# American Institute of Mathematical Sciences

February 2019, 39(2): 995-1017. doi: 10.3934/dcds.2019041

## Topological entropy of free semigroup actions for noncompact sets

 1 School of Mathematics, South China University of Technology, Guangzhou 510641, China 2 School of Computer Science and Engineering, South China University of Technology, Guangzhou 510641, China

* Corresponding author: Dongkui Ma

Received  March 2018 Revised  August 2018 Published  November 2018

Fund Project: Project supported by National Natural Science Foundation of China grant no.11771149, 11671149

In this paper we introduce the topological entropy and lower and upper capacity topological entropies of a free semigroup action, which extends the notion of the topological entropy of a free semigroup action defined by Bufetov [10], by using the Carathéodory-Pesin structure (C-P structure). We provide some properties of these notions and give three main results. The first is the relationship between the upper capacity topological entropy of a skew-product transformation and the upper capacity topological entropy of a free semigroup action with respect to arbitrary subset. The second are a lower and a upper estimations of the topological entropy of a free semigroup action by local entropies. The third is that for any free semigroup action with $m$ generators of Lipschitz maps, topological entropy for any subset is upper bounded by the Hausdorff dimension of the subset multiplied by the maximum logarithm of the Lipschitz constants. The results of this paper generalize results of Bufetov [10], Ma et al. [26], and Misiurewicz [27].

Citation: Yujun Ju, Dongkui Ma, Yupan Wang. Topological entropy of free semigroup actions for noncompact sets. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 995-1017. doi: 10.3934/dcds.2019041
##### References:
 [1] R. Adler, A. Konheim and J. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. doi: 10.2307/1994177. [2] L. Barreira, Ya. Pesin and J. Schmeling, On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity, Chaos, 7 (1997), 27-38. doi: 10.1063/1.166232. [3] A. Biś, Entropies of a semigroup of maps, Discrete Contin. Dyn. Syst, 11 (2004), 639-648. doi: 10.3934/dcds.2004.11.639. [4] A. Biś, Partial variational principle for finitely generated groups of polynomial growth and some foliated spaces, Colloq. Math., 110 (2008), 431-449. doi: 10.4064/cm110-2-7. [5] A. Biś, An analogue of the variational principle for group and pseudogroup actions, Ann. Inst. Fourier., 63 (2013), 839-863. doi: 10.5802/aif.2778. [6] A. Biś and M. Urbański, Some remarks on topological entropy of a semigroup of continuous maps, Cubo, 8 (2006), 63-71. [7] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.2307/1995565. [8] R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.2307/1996403. [9] M. Brin and A. Katok, On local entropy, Geometric Dynamics, Springer, Berlin, Heidelberg, 1007 (1983), 30–38. doi: 10.1007/BFb0061408. [10] A. Bufetov, Topological entropy of free semigroup actions and skew-product transformations, J. Dynam. Control Systems, 5 (1999), 137-143. doi: 10.1023/A:1021796818247. [11] M. Carvalho, F. Rodrigues and P. Varandas, Semigroup actions of expanding maps, J. Stat. Phys., 166 (2017), 114-136. doi: 10.1007/s10955-016-1697-3. [12] M. Carvalho, F. Rodrigues and P. Varandas, A variational principle for free semigroup actions, Advances in Math., 334 (2018), 450-487. doi: 10.1016/j.aim.2018.06.010. [13] M. Carvalho, F. Rodrigues and P. Varandas, Quantitative recurrence for free semigroup actions, Nonlinearity, 31 (2018), 864-886. doi: 10.1088/1361-6544/aa999f. [14] E. C. Chen, T. Küpper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208. doi: 10.1017/S0143385704000872. [15] V. Climenhaga, Multifractal formalism derived from thermodynamics for general dynamical systems, Electron. Res. Announc. Math. Sci., 17 (2010), 1-11. doi: 10.3934/era.2010.17.1. [16] X. Dai, Z. Zhou and X. Geng, Some relations between Hausdorff-dimensions and entropies, Sci. China Ser. A, 41 (1998), 1068-1075. doi: 10.1007/BF02871841. [17] E. I. Dinaburg, The relation between topological entropy and metric entropy, Soviet Math. Dokl., 11 (1970), 13-16. [18] Y. Dong and X. Tian, Multifractal analysis of the new level sets, arXiv: 1510.06514 (2015). [19] S. Friedland, Entropy of graphs, semigroups and groups, London Mathematical Society Lecture Note Series, 228. Cambridge University Press, Cambridge, 1996,319–343. doi: 10.1017/CBO9780511662812.013. [20] E. Ghys, R. Langevin and P. Walczak, Entropie geometrique des feuilletages, Acta Math., 160 (1988), 105-142. doi: 10.1007/BF02392274. [21] S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dyn., 4 (1996), 205-233. [22] X. Lin, D. Ma and Y. Wang, On the measure-theoretic entropy and topological pressure of free semigroup actions, Ergodic Theory Dynam. Systems, 38 (2018), 686-716. doi: 10.1017/etds.2016.41. [23] D. Ma and S. Liu, Some properties of topological pressure of a semigroup of continuous maps, Dyn. Syst, 29 (2014), 1-17. doi: 10.1080/14689367.2013.835387. [24] D. Ma and M. Wu, On Hausdorff dimension and topological entropy, Fractals, 18 (2010), 363-370. doi: 10.1142/S0218348X10004956. [25] D. Ma and M. Wu, Topological pressure and topological entropy of a semigroup of maps, Discrete Contin. Dyn. Syst., 31 (2011), 545-557. doi: 10.3934/dcds.2011.31.545. [26] J. H. Ma and Z. Y. Wen, A Billingsley type theorem for Bowen entropy, C. R. Math. Acad. Sci., Paris, 346 (2008), 503–507. doi: 10.1016/j.crma.2008.03.010. [27] M. Misiurewicz, On Bowen definition of topological entropy, Discrete Contin. Dyn. Syst., 10 (2004), 827-833. doi: 10.3934/dcds.2004.10.827. [28] Y. Pesin, Dimension Theory in Dynamical Systems, Chicago: The university of Chicago Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001. [29] C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, 27 (2007), 929-956. doi: 10.1017/S0143385706000824. [30] F. B. Rodrigues and P. Varandas, Specification and thermodynamical properties of semigroup actions, J. Math. Phys., 57 (2016), 052704, 27 pp. doi: 10.1063/1.4950928. [31] F. Takens and E. Verbitski, Multifractal analysis of local entropies for expansive homeomorphisms with specification, Comm. Math. Phys, 203 (1999), 593-612. doi: 10.1007/s002200050627. [32] F. Takens and E. Verbitski, Multifractal analysis of dimensions and entropies, Regul. Chaotic Dyn., 5 (2000), 361-382. doi: 10.1070/rd2000v005n04ABEH000154. [33] F. Takens and E. Verbitski, On the variational principle for the topological entropy of certain non-compact sets, Ergodic Theory Dynam. Systems, 23 (2003), 317-348. doi: 10.1017/S0143385702000913. [34] Y. Wang and D. Ma, On the topological entropy of a semigroup of continuous maps, J. Math. Anal. Appl., 427 (2015), 1084-1100. doi: 10.1016/j.jmaa.2015.02.082. [35] Y. Wang, D. Ma and X. Lin, On the topological entropy of free semigroup actions, J. Math. Anal. Appl., 435 (2016), 1573-1590. [36] P. Waters, An Introduction to Ergodic Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1982.

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##### References:
 [1] R. Adler, A. Konheim and J. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. doi: 10.2307/1994177. [2] L. Barreira, Ya. Pesin and J. Schmeling, On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity, Chaos, 7 (1997), 27-38. doi: 10.1063/1.166232. [3] A. Biś, Entropies of a semigroup of maps, Discrete Contin. Dyn. Syst, 11 (2004), 639-648. doi: 10.3934/dcds.2004.11.639. [4] A. Biś, Partial variational principle for finitely generated groups of polynomial growth and some foliated spaces, Colloq. Math., 110 (2008), 431-449. doi: 10.4064/cm110-2-7. [5] A. Biś, An analogue of the variational principle for group and pseudogroup actions, Ann. Inst. Fourier., 63 (2013), 839-863. doi: 10.5802/aif.2778. [6] A. Biś and M. Urbański, Some remarks on topological entropy of a semigroup of continuous maps, Cubo, 8 (2006), 63-71. [7] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.2307/1995565. [8] R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.2307/1996403. [9] M. Brin and A. Katok, On local entropy, Geometric Dynamics, Springer, Berlin, Heidelberg, 1007 (1983), 30–38. doi: 10.1007/BFb0061408. [10] A. Bufetov, Topological entropy of free semigroup actions and skew-product transformations, J. Dynam. Control Systems, 5 (1999), 137-143. doi: 10.1023/A:1021796818247. [11] M. Carvalho, F. Rodrigues and P. Varandas, Semigroup actions of expanding maps, J. Stat. Phys., 166 (2017), 114-136. doi: 10.1007/s10955-016-1697-3. [12] M. Carvalho, F. Rodrigues and P. Varandas, A variational principle for free semigroup actions, Advances in Math., 334 (2018), 450-487. doi: 10.1016/j.aim.2018.06.010. [13] M. Carvalho, F. Rodrigues and P. Varandas, Quantitative recurrence for free semigroup actions, Nonlinearity, 31 (2018), 864-886. doi: 10.1088/1361-6544/aa999f. [14] E. C. Chen, T. Küpper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208. doi: 10.1017/S0143385704000872. [15] V. Climenhaga, Multifractal formalism derived from thermodynamics for general dynamical systems, Electron. Res. Announc. Math. Sci., 17 (2010), 1-11. doi: 10.3934/era.2010.17.1. [16] X. Dai, Z. Zhou and X. Geng, Some relations between Hausdorff-dimensions and entropies, Sci. China Ser. A, 41 (1998), 1068-1075. doi: 10.1007/BF02871841. [17] E. I. Dinaburg, The relation between topological entropy and metric entropy, Soviet Math. Dokl., 11 (1970), 13-16. [18] Y. Dong and X. Tian, Multifractal analysis of the new level sets, arXiv: 1510.06514 (2015). [19] S. Friedland, Entropy of graphs, semigroups and groups, London Mathematical Society Lecture Note Series, 228. Cambridge University Press, Cambridge, 1996,319–343. doi: 10.1017/CBO9780511662812.013. [20] E. Ghys, R. Langevin and P. Walczak, Entropie geometrique des feuilletages, Acta Math., 160 (1988), 105-142. doi: 10.1007/BF02392274. [21] S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dyn., 4 (1996), 205-233. [22] X. Lin, D. Ma and Y. Wang, On the measure-theoretic entropy and topological pressure of free semigroup actions, Ergodic Theory Dynam. Systems, 38 (2018), 686-716. doi: 10.1017/etds.2016.41. [23] D. Ma and S. Liu, Some properties of topological pressure of a semigroup of continuous maps, Dyn. Syst, 29 (2014), 1-17. doi: 10.1080/14689367.2013.835387. [24] D. Ma and M. Wu, On Hausdorff dimension and topological entropy, Fractals, 18 (2010), 363-370. doi: 10.1142/S0218348X10004956. [25] D. Ma and M. Wu, Topological pressure and topological entropy of a semigroup of maps, Discrete Contin. Dyn. Syst., 31 (2011), 545-557. doi: 10.3934/dcds.2011.31.545. [26] J. H. Ma and Z. Y. Wen, A Billingsley type theorem for Bowen entropy, C. R. Math. Acad. Sci., Paris, 346 (2008), 503–507. doi: 10.1016/j.crma.2008.03.010. [27] M. Misiurewicz, On Bowen definition of topological entropy, Discrete Contin. Dyn. Syst., 10 (2004), 827-833. doi: 10.3934/dcds.2004.10.827. [28] Y. Pesin, Dimension Theory in Dynamical Systems, Chicago: The university of Chicago Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001. [29] C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, 27 (2007), 929-956. doi: 10.1017/S0143385706000824. [30] F. B. Rodrigues and P. Varandas, Specification and thermodynamical properties of semigroup actions, J. Math. Phys., 57 (2016), 052704, 27 pp. doi: 10.1063/1.4950928. [31] F. Takens and E. Verbitski, Multifractal analysis of local entropies for expansive homeomorphisms with specification, Comm. Math. Phys, 203 (1999), 593-612. doi: 10.1007/s002200050627. [32] F. Takens and E. Verbitski, Multifractal analysis of dimensions and entropies, Regul. Chaotic Dyn., 5 (2000), 361-382. doi: 10.1070/rd2000v005n04ABEH000154. [33] F. Takens and E. Verbitski, On the variational principle for the topological entropy of certain non-compact sets, Ergodic Theory Dynam. Systems, 23 (2003), 317-348. doi: 10.1017/S0143385702000913. [34] Y. Wang and D. Ma, On the topological entropy of a semigroup of continuous maps, J. Math. Anal. Appl., 427 (2015), 1084-1100. doi: 10.1016/j.jmaa.2015.02.082. [35] Y. Wang, D. Ma and X. Lin, On the topological entropy of free semigroup actions, J. Math. Anal. Appl., 435 (2016), 1573-1590. [36] P. Waters, An Introduction to Ergodic Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1982.
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