Topological pressures of a factor map for iterated function systems

Zhongxuan Yang; Xiaojun Huang; Jiajun Zhang; Zhongxuan Yang; Xiaojun Huang; Jiajun Zhang

doi:10.3934/math.2025461

AIMS Mathematics

2025, Volume 10, Issue 4: 10124-10139. doi: 10.3934/math.2025461

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Research article

Topological pressures of a factor map for iterated function systems

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received: 10 September 2024 Revised: 28 February 2025 Accepted: 28 March 2025 Published: 28 April 2025
MSC : 37A05, 37B40, 37D35

In this manuscript, we mainly investigate the topological pressure for iterated function systems on a compact metric space defined by Wang and Liao [Dynam. Syst., 2021, 36(3): 483–506]. Given a factor map, we establish a formula of topological pressure of a factor map, which generalizes Bowen's inequality in [Trans. Amer. Math. Soc., 1971,153: 401–414.] to iterated function systems. Consequently, we further study the power rule of a topological pressure for iterated function systems.
- iterated function system,
- topological pressure,
- Biś's topological entropy,
- factor map
Citation: Zhongxuan Yang, Xiaojun Huang, Jiajun Zhang. Topological pressures of a factor map for iterated function systems[J]. AIMS Mathematics, 2025, 10(4): 10124-10139. doi: 10.3934/math.2025461

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Abstract

In this manuscript, we mainly investigate the topological pressure for iterated function systems on a compact metric space defined by Wang and Liao [Dynam. Syst., 2021, 36(3): 483–506]. Given a factor map, we establish a formula of topological pressure of a factor map, which generalizes Bowen's inequality in [Trans. Amer. Math. Soc., 1971,153: 401–414.] to iterated function systems. Consequently, we further study the power rule of a topological pressure for iterated function systems.

References

[1]	A. Biś, Entropies of a semigroup of maps, Discrete Cont. Dyn. Syst., 11 (2004), 639–648. https://doi.org/10.3934/dcds.2004.11.639 doi: 10.3934/dcds.2004.11.639
[2]	A. Biś, E. Mihailescu, Inverse pressure for finitely generated semigroups, Nonlinear Anal., 222 (2022), 112942. https://doi.org/10.1016/j.na.2022.112942 doi: 10.1016/j.na.2022.112942
[3]	R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Am. Math. Soc., 153 (1971), 401–414. https://doi.org/10.2307/1995565 doi: 10.2307/1995565
[4]	R. Bowen, Topological entropy for noncompact sets, Trans. Am. Math. Soc., 184 (1973), 125–136. https://doi.org/10.2307/1996403 doi: 10.2307/1996403
[5]	A. Bufetov, Topological entropy of free semigroup actions and skew-product transformations, J. Dyn. Control Syst., 5 (1999), 137–143. https://doi.org/10.1023/A:1021796818247 doi: 10.1023/A:1021796818247
[6]	M. Carvalho, F. B. Rodrigues, P. Varandas, A variational principle for free semigroup actions, Adv. Math., 334 (2018), 450–487. https://doi.org/10.1016/j.aim.2018.06.010 doi: 10.1016/j.aim.2018.06.010
[7]	C. Fang, W. Huang, Y. F. Yi, P. F. Zhang, Dimensions of stable sets and scrambled sets in positive finite entropy systems, Ergod. Theor. Dyn. Syst., 32 (2012), 599–628. https://doi.org/10.1017/S0143385710000982 doi: 10.1017/S0143385710000982
[8]	D. J. Feng, W. Huang, Variational principles for topological entropies of subsets, J. Funct. Anal., 263 (2012), 2228–2254. https://doi.org/10.1016/j.jfa.2012.07.010 doi: 10.1016/j.jfa.2012.07.010
[9]	E. Ghys, R. Langevin, P. Walczak, Entropie géométrique des feuilletages, Acta Math., 160 (1988), 105–142. https://doi.org/10.1007/BF02392274 doi: 10.1007/BF02392274
[10]	K. H. Hofmann, L. N. Stojanov, Topological entropy of group and semigroup actions, Adv. Math., 115 (1995), 54–98. https://doi.org/10.1006/aima.1995.1050 doi: 10.1006/aima.1995.1050
[11]	X. J. Huang, X. Wen, F. P. Zeng, Topological pressure of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 8 (2008), 43–48.
[12]	Y. J. Ju, D. K. Ma, Y. P. Wang, Topological entropy of free semigroup actions for noncompact sets, Discrete Cont. Dynam. Syst., 39 (2019), 995–1017. https://doi.org/10.3934/dcds.2019041 doi: 10.3934/dcds.2019041
[13]	Q. Li, E. Chen, X. Y. Zhou, A note of topological pressure for non-compact sets of a factor map, Chaos Soliton. Fract., 49 (2013), 72–77. https://doi.org/10.1016/j.chaos.2013.02.009 doi: 10.1016/j.chaos.2013.02.009
[14]	X. G. Lin, D. K. Ma, Y. P. Wang, On the measure-theoretic entropy and topological pressure of free semigroup actions, Ergod. Theor. Dyn. Syst., 38 (2018), 686–716. https://doi.org/10.1017/etds.2016.41 doi: 10.1017/etds.2016.41
[15]	L. Liu, C. Zhao, Topological pressure of a factor map for nonautonomous dynamical systems, Dynam. Syst., 38 (2023), 353–364. https://doi.org/10.1080/14689367.2023.2182183 doi: 10.1080/14689367.2023.2182183
[16]	D. K. Ma, M. Wu, Topological pressure and topological entropy of a semigroup of maps, Discrete Cont. Dyn. Syst, 31 (2011), 545–557. https://doi.org/10.3934/dcds.2011.31.545 doi: 10.3934/dcds.2011.31.545
[17]	E. Mihailescu, M. Urbański, Inverse pressure estimates and the independence of stable dimension for non-invertible maps, Can. J. Math., 60 (2008), 658–684. https://doi.org/10.4153/CJM-2008-029-2 doi: 10.4153/CJM-2008-029-2
[18]	P. Oprocha, G. H. Zhang, Dimensional entropy over sets and fibres, Nonlinearity, 24 (2011), 2325–2346. https://doi.org/10.1088/0951-7715/24/8/009 doi: 10.1088/0951-7715/24/8/009
[19]	Y. B. Pesin, Dimension theory in dynamical systems, University of Chicago Press, 1997. https://doi.org/10.7208/chicago/9780226662237.001.0001
[20]	Y. B. Pesin, B. S. Pitskel', Topological pressure and the variational principle for noncompact sets, Funct. Anal. Appl., 18 (1984), 307–318. https://doi.org/10.1007/BF01083692 https://doi.org/10.1007/BF01083692 doi: 10.1007/BF01083692
[21]	D. Ruelie, Statistical mechanics on a compact set with $\mathbb{Z}^{v}$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc., 185 (1973), 237–251. https://doi.org/10.1090/S0002-9947-1973-0417391-6 doi: 10.1090/S0002-9947-1973-0417391-6
[22]	J. R. Tang, B. Li, W. C. Cheng, Some properties on topological entropy of free semigroup action, Dynam. Syst., 33 (2017), 54–71. https://doi.org/10.1080/14689367.2017.1298724 doi: 10.1080/14689367.2017.1298724
[23]	P. Walters, An introduction to ergodic theory, New York-Berlin: Springer-Verlag, 79 1982.
[24]	H. Y. Wang, X. Liao, Topological pressure for an iterated function system, Dynam. Syst., 36 (2021), 483–506. https://doi.org/10.1080/14689367.2021.1929081 doi: 10.1080/14689367.2021.1929081
[25]	H. Y. Wang, X. Liao, Q. Liu, P. Zeng, Topological entropy pairs for an iterated function system, J. Math. Anal. Appl., 488 (2020), 124076. https://doi.org/10.1016/j.jmaa.2020.124076 doi: 10.1016/j.jmaa.2020.124076
[26]	Q. Xiao, D. K. Ma, Topological pressure of free semigroup actions for non-compact sets and bowen's equation, I, J. Dyn. Diff. Equat., 35 (2023), 199–236. https://doi.org/10.1007/s10884-021-09983-3 doi: 10.1007/s10884-021-09983-3
[27]	Q. Xiao, D. K. Ma, Variational principle of topological pressure of free semigroup actions for subsets, Qual. Theory Dyn. Syst., 21 (2022), 99. https://doi.org/10.1007/s12346-022-00626-6 doi: 10.1007/s12346-022-00626-6
[28]	C. Zhao, E. Chen, X. C. Hong, X. Y. Zhou, A formula of packing pressure of a factor map, Entropy, 19 (2017), 526. https://doi.org/10.3390/e19100526 doi: 10.3390/e19100526
[29]	C. Zhao, K. X. Yang, Topological pressures of a factor map for free semigroup actions, J. Differ. Equ. Appl., 27 (2021), 389–403. https://doi.org/10.1080/10236198.2021.1900141 doi: 10.1080/10236198.2021.1900141
[30]	X. F. Zhong, Z. J. Chen, Variational principle for topological pressure on subsets of free semigroup actions, Acta Math. Sin.-English Ser., 37 (2021), 1401–1414. https://doi.org/10.1007/s10114-021-0403-9 doi: 10.1007/s10114-021-0403-9

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