Symmetric S-unimodal mappings and positive Liapunov exponents

Tomasz Nowicki

doi:10.1017/S0143385700003199

Abstract

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Symmetric S-unimodal functions with positive Liapunov exponent of the critical value have an invariant measure absolutely continuous with respect to Lebesgue measure.

References

REFERENCES

[1]Collet, P.. Ergodic properties of some unimodal mappings of the interval. Preprint, Institut Mittag-Leffler, Report no. 11, 1984.CrossRef Google Scholar

[2]Collet, P. & Eckmann, J.-P.. Interated Maps on the Interval as Dynamical Systems. Birkhauser: Basel, Boston, Stuttgart, 1980.Google Scholar

[3]Collet, P. & Eckmann, J.-P.. Positive Liapunov exponents and absolute continuity for maps of the interval. Ergod. Th. & Dynam. Sys. 3 (1983), 13–46.CrossRef Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Nowicki, T. and van Strien, S. 1988. Absolutely continuous invariant measures forC 2 unimodal maps satisfying the Collet-Eckmann conditions. Inventiones mathematicae, Vol. 93, Issue. 3, p. 619.

Nowicki, Tomasz 1988. A positive Liapunov exponent for the critical value of an S-unimodal mapping implies uniform hyperbolicity. Ergodic Theory and Dynamical Systems, Vol. 8, Issue. 3, p. 425.

Nowicki, T. and Przytycki, F. 1989. The conjugacy of Collet–Eckmann's map of the interval with the tent map is Holder continuous. Ergodic Theory and Dynamical Systems, Vol. 9, Issue. 2, p. 379.

Góra, P. and Boyarsky, A. 1990. Absolutely continuous invariant measures for the family of maps x → rxe-bxwith application to the Belousov-Zhabotinski reaction. Dynamics and Stability of Systems, Vol. 5, Issue. 2, p. 65.

Cutler, C. D. 1990. A dynamical system with integer information dimension and fractal correlation exponent. Communications in Mathematical Physics, Vol. 129, Issue. 3, p. 621.

van Strien, Sebastian 1990. Hyperbolicity and invariant measures for generalC 2 interval maps satisfying the Misiurewicz condition. Communications in Mathematical Physics, Vol. 128, Issue. 3, p. 437.

Nowicki, Tomasz and van Strien, Sebastian 1990. Hyperbolicity properties of 𝐶² multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions. Transactions of the American Mathematical Society, Vol. 321, Issue. 2, p. 793.

Hofbauer, Franz and Keller, Gerhard 1990. Quadratic maps without asymptotic measure. Communications in Mathematical Physics, Vol. 127, Issue. 2, p. 319.

Nowicki, Tomasz and van Strien, Sebastian 1991. Invariant measures exist under a summability condition for unimodal maps. Inventiones mathematicae, Vol. 105, Issue. 1, p. 123.

Keller, Gerhard and Nowicki, Tomasz 1992. Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps. Communications in Mathematical Physics, Vol. 149, Issue. 1, p. 31.

Benedicks, Michael and Young, Lai-Sang 1992. Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps. Ergodic Theory and Dynamical Systems, Vol. 12, Issue. 1, p. 13.

Young, L. -S. 1992. Decay of correlations for certain quadratic maps. Communications in Mathematical Physics, Vol. 146, Issue. 1, p. 123.

Guckenheimer, John and Johnson, Stewart 1993. From Topology to Computation: Proceedings of the Smalefest. p. 227.

Jakobson, M. and Świạtek, G. 1995. Metric properties of non-renormalizable S-unimodal maps: II. Quasisymmetric conjugacy classes. Ergodic Theory and Dynamical Systems, Vol. 15, Issue. 5, p. 871.

Lanyu, Wang 2001. Topological and metrical conditions for Collet-Eckmann unimodal maps. Acta Mathematicae Applicatae Sinica, Vol. 17, Issue. 3, p. 350.

Bruin, Henk and van Strien, Sebastian 2003. Expansion of derivatives in one-dimensional dynamics. Israel Journal of Mathematics, Vol. 137, Issue. 1, p. 223.

LI, MING-CHIA and MALKIN, MIKHAIL 2003. SMOOTH SYMMETRIC AND LORENZ MODELS FOR UNIMODAL MAPS. International Journal of Bifurcation and Chaos, Vol. 13, Issue. 11, p. 3353.

MOREIRA FREITAS, ANA CRISTINA and FREITAS, JORGE MILHAZES 2008. Extreme values for Benedicks–Carleson quadratic maps. Ergodic Theory and Dynamical Systems, Vol. 28, Issue. 4, p. 1117.

Gao, Rui and Shen, Weixiao 2014. Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps. Discrete & Continuous Dynamical Systems - A, Vol. 34, Issue. 5, p. 2013.

Li, Huaibin 2017. Topological invariance of the Collet–Eckmann condition for one-dimensional maps. Nonlinearity, Vol. 30, Issue. 5, p. 2010.

Article contents

Symmetric S-unimodal mappings and positive Liapunov exponents

Abstract

References

REFERENCES

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