ_φ g}_{j=1}^{∞}$, where $P_φ:L¹ → L¹$ is the Frobenius-Perron operator associated with a nonsingular Markov map defined on a σ-finite measure space, are studied for g ∈ G = f ∈ L¹: f ≥ 0 and ⃦f ⃦ = 1. An operator-theoretic analogue of Rényi’s Condition is introduced. It is proved that under some additional assumptions this condition implies the L¹-convergence of the sequences (a) and (b) to a unique g₀ ∈ G. The general result is applied to some smooth Markov maps in $ℝ^d$. Also the Bernoulli property is proved for a class of smooth Markov maps in $ℝ^d$.

Détail

Publié le : 1998-01-01

Zbl 0902.28012

EUDML-ID : urn:eudml:doc:270740

Bibliographie

[000] [A75] R. L. Adler, Continued fractions and Bernoulli trials, Lecture notes, Ergodic Theory, J. Moser, E. Phillips, and S. Varadhan (eds.), Courant Inst. Math. Sci., New York, 1975.

[001] [A79] R. L. Adler, Afterword (to [Bow79]), Comm. Math. Phys. 69 (1979), 15-17.

[002] [A91] R. L. Adler, Geodesic flows, interval maps, and symbolic dynamics, in: Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, T. Bedford, M. Keane and C. Series (eds.), Oxford Univ. Press, Oxford, 1991, 93-123. | Zbl 0752.58007

[003] [ADU93] J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), 495-548. | Zbl 0789.28010

[004] [AF91] R. L. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc. 25 (1991), 229-334. | Zbl 0802.58037

[005] [Bow79] R. Bowen, Invariant measures for Markov maps of the interval, Comm. Math. Phys. 69 (1979), 1-14.

[006] [BowS79] R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 401-418.

[007] [Br94] H. Bruin, Topological conditions for the existence of invariant measures for unimodal maps, Ergodic Theory Dynam. Systems 14 (1994), 433-451. | Zbl 0812.58052

[008] [Br94a] H. Bruin, Invariant measures of interval maps, Thesis, Technische Universiteit Delft 1994.

[009] [Bu81] P. Bugiel, Notes on invariant measures for Markov maps of the interval, preprint, Jagiellonian University, 1981.

[010] [Bu82] P. Bugiel, Approximation for the measures of ergodic transformations of the real line, Z. Wahrsch. Verw. Gebiete 59 (1982), 27-38. | Zbl 0476.28010

[011] [Bu82a] P. Bugiel, Ergodic properties of maps of Markov type (in ℝ¹), Thesis, Jagiellonian University, Kraków, 1982.

[012] [Bu85] P. Bugiel, A note on invariant measures for Markov maps of an interval, Z. Wahrsch. Verw. Gebiete 70 (1985), 345-349. | Zbl 0606.28013

[013] [Bu85a] P. Bugiel, On a class of piecewise monotonic transformations admitting smooth invariant measures, Prace Inf. Z.1, Zesz. Nauk. UJ, 726 (1985), 49-60.

[014] [Bu86] P. Bugiel, On the number of all absolutely continuous, ergodic measures of Markov type transformations defined on an interval, Math. Nachr. 129 (1986), 261-268. | Zbl 0622.28014

[015] [Bu87] P. Bugiel, Correction and addendum to 'A note on invariant measures for Markov maps of an interval.' Z. Wahrsch. Verw. Gebiete 70, 345-349 (1985), Probab. Theory Related Fields 76 (1987), 255-256. | Zbl 0606.28013

[016] [Bu91] P. Bugiel, On the convergence of iterates of the Frobenius-Perron operator associated with a Markov map defined on an interval. The lower-function approach, Ann. Polon. Math. 53 (1991), 131-137. | Zbl 0731.47031

[017] [Bu91a] P. Bugiel, Ergodic properties of Markov maps in $ℝ^d$, Probab. Theory Related Fields 88 (1991), 483-496. | Zbl 0723.60089

[018] [Bu92] P. Bugiel, Invariant measures of C²-diffeomorphisms of Markov type in $ℝ^d$, Monatsh. Math. 113 (1992), 255-263. | Zbl 0765.58014

[019] [Bu93] P. Bugiel, On a Bernoulli property of some piecewise C²-diffeomorphisms in $ℝ^d$, Monatsh. Math. 116 (1993), 99-110. | Zbl 0792.58024

[020] [Bu96] P. Bugiel, Application of the Ulam’s procedure for approximating invariant measures to the case of $C^{1+α}$-smooth Markov maps in $ℝ^d$, Univ. Iagell. Acta Math. 34 (1997), 7-27. | Zbl 0947.37011

[021] [DS63] N. Dunford and J. T. Schwartz, Linear Operators, Part I, General Theory, Wiley, New York, 1963.

[022] [FO70] N. Friedman and D. Ornstein, On isomorphism of weak Bernoulli transformations, Adv. Math. 5 (1970), 365-394. | Zbl 0203.05801

[023] [G69] G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Transl. Math. Monographs 26, Amer. Math. Soc., 1969.

[024] [H77] M. Halfant, Analytic properties of Rényi's invariant density, Israel J. Math. 27 (1977), 1-20. | Zbl 0363.28013

[025] [IMT71] S. Ito, H. Murata and H. Totoki, Remarks on the isomorphism theorems for weak Bernoulli transformations in general case, Publ. Res. Inst. Math. Sci. 7 (1971/72), 541-580. | Zbl 0246.28012

[026] [IR78] I. A. Ibragimov and Ya. A. Rozanov, Gaussian Random Processes, Springer, Berlin, 1978.

[027] [I87] M. Iosifescu, Mixing properties for f-expansions, in: Probab. Theory and Math. Statist., Vol. II, Yu. V. Prohorov et al. (eds.), VNU Science Press, 1987, 1-8.

[028] [IG91] M. Iosifescu and S. Grigorescu, Dependence with Complete Connections and its Applications, Cambridge Tracts in Math. 96, Cambridge Univ. Press, Cambridge, 1990.

[029] [JGB94] M. Jabłoński, P. Góra and A. Boyarsky, A general existence theorem for absolutely continuous invariant measures on bounded and unbounded intervals, preprint, 1994, to appear. | Zbl 0895.28005

[030] [K90] G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory Dynam. Systems 10 (1990), 717-744. | Zbl 0715.58020

[031] [LY82] A. Lasota and J. Yorke, Exact dynamical systems and the Frobenius-Perron operator, Trans. Amer. Math. Soc. 273 (1982), 375-384. | Zbl 0524.28021

[032] [Li71] M. Lin, Mixing for Markov operators, Z. Wahrsch. Verw. Gebiete 16 (1971), 231-242. | Zbl 0212.49301

[033] [M71] G. Maruyama, Some aspects of Ornstein's theory of isomorphism problems in ergodic theory, Publ. Res. Inst. Math. Sci. 7 (1971/72), 511-539.

[034] [Ma87] R. Ma né, Ergodic Theory and Differentiable Dynamics, Ergeb. Math. Kuznzgeb. 8, Springer, Berlin, 1987.

[035] [MS93] W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergeb. Math. Grenzgeb. 25, Springer, Berlin, 1993.

[036] [P80] G. Pianigiani, First return map and invariant measures, Israel J. Math. 35 (1980), 32-47. | Zbl 0445.28016

[037] [R56] O. Rechard, Invariant measures for many-one transformations, Duke Math. J. 23 (1956), 477-488. | Zbl 0070.28001

[038] [Re57] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493. | Zbl 0079.08901

[039] [Ro61] V. A. Rokhlin, Exact endomorphisms of Lebesgue spaces, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 490-530 (in Russian); English transl.: Amer. Math. Soc. Transl. (2) 39 (1964), 1-36.

[040] [Ry83] M. R. Rychlik, Bounded variation and invariant measures, Studia Math. 76 (1983), 69-80. | Zbl 0575.28011

[041] [Sch89] F. Schweiger, Ergodic theory of fibred systems, Institut für Mathematik der Universität Salzburg, Salzburg, 1989.

[042] [Se79] C. Series, Additional comments (to [Bow79]), Comm. Math. Phys. 69 (1979), 17.

[043] [Sz84] W. Szlenk, An Introduction to the Theory of Smooth Dynamical Systems, PWN, Warszawa, 1984.

[044] [U60] S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Appl. Math., Interscience, New York, 1960.