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

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

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.

Keywords: Thermodynamic formalism, variational principle, discontinuous potential, discontinuous geometric potential, cookie-cutter, Hausdorff dimension.
Mathematics Subject Classification: Primary: 37B99; Secondary: 28A80.
Citation: Marc Rauch. Variational principles for the topological pressure of measurable potentials. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : 367-394. doi: 10.3934/dcdss.2017018
References:
[1]

C. Aliprantis and K. Border, Infinite Dimensional Analysis Third edition, Springer-Verlag, Berlin, 2006.

[2]

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.

[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. doi: 10.1007/BFb0061408.

[4]

Y.-L. Cao, D.-J. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657. 

[5]

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.

[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 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.

[8]

K. Falconer, Fractal Geometry Second edition, Wiley, Chichester, 2003. doi: 10.1002/0470013850.

[9]

K. Falconer, Techniques in Fractal Geometry Wiley, Chichester, 1997.

[10]

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.

[11]

D.-J. Feng and W. Huang, Variational principles for weighted topological pressure, J. Math. Pures Appl.(9), 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. doi: 10.1017/CBO9781107359987.

[14]

A. Klenke, Probability Theory First edition, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84800-048-3.

[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. doi: 10.7208/chicago/9780226662237.001.0001.

[17]

Ya. B. Pesin and 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.

show all references

References:
[1]

C. Aliprantis and K. Border, Infinite Dimensional Analysis Third edition, Springer-Verlag, Berlin, 2006.

[2]

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.

[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. doi: 10.1007/BFb0061408.

[4]

Y.-L. Cao, D.-J. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657. 

[5]

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.

[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 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.

[8]

K. Falconer, Fractal Geometry Second edition, Wiley, Chichester, 2003. doi: 10.1002/0470013850.

[9]

K. Falconer, Techniques in Fractal Geometry Wiley, Chichester, 1997.

[10]

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.

[11]

D.-J. Feng and W. Huang, Variational principles for weighted topological pressure, J. Math. Pures Appl.(9), 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. doi: 10.1017/CBO9781107359987.

[14]

A. Klenke, Probability Theory First edition, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84800-048-3.

[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. doi: 10.7208/chicago/9780226662237.001.0001.

[17]

Ya. B. Pesin and 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.

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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