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

Eduardo Garibaldi; Irene Inoquio-Renteria

doi:10.3934/dcds.2020115

Article Contents

2020, Volume 40, Issue 4: 2315-2333. Doi: 10.3934/dcds.2020115

This issue Previous Article A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies Next Article Necessary conditions for tiling finitely generated amenable groups

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 online: April 2020

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

Abstract / Introduction Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 26A12, 37E05, 49L25, 49N15.

Citation:

Full Text(HTML)

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

Download: Full-size image PowerPoint slide

Related Papers

Cited by

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.