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April  2020, 40(4): 2315-2333. doi: 10.3934/dcds.2020115

Dynamical obstruction to the existence of continuous sub-actions for interval maps with regularly varying property

Eduardo Garibaldi 1, and  Irene Inoquio-Renteria 2,

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

Received  June 2019 Published  January 2020

Fund Project: CNPq grant 304792/2017-9
FONDECYT 11130341 and BCH-CONICYT grant 74170014.

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.

Keywords: Ergodic optimization, sub-actions, modulus of continuity, non-uniformly expanding dynamics, intermittent maps.
Mathematics Subject Classification: 26A12, 37E05, 49L25, 49N15.
Citation: Eduardo Garibaldi, Irene Inoquio-Renteria. Dynamical obstruction to the existence of continuous sub-actions for interval maps with regularly varying property. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2315-2333. doi: 10.3934/dcds.2020115
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.

Figure 1.  $ d^{-}: = d(w_{n_k},w_{n_{k-1}}) $, $ d^{+}: = d(w_{n_k},w_{n_{k+1}}) $
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