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Derivatives of slippery Devil's staircases
Variational principles for the topological pressure of measurable potentials
Mathematical Institute, University of Jena, Ernst-Abbe-Platz 2,07745 Jena, Germany |
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.
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. |
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