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A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies
Dynamical obstruction to the existence of continuous sub-actions for interval maps with regularly varying property
1. | Department of Mathematics, University of Campinas, 13083-859 Campinas, Brazil |
2. | Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, casilla 567 Valdivia, Chile |
For transformations with regularly varying property, we identify a class of moduli of continuity related to the local behavior of the dynamics near a fixed point, and we prove that this class is not compatible with the existence of continuous sub-actions. The dynamical obstruction is given merely by a local property. As a natural complement, we also deal with the question of the existence of continuous sub-actions focusing on a particular dynamic setting. Applications of both results include interval maps that are expanding outside a neutral fixed point, as Manneville-Pomeau and Farey maps.
References:
[1] |
J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50. American Mathematical Society, Providence, RI, 1997.
doi: 10.1090/surv/050. |
[2] |
T. Bousch and O. Jenkinson,
Cohomology classes of dynamically non-negative $C^k$ functions, Inventiones Mathematicae, 148 (2002), 207-217.
doi: 10.1007/s002220100194. |
[3] |
F. M. Branco,
Subactions and maximizing measures for one-dimensional transformations with a critical point, Discrete Contin. Dyn. Syst., 17 (2007), 271-280.
doi: 10.3934/dcds.2007.17.271. |
[4] |
S. D. Branton,
Sub-actions for Young towers, Discrete and Continuous Dynamical Systems, 22 (2008), 541-556.
doi: 10.3934/dcds.2008.22.541. |
[5] |
G. Contreras, A. O. Lopes and Ph. Thieullen,
Lyapunov minimizing measures for expanding maps of the circle, Ergodic Theory Dynam. Systems, 21 (2001), 1379-1409.
doi: 10.1017/S0143385701001663. |
[6] |
E. Garibaldi, Ergodic Optimization in the Expanding Case: Concepts, Tools and Applications, SpringerBriefs in Mathematics, Springer, Cham, 2017.
doi: 10.1007/978-3-319-66643-3. |
[7] |
E. Garibaldi, A. O. Lopes and Ph. Thieullen,
On calibrated and separating sub-actions, Bull. Braz. Math. Soc. (N.S.), 40 (2009), 577-602.
doi: 10.1007/s00574-009-0028-6. |
[8] |
M. Holland,
Slowly mixing systems and intermittency maps, Ergodic Theory and Dynamical Systems, 25 (2005), 133-159.
doi: 10.1017/S0143385704000343. |
[9] |
O. Jenkinson,
Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197-224.
doi: 10.3934/dcds.2006.15.197. |
[10] |
O. Jenkinson,
Ergodic optimization in dynamical systems, Ergodic Theory and Dynamical Systems, 39 (2019), 2593-2618.
doi: 10.1017/etds.2017.142. |
[11] |
J. Karamata,
Sur un mode de croissance régulière. Théorèmes fondamentaux, Bull. Soc. Math. France, 61 (1933), 55-62.
|
[12] |
A. O. Lopes, V. A. Rosas and R. O. Ruggiero,
Cohomology and subcohomology problems for expansive, non Anosov geodesic flows, Discrete Contin. Dyn. Syst., 17 (2007), 403-422.
doi: 10.3934/dcds.2007.17.403. |
[13] |
A. O. Lopes and Ph. Thieullen, Sub-actions for Anosov diffeomorphisms. Geometric methods in dynamics. Ⅱ, Astérisque, (2003), 135–146. |
[14] |
A. O. Lopes and Ph. Thieullen,
Sub-actions for Anosov flows, Ergodic Theory and Dynamical Systems, 25 (2005), 605-628.
doi: 10.1017/S0143385704000732. |
[15] |
A. V. Medvedev,
On a concave differentiable majorant of a modulus of continuity, Real Anal. Exchange, 27 (2001/02), 123-129.
|
[16] |
I. D. Morris,
A sufficient condition for the subordination principle in ergodic optimization, Bulletin of the London Mathematical Society, 39 (2007), 214-220.
doi: 10.1112/blms/bdl030. |
[17] |
I. D. Morris,
The Mañé-Conze-Guivarc'h lemma for intermittent maps of the circle, Ergodc Theory and Dynamical Systems, 29 (2009), 1603-1611.
doi: 10.1017/S0143385708000837. |
[18] |
E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics, Vol. 508. Springer-Verlag, Berlin-New York, 1976. |
[19] |
R. R. Souza,
Sub-actions for weakly hyperbolic one-dimensional systems, Dynamical System, 18 (2003), 165-179.
doi: 10.1080/1468936031000136126. |
show all references
References:
[1] |
J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50. American Mathematical Society, Providence, RI, 1997.
doi: 10.1090/surv/050. |
[2] |
T. Bousch and O. Jenkinson,
Cohomology classes of dynamically non-negative $C^k$ functions, Inventiones Mathematicae, 148 (2002), 207-217.
doi: 10.1007/s002220100194. |
[3] |
F. M. Branco,
Subactions and maximizing measures for one-dimensional transformations with a critical point, Discrete Contin. Dyn. Syst., 17 (2007), 271-280.
doi: 10.3934/dcds.2007.17.271. |
[4] |
S. D. Branton,
Sub-actions for Young towers, Discrete and Continuous Dynamical Systems, 22 (2008), 541-556.
doi: 10.3934/dcds.2008.22.541. |
[5] |
G. Contreras, A. O. Lopes and Ph. Thieullen,
Lyapunov minimizing measures for expanding maps of the circle, Ergodic Theory Dynam. Systems, 21 (2001), 1379-1409.
doi: 10.1017/S0143385701001663. |
[6] |
E. Garibaldi, Ergodic Optimization in the Expanding Case: Concepts, Tools and Applications, SpringerBriefs in Mathematics, Springer, Cham, 2017.
doi: 10.1007/978-3-319-66643-3. |
[7] |
E. Garibaldi, A. O. Lopes and Ph. Thieullen,
On calibrated and separating sub-actions, Bull. Braz. Math. Soc. (N.S.), 40 (2009), 577-602.
doi: 10.1007/s00574-009-0028-6. |
[8] |
M. Holland,
Slowly mixing systems and intermittency maps, Ergodic Theory and Dynamical Systems, 25 (2005), 133-159.
doi: 10.1017/S0143385704000343. |
[9] |
O. Jenkinson,
Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197-224.
doi: 10.3934/dcds.2006.15.197. |
[10] |
O. Jenkinson,
Ergodic optimization in dynamical systems, Ergodic Theory and Dynamical Systems, 39 (2019), 2593-2618.
doi: 10.1017/etds.2017.142. |
[11] |
J. Karamata,
Sur un mode de croissance régulière. Théorèmes fondamentaux, Bull. Soc. Math. France, 61 (1933), 55-62.
|
[12] |
A. O. Lopes, V. A. Rosas and R. O. Ruggiero,
Cohomology and subcohomology problems for expansive, non Anosov geodesic flows, Discrete Contin. Dyn. Syst., 17 (2007), 403-422.
doi: 10.3934/dcds.2007.17.403. |
[13] |
A. O. Lopes and Ph. Thieullen, Sub-actions for Anosov diffeomorphisms. Geometric methods in dynamics. Ⅱ, Astérisque, (2003), 135–146. |
[14] |
A. O. Lopes and Ph. Thieullen,
Sub-actions for Anosov flows, Ergodic Theory and Dynamical Systems, 25 (2005), 605-628.
doi: 10.1017/S0143385704000732. |
[15] |
A. V. Medvedev,
On a concave differentiable majorant of a modulus of continuity, Real Anal. Exchange, 27 (2001/02), 123-129.
|
[16] |
I. D. Morris,
A sufficient condition for the subordination principle in ergodic optimization, Bulletin of the London Mathematical Society, 39 (2007), 214-220.
doi: 10.1112/blms/bdl030. |
[17] |
I. D. Morris,
The Mañé-Conze-Guivarc'h lemma for intermittent maps of the circle, Ergodc Theory and Dynamical Systems, 29 (2009), 1603-1611.
doi: 10.1017/S0143385708000837. |
[18] |
E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics, Vol. 508. Springer-Verlag, Berlin-New York, 1976. |
[19] |
R. R. Souza,
Sub-actions for weakly hyperbolic one-dimensional systems, Dynamical System, 18 (2003), 165-179.
doi: 10.1080/1468936031000136126. |
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