American Institute of Mathematical Sciences

Advanced
July  2008, 20(3): 713-724. doi: 10.3934/dcds.2008.20.713

A continuous Bowen-Mane type phenomenon

Esteban Muñoz-Young 1, , Andrés Navas 2, , Enrique Pujals 3, and  Carlos H. Vásquez 1,

1. 

Departamento de Matemática, Universidad Católica del Norte, Av. Angamos 0610, Antofagasta, Chile, Chile
2. 

Departamento de Matemática, Fac. de Ciencias, Universidad de Santiago, Alameda 3363, Santiago, Chile
3. 

Instituto Nacional de Matemática Pura e Aplicada, IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil

Received  January 2007 Revised  August 2007 Published  December 2007

In this work we exhibit a one-parameter family of $C^1$-diffeomorphisms $F_\alpha$ of the 2-sphere, where $\alpha>1$, such that the equator $\S^1$ is an attracting set for every $F_\alpha$ and $F_\alpha|_{\S^1}$ is the identity. For $\alpha>2$ the Lebesgue measure on the equator is a non ergodic physical measure having uncountably many ergodic components. On the other hand, for $1<\alpha\leq 2$ there is no physical measure for $F_\alpha$. If $\alpha<2$ this follows directly from the fact that the $\omega$-limit of almost every point is a single point on the equator (and the basin of each of these points has zero Lebesgue measure). This is no longer true for $\alpha=2$, and the non existence of physical measure in this critical case is a more subtle issue.
Keywords: ergodic components., Physical measures.
Mathematics Subject Classification: Primary: 37C40. Secondary: 37A1.
Citation: Esteban Muñoz-Young, Andrés Navas, Enrique Pujals, Carlos H. Vásquez. A continuous Bowen-Mane type phenomenon. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 713-724. doi: 10.3934/dcds.2008.20.713
[1]

Vítor Araújo, Ali Tahzibi. Physical measures at the boundary of hyperbolic maps. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 849-876. doi: 10.3934/dcds.2008.20.849
[2]

Vítor Araújo. Semicontinuity of entropy, existence of equilibrium states and continuity of physical measures. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 371-386. doi: 10.3934/dcds.2007.17.371
[3]

Xavier Bressaud. Expanding interval maps with intermittent behaviour, physical measures and time scales. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 517-546. doi: 10.3934/dcds.2004.11.517
[4]

Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829
[5]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
[6]

Radu Saghin. On the number of ergodic minimizing measures for Lagrangian flows. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 501-507. doi: 10.3934/dcds.2007.17.501
[7]

Wen Huang, Leiye Xu, Shengnan Xu. Ergodic measures of intermediate entropy for affine transformations of nilmanifolds. Electronic Research Archive, 2021, 29 (4) : 2819-2827. doi: 10.3934/era.2021015
[8]

Marzie Zaj, Abbas Fakhari, Fatemeh Helen Ghane, Azam Ehsani. Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1777-1807. doi: 10.3934/dcds.2018073
[9]

Jialu Fang, Yongluo Cao, Yun Zhao. Measure theoretic pressure and dimension formula for non-ergodic measures. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2767-2789. doi: 10.3934/dcds.2020149
[10]

Andrew D. Lewis. The physical foundations of geometric mechanics. Journal of Geometric Mechanics, 2017, 9 (4) : 487-574. doi: 10.3934/jgm.2017019
[11]

Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929
[12]

Nils Raabe, Claus Weihs. Physical statistical modelling of bending vibrations. Conference Publications, 2011, 2011 (Special) : 1214-1223. doi: 10.3934/proc.2011.2011.1214
[13]

Oliver Jenkinson. Ergodic Optimization. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 197-224. doi: 10.3934/dcds.2006.15.197
[14]

Marta Lewicka, Piotr B. Mucha. A local existence result for a system of viscoelasticity with physical viscosity. Evolution Equations and Control Theory, 2013, 2 (2) : 337-353. doi: 10.3934/eect.2013.2.337
[15]

Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513
[16]

Xiaodong Liu. The factorization method for scatterers with different physical properties. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 563-577. doi: 10.3934/dcdss.2015.8.563
[17]

Nikolai Chernov. The work of Dmitry Dolgopyat on physical models with moving particles. Journal of Modern Dynamics, 2010, 4 (2) : 243-255. doi: 10.3934/jmd.2010.4.243
[18]

Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757
[19]

Roy Adler, Bruce Kitchens, Michael Shub. Stably ergodic skew products. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 349-350. doi: 10.3934/dcds.1996.2.349
[20]

Alexandre I. Danilenko, Mariusz Lemańczyk. Spectral multiplicities for ergodic flows. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4271-4289. doi: 10.3934/dcds.2013.33.4271

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy