American Institute of Mathematical Sciences

Advanced
November  2009, 24(4): 1409-1435. doi: 10.3934/dcds.2009.24.1409

Variational principles of pressure

Guohua Zhang 1,

1. 

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

Received  May 2008 Revised  December 2008 Published  May 2009

Given a topological dynamical system $(X, T)$, a Borel cover $\mathcal{U}$ of $X$ and a sub-additive sequence $\mathcal{F}$ of real-valued continuous functions on $X$, two notions of measure-theoretical pressure $P_\mu^- (T, \mathcal{U}, \mathcal{F})$ and $P_\mu^+ (T, \mathcal{U}, \mathcal{F})$ for an invariant Borel probability measure $\mu$ are introduced. When $\mathcal{U}$ is an open cover, a local variational principle between topological and measure-theoretical pressure is proved; it is also established the upper semi-continuity of P•+$(T, \mathcal{U}, \mathcal{F})$ and P•+$(T, \mathcal{U}, \mathcal{F})$ on the space of all invariant Borel probability measures. The notions of measure-theoretical pressure $P_\mu^- (T, X, \mathcal{F})$ and $P_\mu^+ (T, X, \mathcal{F})$ for an invariant Borel probability measure $\mu$ are also introduced. A global variational principle between topological and measure-theoretical pressure is also obtained.
Keywords: upper semi-continuity, measure-theoretical pressure, variational principle., topological pressure, sub-additive sequence.
Mathematics Subject Classification: Primary: 37A05, 37A3.
Citation: Guohua Zhang. Variational principles of pressure. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1409-1435. doi: 10.3934/dcds.2009.24.1409
[1]

Xiaojun Huang, Zhiqiang Li, Yunhua Zhou. A variational principle of topological pressure on subsets for amenable group actions. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2687-2703. doi: 10.3934/dcds.2020146
[2]

Xiaoming Wang. Upper semi-continuity of stationary statistical properties of dissipative systems. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 521-540. doi: 10.3934/dcds.2009.23.521
[3]

Marc Rauch. Variational principles for the topological pressure of measurable potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 367-394. doi: 10.3934/dcdss.2017018
[4]

Yongluo Cao, De-Jun Feng, Wen Huang. The thermodynamic formalism for sub-additive potentials. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 639-657. doi: 10.3934/dcds.2008.20.639
[5]

Pengyu Chen, Xuping Zhang. Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4325-4357. doi: 10.3934/dcdsb.2020290
[6]

Wen-Chiao Cheng, Yun Zhao, Yongluo Cao. Pressures for asymptotically sub-additive potentials under a mistake function. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 487-497. doi: 10.3934/dcds.2012.32.487
[7]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[8]

Xueting Tian. Topological pressure for the completely irregular set of birkhoff averages. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2745-2763. doi: 10.3934/dcds.2017118
[9]

Ugur G. Abdulla. Wiener's criterion at $\infty$ for the heat equation and its measure-theoretical counterpart. Electronic Research Announcements, 2008, 15: 44-51. doi: 10.3934/era.2008.15.44
[10]

Jialu Fang, Yongluo Cao, Yun Zhao. Measure theoretic pressure and dimension formula for non-ergodic measures. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2767-2789. doi: 10.3934/dcds.2020149
[11]

Kyungwoo Song, Yuxi Zheng. Semi-hyperbolic patches of solutions of the pressure gradient system. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1365-1380. doi: 10.3934/dcds.2009.24.1365
[12]

Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1091-1102. doi: 10.3934/cpaa.2021008
[13]

Ruxandra Stavre. Optimization of the blood pressure with the control in coefficients. Evolution Equations & Control Theory, 2020, 9 (1) : 131-151. doi: 10.3934/eect.2020019
[14]

M. Bulíček, Josef Málek, Dalibor Pražák. On the dimension of the attractor for a class of fluids with pressure dependent viscosities. Communications on Pure & Applied Analysis, 2005, 4 (4) : 805-822. doi: 10.3934/cpaa.2005.4.805
[15]

Vishal Vasan, Katie Oliveras. Pressure beneath a traveling wave with constant vorticity. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3219-3239. doi: 10.3934/dcds.2014.34.3219
[16]

Alessandro Bertuzzi, Antonio Fasano, Alberto Gandolfi, Carmela Sinisgalli. Interstitial Pressure And Fluid Motion In Tumor Cords. Mathematical Biosciences & Engineering, 2005, 2 (3) : 445-460. doi: 10.3934/mbe.2005.2.445
[17]

Baojun Song, Melissa Castillo-Garsow, Karen R. Ríos-Soto, Marcin Mejran, Leilani Henso, Carlos Castillo-Chavez. Raves, clubs and ecstasy: the impact of peer pressure. Mathematical Biosciences & Engineering, 2006, 3 (1) : 249-266. doi: 10.3934/mbe.2006.3.249
[18]

De-Jun Feng, Antti Käenmäki. Equilibrium states of the pressure function for products of matrices. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 699-708. doi: 10.3934/dcds.2011.30.699
[19]

Ricardo Almeida, Agnieszka B. Malinowska. Fractional variational principle of Herglotz. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2367-2381. doi: 10.3934/dcdsb.2014.19.2367
[20]

Rafael Alcaraz Barrera. Topological and ergodic properties of symmetric sub-shifts. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4459-4486. doi: 10.3934/dcds.2014.34.4459

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (18)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences