June  2008, 21(2): 367-392. doi: 10.3934/dcds.2008.21.367

Entropy formula for endomorphisms: Relations between entropy, exponents and dimension

1. 

School of Mathematical Sciences, Peking University, Beijing 100871, China

2. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  February 2007 Revised  December 2007 Published  March 2008

We present an entropy formula of Ledrappier-Young type for invariant measures (maybe non-SRB) of $ C^2 $ endomorphisms (maybe non-invertible and with singularities) on a compact manifold via their inverse limit spaces. This result may be considered as the most general form of entropy formula for a deterministic system with an invariant measure, and a preliminary step to Eckmann-Ruelle conjecture. As an important application, we have proved the exact dimensionality of ergodic measures invariant under expanding maps.
Citation: Min Qian, Jian-Sheng Xie. Entropy formula for endomorphisms: Relations between entropy, exponents and dimension. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 367-392. doi: 10.3934/dcds.2008.21.367
[1]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215

[2]

Alejo Barrio Blaya, Víctor Jiménez López. On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 433-466. doi: 10.3934/dcds.2012.32.433

[3]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91

[4]

Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957

[5]

Zoltán Buczolich, Gabriella Keszthelyi. Isentropes and Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 1989-2009. doi: 10.3934/dcds.2020102

[6]

Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107

[7]

Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433

[8]

Fumihiko Nakamura, Yushi Nakano, Hisayoshi Toyokawa. Lyapunov exponents for random maps. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022058

[9]

Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004

[10]

Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287

[11]

Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341

[12]

Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192

[13]

Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861

[14]

Lucas Backes, Aaron Brown, Clark Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies. Journal of Modern Dynamics, 2018, 12: 223-260. doi: 10.3934/jmd.2018009

[15]

Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619

[16]

Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098

[17]

Jianyu Chen. On essential coexistence of zero and nonzero Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4149-4170. doi: 10.3934/dcds.2012.32.4149

[18]

Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187

[19]

Andrey Gogolev, Ali Tahzibi. Center Lyapunov exponents in partially hyperbolic dynamics. Journal of Modern Dynamics, 2014, 8 (3&4) : 549-576. doi: 10.3934/jmd.2014.8.549

[20]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]