# American Institute of Mathematical Sciences

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

 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.

Citation: Eduardo Garibaldi, Irene Inoquio-Renteria. Dynamical obstruction to the existence of continuous sub-actions for interval maps with regularly varying property. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] M. Holland, Slowly mixing systems and intermittency maps, Ergodic Theory and Dynamical Systems, 25 (2005), 133-159.  doi: 10.1017/S0143385704000343.  Google Scholar [9] O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197-224.  doi: 10.3934/dcds.2006.15.197.  Google Scholar [10] O. Jenkinson, Ergodic optimization in dynamical systems, Ergodic Theory and Dynamical Systems, 39 (2019), 2593-2618.  doi: 10.1017/etds.2017.142.  Google Scholar [11] J. Karamata, Sur un mode de croissance régulière. Théorèmes fondamentaux, Bull. Soc. Math. France, 61 (1933), 55-62.   Google Scholar [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.  Google Scholar [13] A. O. Lopes and Ph. Thieullen, Sub-actions for Anosov diffeomorphisms. Geometric methods in dynamics. Ⅱ, Astérisque, (2003), 135–146.  Google Scholar [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.  Google Scholar [15] A. V. Medvedev, On a concave differentiable majorant of a modulus of continuity, Real Anal. Exchange, 27 (2001/02), 123-129.   Google Scholar [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.  Google Scholar [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.  Google Scholar [18] E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics, Vol. 508. Springer-Verlag, Berlin-New York, 1976.  Google Scholar [19] R. R. Souza, Sub-actions for weakly hyperbolic one-dimensional systems, Dynamical System, 18 (2003), 165-179.  doi: 10.1080/1468936031000136126.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] M. Holland, Slowly mixing systems and intermittency maps, Ergodic Theory and Dynamical Systems, 25 (2005), 133-159.  doi: 10.1017/S0143385704000343.  Google Scholar [9] O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst., 15 (2006), 197-224.  doi: 10.3934/dcds.2006.15.197.  Google Scholar [10] O. Jenkinson, Ergodic optimization in dynamical systems, Ergodic Theory and Dynamical Systems, 39 (2019), 2593-2618.  doi: 10.1017/etds.2017.142.  Google Scholar [11] J. Karamata, Sur un mode de croissance régulière. Théorèmes fondamentaux, Bull. Soc. Math. France, 61 (1933), 55-62.   Google Scholar [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.  Google Scholar [13] A. O. Lopes and Ph. Thieullen, Sub-actions for Anosov diffeomorphisms. Geometric methods in dynamics. Ⅱ, Astérisque, (2003), 135–146.  Google Scholar [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.  Google Scholar [15] A. V. Medvedev, On a concave differentiable majorant of a modulus of continuity, Real Anal. Exchange, 27 (2001/02), 123-129.   Google Scholar [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.  Google Scholar [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.  Google Scholar [18] E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics, Vol. 508. Springer-Verlag, Berlin-New York, 1976.  Google Scholar [19] R. R. Souza, Sub-actions for weakly hyperbolic one-dimensional systems, Dynamical System, 18 (2003), 165-179.  doi: 10.1080/1468936031000136126.  Google Scholar
$d^{-}: = d(w_{n_k},w_{n_{k-1}})$, $d^{+}: = d(w_{n_k},w_{n_{k+1}})$
 [1] Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021018 [2] Qiang Long, Xue Wu, Changzhi Wu. Non-dominated sorting methods for multi-objective optimization: Review and numerical comparison. Journal of Industrial & Management Optimization, 2021, 17 (2) : 1001-1023. doi: 10.3934/jimo.2020009 [3] Hedy Attouch, Aïcha Balhag, Zaki Chbani, Hassan Riahi. Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021010 [4] Andreas Koutsogiannis. Multiple ergodic averages for tempered functions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1177-1205. doi: 10.3934/dcds.2020314 [5] Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269 [6] Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325 [7] Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320 [8] Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $\mathbb{R}^{N}$ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020376 [9] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [10] Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020409 [11] Xiaoming Wang. Upper semi-continuity of stationary statistical properties of dissipative systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 521-540. doi: 10.3934/dcds.2009.23.521 [12] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [13] Claudio Bonanno, Marco Lenci. Pomeau-Manneville maps are global-local mixing. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1051-1069. doi: 10.3934/dcds.2020309 [14] Aihua Fan, Jörg Schmeling, Weixiao Shen. $L^\infty$-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363 [15] Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266 [16] Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395 [17] Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $p$-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293 [18] Wenqin Zhang, Zhengchun Zhou, Udaya Parampalli, Vladimir Sidorenko. Capacity-achieving private information retrieval scheme with a smaller sub-packetization. Advances in Mathematics of Communications, 2021, 15 (2) : 347-363. doi: 10.3934/amc.2020070 [19] Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050 [20] Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from ${{\mathbb{R}}^{2}}$ to ${{\mathbb{S}}^{2}}$. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3215-3233. doi: 10.3934/dcds.2019228

2019 Impact Factor: 1.338