Variational principles for the topological pressure of measurable potentials

Marc Rauch

doi:10.3934/dcdss.2017018

Article Contents

2017, Volume 10, Issue 2: 367-394. Doi: 10.3934/dcdss.2017018

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Variational principles for the topological pressure of measurable potentials

Marc Rauch^,

Mathematical Institute, University of Jena, Ernst-Abbe-Platz 2,07745 Jena, Germany

* Corresponding author: Marc Rauch
* Corresponding author: Marc Rauch

Received: October 2015

Revised: November 2016

Published: February 2018

This work was supported by the Deutsche Forschungsgemeinschhaft

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Abstract

We introduce notions of topological pressure for measurable potentials and prove corresponding variational principles. The formalism is then used to establish a Bowen formula for the Hausdorff dimension of cookie-cutters with discontinuous geometric potentials.

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Mathematics Subject Classification: Primary: 37B99; Secondary: 28A80.

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Figure 1. A cookie cutter with discontinuous geometric potentials

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Figure 2. The function $T_i(x)$ for $\epsilon=1/8$

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Figure 3. The cookie-cutters $T_n$ approaching the limit cookie-cutter $T_\infty$

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References

	C. Aliprantis and K. Border, Infinite Dimensional Analysis Third edition, Springer-Verlag, Berlin, 2006.
	J. Barral and D.-J. Feng , Weighted thermodynamic formalism on subshifts and applications, Asian J. Math., 16 (2012) , 319-352. doi: 10.4310/AJM.2012.v16.n2.a8.
	M. Brin and A. Katok, On local entropy, Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Math. , Springer-Verlag, Berlin, 1007 (1983), 30–38. doi: 10.1007/BFb0061408.
	Y.-L. Cao , D.-J. Feng and W. Huang , The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008) , 639-657.
	J. Chen and Ya. B. Pesin , Dimension of non-conformal repellers: A survey, Nonlinearity, 23 (2010) , R93-R114. doi: 10.1088/0951-7715/23/4/R01.
	V. Climenhaga , Bowen's equation in the non-uniform setting, Ergodic Theory Dynam. Systems, 31 (2011) , 1163-1182. doi: 10.1017/S0143385710000362.
	T. Downarowicz and G. H. Zhang , Modeling potential as fiber entropy and pressure as entropy, Ergodic Theory Dynam. Systems, 35 (2015) , 1165-1186. doi: 10.1017/etds.2013.95.
	K. Falconer, Fractal Geometry Second edition, Wiley, Chichester, 2003. doi: 10.1002/0470013850.
	K. Falconer, Techniques in Fractal Geometry Wiley, Chichester, 1997.
	D.-J. Feng and W. Huang , Variational principles for topological entropies of subsets, J. Funct. Anal., 263 (2012) , 2228-2254. doi: 10.1016/j.jfa.2012.07.010.
	D.-J. Feng and W. Huang , Variational principles for weighted topological pressure, J. Math. Pures Appl., 106 (2016) , 411-452. doi: 10.1016/j.matpur.2016.02.016.
	F. Hofbauer , The box dimension of completely invariant subsets for expanding piecewise monotonic transformations, Monatsh. Math., 121 (1996) , 199-211. doi: 10.1007/BF01298950.
	G. Keller, Equilibrium States in Ergodic Theory Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781107359987.
	A. Klenke, Probability Theory First edition, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84800-048-3.
	A. Mummert , A variational principle for discontinuous potentials, Ergodic Theory Dynam. Systems, 27 (2007) , 583-594. doi: 10.1017/S0143385706000642.
	Ya. B. Pesin, Dimension Theory in Dynamical Systems Chicago Lectures in Mathematics, Contemporary views and applications, University of Chicago Press, Chicago, IL, 1997. doi: 10.7208/chicago/9780226662237.001.0001.
	Ya. B. Pesin and B. S. Pitskel' , Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen., 18 (1984) , 50-63.
	P. Walters , A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975) , 937-971. doi: 10.2307/2373682.
	P. Walters, An Introduction to Ergodic Theory First edition, Springer-Verlag, New York, 1982.