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Shrinking targets in parametrised families

Published online by Cambridge University Press:  28 November 2017

MAGNUS ASPENBERG  and
TOMAS PERSSON
MAGNUS ASPENBERG
Affiliation:
Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden. e-mail: magnusa@maths.lth.se, tomasp@maths.lth.se
TOMAS PERSSON
Affiliation:
Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden. e-mail: magnusa@maths.lth.se, tomasp@maths.lth.se
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Abstract

We consider certain parametrised families of piecewise expanding maps on the interval, and estimate and sometimes calculate the Hausdorff dimension of the set of parameters for which the orbit of a fixed point has a certain shrinking target property. This generalises several similar results for β-transformations to more general non-linear families. The proofs are based on a result by Schnellmann on typicality in parametrised families.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 2 , March 2019 , pp. 265 - 295
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Aspenberg, M. Rational Misiurewicz maps are rare. Comm. Math. Phys. 291 (2009), 645–658.Google Scholar
[2] Aspenberg, M. The Collet–Eckmann condition for rational functions on the Riemann sphere. Math. Z. 273 (2013), 935–980.Google Scholar
[3] Benedicks, M. and Carleson, L. On iterations of 1-ax 2 on (-1,1). Ann. Math. (2) 122 (1985), no. 1, 1–25.Google Scholar
[4] Benedicks, M. and Carleson, L. On the dynamics of the Hénon map. Ann. Math. (2) 133 (1991), no. 1, 73–169.Google Scholar
[5] Bugeaud, Y., Liao, L. Uniform Diophantine approximation related to b-ary and β-expansions. Ergodic Theory Dynam. Syst. 36 (2016), no. 1, 1–22.Google Scholar
[6] Bugeaud, Y. and Wang, B. Distribution of full cylinders and the Diophantine properties of the orbits in β-expansions. J. Fract. Geom. 1 (2014), no. 2, 221–241.Google Scholar
[7] Carleson, L. and Gamelin, T.. Complex Dynamics (springer, New York, 1993).Google Scholar
[8] Fan, A.–H., Schmeling, J. and Troubetzkoy, S.. A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation. Proc. London Math. Soc. 107 (2013), 1173–1219.Google Scholar
[9] Ge, Y. and Lü, F. A note on inhomogeneous Diophantine approximation in beta-dynamical system. Bull. Austral. Math. Soc. 91 (2015), no. 1, 34–40.Google Scholar
[10] Góra, P. Invariant densities for generalized β-maps. Ergodic Theory Dynam. Syst. 27 (2007), 1583–1598.Google Scholar
[11] Hill, R. and Velani, S. The ergodic theory of shrinking targets. Inven. Math. 119 (1995), no. 1, 175–198.Google Scholar
[12] Hofbauer, F. Local dimension for piecewise monotonic maps on the interval. Ergodic Theory Dynam. Syst. 15 (1995), no. 6, 1119–1142.Google Scholar
[13] Ito, S. and Sadahiro, T. Beta-expansions with negative bases. Integers 9 (2009), 239–259.Google Scholar
[14] Ledrappier, F. Some properties of absolutely continuous invariant measures on an interval. Ergodic Theory Dynam. Syst. 1 (1981), no. 1, 77–93.Google Scholar
[15] Li, B., Persson, T., Wang, B. and Wu, J.. Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions. Math. Z. volume 276, Issue 3–4, (2014), 799–827.Google Scholar
[16] Liao, L. and Seuret, S. Diophantine approximation by orbits of expanding Markov maps. Ergodic Theory Dynam. Syst. 33 (2013), no. 2, 585–608.Google Scholar
[17] Liao, L. and Steiner, W. Dynamical properties of the negative beta-transformation. Ergodic Theory Dynam. Syst. 32 (2012), no. 5, 1673–1690.Google Scholar
[18] Liverani, C. Decay of correlations for piecewise expanding maps. J. Statis. Phys. 78 (1995), no. 3/4, 1111–1129.Google Scholar
[19] Lü, F. and Wu, J. Diophantine analysis in beta-dynamical systems and Hausdorff dimensions. Adv. Math. 290 (2016), 919–937.Google Scholar
[20] Persson, T. Typical points and families of expanding interval mappings. Discrete Contin. Dynam. Syst. 37 (2017), no. 7, 4019–4034.Google Scholar
[21] Persson, T. and Rams, M. On Shrinking Targets for Piecewise Expanding Interval Maps. Ergodic Theory Dynam. Syst. 37 (2017), no. 2, 646–663.Google Scholar
[22] Persson, T. and Schmeling, J. Dyadic Diophantine Approximation and Katok's Horseshoe Approximation. Acta Arithmetica 132 (2008), 205–230.Google Scholar
[23] Schnellmann, D. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete Contin. Dynam. Syst. 31 (2011), no. 3, 877–911.Google Scholar
[24] Wong, S. Some metric properties of piecewise monotonic mappings of the unit interval. Trans. Amer. Math. Soc. 246 (1978), 493–500.Google Scholar