# American Institue of Mathematical Sciences

2017, 10(2): 367-394. doi: 10.3934/dcdss.2017018

## Variational principles for the topological pressure of measurable potentials

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

* Corresponding author: Marc Rauch

Received  October 2015 Revised  November 2016 Published  January 2017

Fund Project: This work was supported by the Deutsche Forschungsgemeinschhaft

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.

Citation: Marc Rauch. Variational principles for the topological pressure of measurable potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 367-394. doi: 10.3934/dcdss.2017018
##### References:
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##### References:
 [1] C. Aliprantis and K. Border, Infinite Dimensional Analysis Third edition, Springer-Verlag, Berlin, 2006. [2] J. Barral, 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. [3] 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. [4] Y.-L. Cao, D.-J. Feng, W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657. [5] J. Chen, Ya. B. Pesin, Dimension of non-conformal repellers: A survey, Nonlinearity, 23 (2010), R93-R114. doi: 10.1088/0951-7715/23/4/R01. [6] V. Climenhaga, Bowen's equation in the non-uniform setting, Ergodic Theory Dynam. Systems, 31 (2011), 1163-1182. doi: 10.1017/S0143385710000362. [7] T. Downarowicz, 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. [8] K. Falconer, Fractal Geometry Second edition, Wiley, Chichester, 2003. [9] K. Falconer, Techniques in Fractal Geometry Wiley, Chichester, 1997. [10] D.-J. Feng, W. Huang, Variational principles for topological entropies of subsets, J. Funct. Anal., 263 (2012), 2228-2254. doi: 10.1016/j.jfa.2012.07.010. [11] D.-J. Feng, W. Huang, Variational principles for weighted topological pressure, J. Math. Pures Appl., 106 (2016), 411-452. doi: 10.1016/j.matpur.2016.02.016. [12] F. Hofbauer, The box dimension of completely invariant subsets for expanding piecewise monotonic transformations, Monatsh. Math., 121 (1996), 199-211. doi: 10.1007/BF01298950. [13] G. Keller, Equilibrium States in Ergodic Theory Cambridge University Press, Cambridge, 1998. [14] A. Klenke, Probability Theory First edition, Springer-Verlag, London, 2008. [15] A. Mummert, A variational principle for discontinuous potentials, Ergodic Theory Dynam. Systems, 27 (2007), 583-594. doi: 10.1017/S0143385706000642. [16] Ya. B. Pesin, Dimension Theory in Dynamical Systems Chicago Lectures in Mathematics, Contemporary views and applications, University of Chicago Press, Chicago, IL, 1997. [17] Ya. B. Pesin, B. S. Pitskel', Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen., 18 (1984), 50-63. [18] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971. doi: 10.2307/2373682. [19] P. Walters, An Introduction to Ergodic Theory First edition, Springer-Verlag, New York, 1982.
A cookie cutter with discontinuous geometric potentials
The function $T_i(x)$ for $\epsilon=1/8$
The cookie-cutters $T_n$ approaching the limit cookie-cutter $T_\infty$
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