×

zbMATH — the first resource for mathematics

Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps. (English) Zbl 0763.58024
The objects of study in this paper are unimodular interval maps \(T:[0,1]\to[0,1]\), assumed to be \(C^ 3\) and to satisfy the so-called Collet-Eckmann condition \(\liminf_{n\to\infty}(DT^ n(Tc))^{1/n}\), where \(c\) is the assumed unique non-degenerate critical point of \(T\) (of order 1). For such maps and assuming additionally that \(T\) has an invariant probability density, two main aspects are investigated: mixing properties and the distribution of typical periodic orbits.
Using properties of the Perron Frobenius operator (quasi-compactness) as well as analytic properties of the zeta function an exponential mixing property is proven and a description is given of the long term behaviour of generic trajectories.

MSC:
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] [BK] Baladi, V., Keller, G.: Zeta-functions and transfer operators for piecewise monotone transformations. Commun. Math. Phys.127, 459–478 (1990) · Zbl 0703.58048 · doi:10.1007/BF02104498
[2] [BC] Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. Math.133, 73–169 (1991) · Zbl 0724.58042 · doi:10.2307/2944326
[3] [BL1] Blokh, A.M., Lyubich, M.Yu.: Attractors of maps of the interval. Funct. Anal. Appl.21(2), 70–71 (1987) (Russian) · Zbl 0653.58022 · doi:10.1007/BF01078030
[4] [BL2] Blokh, A.M., Lyubich, M.Yu.: Ergodic properties of transformations of an interval. Funct. Anal. Appl.23(1), 59–60 (1989) (Russian) · Zbl 0704.58015 · doi:10.1007/BF01078573
[5] [BL3] Blokh, A.M., Lyubich, M.Yu.: Measurable dynamics of S-unimodal maps of the interval. Preprint, Stony Brook, 1990
[6] [CE] Collet, P., Eckmann, J.-P.: Positive Liapounov exponents and absolute continuity for maps of the interval. Ergodic Theory Dyn. Syst.3, 13–46 (1983) · Zbl 0532.28014 · doi:10.1017/S0143385700001802
[7] [CG] Cox, J.T., Griffeath, D.: Large deviations for Poisson systems of independent random walks. Z. Wahrscheinlichkeitstheorie Verw. Gebiete66, 543–558 (1984) · Zbl 0551.60028 · doi:10.1007/BF00531890
[8] [GJ] Guckenheimer, J., Johnson, S.: Distortion of S-unimodal maps. Ann. Math.132, 73–130 (1990) · Zbl 0708.58007 · doi:10.2307/1971501
[9] [Ha] Haydn, N.T.A.: Meromorphic extension of the zeta function for Axiom A flows. Ergodic Theory Dyn. Syst.10, 347–360 (1990) · Zbl 0694.58035 · doi:10.1017/S0143385700005587
[10] [Ho1] Hofbauer, F.: On intrinsic ergodicity for piecewise monotonic transformations. Ergodic Theory Dyn. Syst.5, 237–256 (1985) · Zbl 0572.54036 · doi:10.1017/S014338570000287X
[11] [Ho2] Hofbauer, F.: Piecewise invertible dynamical systems. Probab. Theoret. Rel. Fields72, 359–386 (1986) · Zbl 0591.60064 · doi:10.1007/BF00334191
[12] [HK1] Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z.180, 119–140 (1982) · Zbl 0485.28016 · doi:10.1007/BF01215004
[13] [HK2] Hofbauer, F., Keller, G.: Zeta-functions and transfer-operators for piecewise linear transformations. J. Reine Angew. Math.352, 100–113 (1984) · Zbl 0533.28011 · doi:10.1515/crll.1984.352.100
[14] [HK3] Hofbauer, F., Keller, G.: Some remarks about recent results on S-unimodal maps, Annales de l’Institut Henri Poincaré, Physique Théorique53, 413–425 (1990)
[15] [Ka] Kato, T.: Perturbation Theory for Linear Operators. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0148.12601
[16] [K1] Keller, G.: Un théorème de la limite centrale pour une classe de transformations monotones par morceaux. C.R. Acad. Sci. Paris, Série A291, 155–158 (1980) · Zbl 0446.60013
[17] [K2] Keller, G.: Markov extensions, zeta-functions, and Fredholm theory for piecewise invertible dynamical systems. Trans. Am. Math. Soc.314, 433–497 (1989) · Zbl 0686.58027 · doi:10.1090/S0002-9947-1989-1005524-4
[18] [K3] Keller, G.: Lifting measures to Markov extensions. Monatsh. Math.108, 183–200 (1989) · Zbl 0712.28008 · doi:10.1007/BF01308670
[19] [K4] Keller, G.: Exponents, attractors, and Hopf decompositions for interval maps, Ergodic Theory Dyn. Syst.10, 717–744 (1990) · Zbl 0715.58020 · doi:10.1017/S0143385700005861
[20] [K5] Keller, G.: On the distribution of periodic orbits for interval maps, in ”Stochastic Modelling in Biology”, Tautu, P. (ed.) pp. 412–419. Singapore: World Scientific 1990
[21] [La] Lalley, S.P.: Distribution of periodic orbits of symbolic and Axiom A flows. Adv. Appl. Math.8, 154–193 (1987) · Zbl 0637.58013 · doi:10.1016/0196-8858(87)90012-1
[22] [LM] Lasota, A., Mackey, M.C.: Probabilistic Properties of Deterministic Systems. Cambridge: Cambridge Univ. Press 1985 · Zbl 0606.58002
[23] [Le] Ledrappier, F.: Some properties of absolutely continuous invariant measures on an interval. Ergodic Theory Dyn. Syst.1, 77–93 (1981) · Zbl 0487.28015 · doi:10.1017/S0143385700001176
[24] [Ma] Martens, M.: Interval Dynamics. Thesis, University of Delft (1990)
[25] [MS] de Melo, W., van Strien, S.: A structure theorem in one-dimensional dynamics. Ann. Math.129, 519–546 (1989) · Zbl 0737.58020 · doi:10.2307/1971516
[26] [Mi] Milnor, J.: On the concept of attractor. Commun. Math. Phys.99, 177–195 (1985) · Zbl 0595.58028 · doi:10.1007/BF01212280
[27] [N1] Nowicki, T.: On some dynamical properties of S-unimodal maps on an interval. Fundamenta Math.126, 27–43 (1985) · Zbl 0608.58030
[28] [N2] Nowicki, T.: Symmetric S-unimodal mappings and positive Liapunov exponents. Ergodic Theory Dyn. Syst.5, 611–616 (1985) · Zbl 0615.28009 · doi:10.1017/S0143385700003199
[29] [N3] Nowicki, T.: A positive Liapunov exponent for the critical value of an S-unimodal mapping implies uniform hyperbolicity. Ergodic Theory Dyn. Syst.8, 425–435 (1988) · Zbl 0638.58021 · doi:10.1017/S0143385700004569
[30] [N4] Nowicki, T.: Some dynamical properties of S-unimodal maps. Preprint (1991) to appear in Fundamenta Math.
[31] [NvS1] Nowicki, T., van Strien, S.: Hyperbolicity properties of C2 multimodal Collet-Eckmann maps without Schwarzian derivative assumptions. Trans. Am. Math. Soc.321, 793–810 (1990) · Zbl 0731.58021 · doi:10.2307/2001586
[32] [NvS2] Nowicki, T., van Strien, S.: Invariant measures exist under a summability condition for unimodal maps. Inv. Math.105, 123–136 (1991) · Zbl 0736.58030 · doi:10.1007/BF01232258
[33] [PP] Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque187–188 (1990) · Zbl 0726.58003
[34] [PS] Plachky, D., Steinebach, J.: A theorem about probabilities of large deviations with an application to queuing theory. Period. Math. Hungar.6, 343–345 (1975) · Zbl 0322.60027 · doi:10.1007/BF02017929
[35] [PUZ] Przytycki, F., Urbański, M., Zdunik, A.: Harmonic, Gibbs and Hausdorff measures on repellors for holomorphic maps I. Ann. Math.130, 1–40 (1989) · Zbl 0703.58036 · doi:10.2307/1971475
[36] [Rou] Rousseau-Egele, J.: Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux. Ann. Probab.11, 772–788 (1983) · Zbl 0518.60033 · doi:10.1214/aop/1176993522
[37] [Rud] Rudin, W.: Functional Analysis. New York: McGraw-Hill 1973 · Zbl 0253.46001
[38] [Rue] Ruelle, D.: Zeta-functions for expanding maps and Anosov-flows. Inv. Math.34, 231–242 (1976) · Zbl 0329.58014 · doi:10.1007/BF01403069
[39] [Ry] Rychlik, M.: Bounded variation and invariant measures. Studia Math.LXXVI, 69–80 (1983) · Zbl 0575.28011
[40] [vS] van Strien, S.: On the creation of horseshoes. Lecture Notes in Math. vol.898, pp. 316–351. Berlin, Heidelberg, New York: Springer 1981
[41] [Sz] Szewc, B.: Perron-Frobenius operator in spaces of smooth functions on an interval. Ergodic Theory Dyn. Syst.4, 613–641 (1984) · Zbl 0552.58019 · doi:10.1017/S0143385700002686
[42] [Yo] Young, L.S.: Decay of correlations for certain quadratic maps. Preprint (1991)
[43] [Zi1] Ziemian, K.: Almost sure invariance principle for some maps of an interval. Ergodic Theory Dyn. Syst.5, 625–640 (1985) · Zbl 0604.60031 · doi:10.1017/S0143385700003217
[44] [Zi2] Ziemian, K.: Refinement of the Shannon-McMillan-Breiman Theorem for some maps of an interval. Studia Math.XCIII, 271–285 (1989) · Zbl 0691.28006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.