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. AdlerA. Konheim and J. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.2307/1994177.  Google Scholar

[2]

L. BarreiraYa. 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.  Google Scholar

[3]

A. Biś, Entropies of a semigroup of maps, Discrete Contin. Dyn. Syst, 11 (2004), 639-648.  doi: 10.3934/dcds.2004.11.639.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Biś and M. Urbański, Some remarks on topological entropy of a semigroup of continuous maps, Cubo, 8 (2006), 63-71.   Google Scholar

[7]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.2307/1995565.  Google Scholar

[8]

R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.2307/1996403.  Google Scholar

[9]

M. Brin and A. Katok, On local entropy, Geometric Dynamics, Springer, Berlin, Heidelberg, 1007 (1983), 30–38. doi: 10.1007/BFb0061408.  Google Scholar

[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.  Google Scholar

[11]

M. CarvalhoF. Rodrigues and P. Varandas, Semigroup actions of expanding maps, J. Stat. Phys., 166 (2017), 114-136.  doi: 10.1007/s10955-016-1697-3.  Google Scholar

[12]

M. CarvalhoF. 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.  Google Scholar

[13]

M. CarvalhoF. Rodrigues and P. Varandas, Quantitative recurrence for free semigroup actions, Nonlinearity, 31 (2018), 864-886.  doi: 10.1088/1361-6544/aa999f.  Google Scholar

[14]

E. C. ChenT. Küpper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208.  doi: 10.1017/S0143385704000872.  Google Scholar

[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.  Google Scholar

[16]

X. DaiZ. Zhou and X. Geng, Some relations between Hausdorff-dimensions and entropies, Sci. China Ser. A, 41 (1998), 1068-1075.  doi: 10.1007/BF02871841.  Google Scholar

[17]

E. I. Dinaburg, The relation between topological entropy and metric entropy, Soviet Math. Dokl., 11 (1970), 13-16.   Google Scholar

[18]

Y. Dong and X. Tian, Multifractal analysis of the new level sets, arXiv: 1510.06514 (2015). Google Scholar

[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.  Google Scholar

[20]

E. GhysR. Langevin and P. Walczak, Entropie geometrique des feuilletages, Acta Math., 160 (1988), 105-142.  doi: 10.1007/BF02392274.  Google Scholar

[21]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dyn., 4 (1996), 205-233.   Google Scholar

[22]

X. LinD. 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.  Google Scholar

[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.  Google Scholar

[24]

D. Ma and M. Wu, On Hausdorff dimension and topological entropy, Fractals, 18 (2010), 363-370.  doi: 10.1142/S0218348X10004956.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. Misiurewicz, On Bowen definition of topological entropy, Discrete Contin. Dyn. Syst., 10 (2004), 827-833.  doi: 10.3934/dcds.2004.10.827.  Google Scholar

[28]

Y. Pesin, Dimension Theory in Dynamical Systems, Chicago: The university of Chicago Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

F. Takens and E. Verbitski, Multifractal analysis of dimensions and entropies, Regul. Chaotic Dyn., 5 (2000), 361-382.  doi: 10.1070/rd2000v005n04ABEH000154.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

Y. WangD. Ma and X. Lin, On the topological entropy of free semigroup actions, J. Math. Anal. Appl., 435 (2016), 1573-1590.   Google Scholar

[36]

P. Waters, An Introduction to Ergodic Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1982.  Google Scholar

show all references

References:
[1]

R. AdlerA. Konheim and J. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.2307/1994177.  Google Scholar

[2]

L. BarreiraYa. 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.  Google Scholar

[3]

A. Biś, Entropies of a semigroup of maps, Discrete Contin. Dyn. Syst, 11 (2004), 639-648.  doi: 10.3934/dcds.2004.11.639.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Biś and M. Urbański, Some remarks on topological entropy of a semigroup of continuous maps, Cubo, 8 (2006), 63-71.   Google Scholar

[7]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.2307/1995565.  Google Scholar

[8]

R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.2307/1996403.  Google Scholar

[9]

M. Brin and A. Katok, On local entropy, Geometric Dynamics, Springer, Berlin, Heidelberg, 1007 (1983), 30–38. doi: 10.1007/BFb0061408.  Google Scholar

[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.  Google Scholar

[11]

M. CarvalhoF. Rodrigues and P. Varandas, Semigroup actions of expanding maps, J. Stat. Phys., 166 (2017), 114-136.  doi: 10.1007/s10955-016-1697-3.  Google Scholar

[12]

M. CarvalhoF. 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.  Google Scholar

[13]

M. CarvalhoF. Rodrigues and P. Varandas, Quantitative recurrence for free semigroup actions, Nonlinearity, 31 (2018), 864-886.  doi: 10.1088/1361-6544/aa999f.  Google Scholar

[14]

E. C. ChenT. Küpper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208.  doi: 10.1017/S0143385704000872.  Google Scholar

[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.  Google Scholar

[16]

X. DaiZ. Zhou and X. Geng, Some relations between Hausdorff-dimensions and entropies, Sci. China Ser. A, 41 (1998), 1068-1075.  doi: 10.1007/BF02871841.  Google Scholar

[17]

E. I. Dinaburg, The relation between topological entropy and metric entropy, Soviet Math. Dokl., 11 (1970), 13-16.   Google Scholar

[18]

Y. Dong and X. Tian, Multifractal analysis of the new level sets, arXiv: 1510.06514 (2015). Google Scholar

[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.  Google Scholar

[20]

E. GhysR. Langevin and P. Walczak, Entropie geometrique des feuilletages, Acta Math., 160 (1988), 105-142.  doi: 10.1007/BF02392274.  Google Scholar

[21]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dyn., 4 (1996), 205-233.   Google Scholar

[22]

X. LinD. 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.  Google Scholar

[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.  Google Scholar

[24]

D. Ma and M. Wu, On Hausdorff dimension and topological entropy, Fractals, 18 (2010), 363-370.  doi: 10.1142/S0218348X10004956.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. Misiurewicz, On Bowen definition of topological entropy, Discrete Contin. Dyn. Syst., 10 (2004), 827-833.  doi: 10.3934/dcds.2004.10.827.  Google Scholar

[28]

Y. Pesin, Dimension Theory in Dynamical Systems, Chicago: The university of Chicago Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

F. Takens and E. Verbitski, Multifractal analysis of dimensions and entropies, Regul. Chaotic Dyn., 5 (2000), 361-382.  doi: 10.1070/rd2000v005n04ABEH000154.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

Y. WangD. Ma and X. Lin, On the topological entropy of free semigroup actions, J. Math. Anal. Appl., 435 (2016), 1573-1590.   Google Scholar

[36]

P. Waters, An Introduction to Ergodic Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1982.  Google Scholar

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