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A Billingsley-type theorem for the pressure of an action of an amenable group
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 |
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 [
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. 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. |
[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. Google Scholar |
[36] |
P. Waters,
An Introduction to Ergodic Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1982. |
show all references
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. 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. |
[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. Google Scholar |
[36] |
P. Waters,
An Introduction to Ergodic Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1982. |
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