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Badly approximable and nonrecurrent sets for expanding Markov maps. (English) Zbl 1489.37020

Summary: We consider the asymptotic behaviors of the orbits of an expanding Markov system \(([0,1], f)\), and prove that the badly approximable set \[ \{x\in[0, 1): \liminf_{n \to \infty}|f^n(x)- y_n|>0\}, \] is of full Hausdorff dimension for any given sequence \(\{y_n\}_{n \geq 0}\subset[0, 1]\). Consequently, the Hansdorff dimension of the set of nonrecurrent points in the sense that \(\{x\in[0, 1]: \liminf_{n \to \infty}|f^n(x)-x|>0\}\) is also full. The results can be applied to \(\beta\)-transformations, Gauss maps and Lüroth maps, etc.

MSC:

37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37E05 Dynamical systems involving maps of the interval
37H12 Random iteration
37B35 Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
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[1] Furstenberg, H., Recurrence in Ergodic Theory and Combinatorial Number Theory (Princeton University Press, 1981). · Zbl 0459.28023
[2] Urbański, M., The Hausdorff dimension of the set of points with nondense orbit under a hyperdic dynamical system, Nonlinearity4(2) (1991) 385-397. · Zbl 0725.58031
[3] Li, B. and Chen, Y.-C., Chaotic and topological properties of \(\beta \)-transformations, J. Math. Anal. Appl.383 (2011) 585-596. · Zbl 1223.37017
[4] Tseng, J., Schmidt games and Markov paititions, Nonlinearity22(3) (2009) 525-543. · Zbl 1167.37018
[5] Färm, D., Persson, T. and Schmeling, J., Dimension of countable intersections of some sets arising in expansions in non-integer bases, Fund. Math.209(2) (2010) 157-176. · Zbl 1211.37047
[6] Hu, H. and Yu, Y.-L., On Schmidt’s game and the set of points with non-dense orbits under a class of expanding maps, J. Math. Ana. Appl.418(2) (2014) 906-920. · Zbl 1345.37038
[7] Yang, Q.-Q., Li, B., Liu, W.-B. and Chen, Y.-C., On the distal and asymptotic sets for \(\beta \)-transformations, J. Math. Anal. Appl.464 (2018) 188-200. · Zbl 1386.37011
[8] Liu, W.-B. and Wang, S.-L., On the distal and asymptotic sets for continued fractions, Fractals27(8) (2019) 1-10. · Zbl 1434.11154
[9] Yang, Q.-Q. and Wang, S.-L., Metric and dimensional properties of the badly approximable set for beta-transformations, Fractals27(3) (2019) 1-9. · Zbl 1433.28034
[10] Maňé, R., Ergodic Theory and Differentiable Dynamics (Springer-Verlag1987). · Zbl 0616.28007
[11] Falconer, K. J., Fractal Geometry: Mathematical Foundations and Applications (John Wiley and Sons, Chichester, 1990). · Zbl 0689.28003
[12] Hanus, P., Mauldin, R. D. and Urbański, M., Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems, Acta Math. Hungar.96 (2002) 27-98. · Zbl 1012.28007
[13] Mauldin, R. D. and Urbański, M., Dimensions and Measures in Infinite Iterated Function Systems, Proc. London Math. Soc.73(3) (2016) 105-154. · Zbl 0852.28005
[14] Renyi, A., Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar.8 (1957) 477-493. · Zbl 0079.08901
[15] Li, Y.-Q. and Li, B., Distributions of full and non-full words in beta-expansions, J. Number Theory190 (2018) 311-332. · Zbl 1442.11115
[16] Schmeling, J., Symbolic dynamics for \(\beta \)-shifts and self-normal numbers, Ergodic Theory Dynam. Syst.17(3) (1997) 675-694. · Zbl 0908.58017
[17] Li, B. and Wu, J., Beta-expansion and continued fraction expansion, J. Math. Anal. Appl.339(2) (2008) 1322-1331. · Zbl 1137.11053
[18] Fan, A.-H. and Wang, B.-W., On the lengths of basic intervals in beta expansions, Nonlinearity25(5) (2012) 1329-1343. · Zbl 1256.11044
[19] Tan, B. and Wang, B.-W., Quantitative recurrence properties for beta-dynamical system, Adv. Math.228 (2011) 2071-2097. · Zbl 1284.11113
[20] Ban, J.-C. and Li, B., The multifractal spectra for the recurrence rates of beta-transformations, J. Math. Anal. Appl.420 (2014) 1662-1679. · Zbl 1347.37031
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