Skip to main content Accessibility help
×
Home
Hostname: page-component-5c569c448b-bmzkg Total loading time: 0.163 Render date: 2022-07-06T05:26:41.342Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

New approach to weighted topological entropy and pressure

Published online by Cambridge University Press:  28 January 2022

MASAKI TSUKAMOTO*
Affiliation:
Department of Mathematics, Kyushu University, Moto-oka 744, Nishi-ku, Fukuoka 819-0395, Japan

Abstract

Motivated by fractal geometry of self-affine carpets and sponges, Feng and Huang [J. Math. Pures Appl.106(9) (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems, and proved variational principles for them. We introduce a new approach to this theory. Our new definitions of weighted topological entropy and pressure are very different from the original definitions of Feng and Huang. The equivalence of the two definitions seems highly non-trivial. Their equivalence can be seen as a generalization of the dimension formula for the Bedford–McMullen carpet in purely topological terms.

Type
Original Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bedford, T.. Crinkly curves, Markov partitions and box dimension in self-similarsets. PhD Thesis, University of Warwick, 1984.Google Scholar
Barral, J. and Feng, D.-J.. Weighted thermodynamic formalism and applications. Preprint, 2009,arXiv:0909.4247.Google Scholar
Barral, J. and Feng, D.-J.. Weighted thermodynamic formalism on subshifts and applications.Asian J. Math. 16 (2012), 319352.10.4310/AJM.2012.v16.n2.a8CrossRefGoogle Scholar
Bowen, R.. Topological entropy for noncompactsubsets. Trans. Amer. Math. Soc. 184 (1973),125136.10.1090/S0002-9947-1973-0338317-XCrossRefGoogle Scholar
Downarowicz, T. and Huczek, D.. Zero-dimensional principal extensions. Acta Appl.Math. 126 (2013), 117129.10.1007/s10440-013-9810-yCrossRefGoogle Scholar
Dinaburg, E. I.. A correlation between topological entropy andmetric entropy. Dokl. Akad. Nauk SSSR 190 (1970),1922.Google Scholar
Downarowicz, T.. Entropy in Dynamical Systems.Cambridge University Press, Cambridge, 2011.10.1017/CBO9780511976155CrossRefGoogle Scholar
Feng, D.-J.. Equilibrium states for factor maps betweensubshifts. Adv. Math. 226 (2011),24702502.10.1016/j.aim.2010.09.012CrossRefGoogle Scholar
Feng, D.-J. and Huang, W.. Variational principle for weighted topological pressure. J.Math. Pures Appl. (9) 106 (2016), 411452.10.1016/j.matpur.2016.02.016CrossRefGoogle Scholar
Goodman, T. N. T.. Relating topological entropy and measureentropy. Bull. Lond. Math. Soc. 3 (1971),176180.10.1112/blms/3.2.176CrossRefGoogle Scholar
Goodwyn, L. W.. Topological entropy bounds measure-theoreticentropy. Proc. Amer. Math. Soc. 23 (1969),679688.10.1090/S0002-9939-1969-0247030-3CrossRefGoogle Scholar
Kenyon, R. and Peres, Y.. Measures of full dimension on affine-invariant sets. Ergod.Th. & Dynam. Sys. 16 (1996), 307323.10.1017/S0143385700008828CrossRefGoogle Scholar
Kenyon, R. and Peres, Y.. Hausdorff dimensions of sofic affine-invariant sets. Israel J.Math. 94 (1996), 157178.10.1007/BF02762702CrossRefGoogle Scholar
Lindenstrauss, E. and Tsukamoto, M.. Double variational principle for mean dimension. Geom.Funct. Anal. 29 (2019), 10481109.10.1007/s00039-019-00501-8CrossRefGoogle Scholar
Ledrappier, F. and Walters, P.. A relativised variational principle for continuous transformations.J. Lond. Math. Soc. (2) 16 (1977),568576.10.1112/jlms/s2-16.3.568CrossRefGoogle Scholar
McMullen, C.. The Hausdorff dimension of general Sierpinskicarpets. Nagoya Math. J. 96 (1984),19.10.1017/S0027763000021085CrossRefGoogle Scholar
Misiurewicz, M.. A short proof of the variational principle for${\mathbb{Z}}_{+}^N$ actions on a compact space. Int. Conf.on Dyn. Syst. Math. Phys. (Rennes, 1975) (Astérisque, 40). SociétéMathématique de France, Paris, 1976, pp.145157.Google Scholar
Ruelle, D.. Statistical mechanics on a compact set with${Z}^{\nu}$ action satisfying expansiveness and specification. Trans. Amer. Math. Soc. 185 (1973), 237251.10.2307/1996437CrossRefGoogle Scholar
Walters, P.. A variational principle for the pressure of continuoustransformations. Amer. J. Math. 17 (1975),937971.10.2307/2373682CrossRefGoogle Scholar
Walters, P.. An Introduction to Ergodic Theory.Springer-Verlag, New York, 1982.10.1007/978-1-4612-5775-2CrossRefGoogle Scholar
Yayama, Y.. Existence of a measurable saturated compensationfunction between subshifts and its applications. Ergod. Th. & Dynam. Sys. 31(2011), 15631589.10.1017/S0143385710000404CrossRefGoogle Scholar
Yayama, Y.. Application of a relative variational principle todimension of nonconformal expanding maps. Stoch. Dyn. 11 (2011),643679.10.1142/S0219493711003486CrossRefGoogle Scholar

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

New approach to weighted topological entropy and pressure
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

New approach to weighted topological entropy and pressure
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

New approach to weighted topological entropy and pressure
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *