American Institute of Mathematical Sciences

Advanced
April  2007, 1(2): 205-253. doi: 10.3934/jmd.2007.1.205

Construction of ergodic cocycles that are fundamental solutions to linear systems of a special form

Mahesh G. Nerurkar 1, and  Héctor J. Sussmann 2,

1. 

Department of Mathematics, Rutgers University, Camden NJ 08102, United States
2. 

Department of Mathematics, Rutgers University, Piscataway, NJ 08854, United States

Received  May 2006 Published  January 2007

If $T=\{T_t\}_{t\in\mathbb R}$ is an aperiodic measure-preserving jointly continuous flow on a compact metric space $\Omega$ endowed with a Borel probability measure $m$, and $G$ is a compact Lie group with Lie algebra $L$, then to each continuous map $A: \Omega \to L$ associate the solution $\Omega\times\mathbb R$ ∋ $(\omega,t)\mapsto X^A(\omega,t)\in G$ of the family of time-dependent initial-value problems $X'(t) = A(T_t\omega)X(t)$, $X(0) =$ identity, $X(t) \in G$ for $\omega\in \Omega$. The corresponding skew-product flow $T^A=\{T_t^A\}_{t\in\mathbb R}$ on $G\times\Omega$ is then defined by letting $T^A_t(g,\omega ) = (X^A(\omega ,t)g,T_t\omega)$ for $(g,\omega)\in G\times\Omega$, $t\in\mathbb R$. The flow $T^A$ is measure-preserving on $(G\times \Omega,\nu_{_G}\otimes m)$ (where $\nu_{_G}$ is normalized Haar measure on $G$) and jointly continuous. For a given closed convex subset $S$ of $L$, we study the set $C_{erg}(\Omega ,S)$ of all continuous maps $A: \Omega\to S$ for which the flow $T^A$ is ergodic. We develop a new technique to determine a necessary and sufficient condition for the set $C_{erg}(\Omega ,S)$ to be residual. Since the dimension of $S$ can be much smaller than that of $L$, our result proves that ergodicity is typical even within very "thin'' classes of cocycles. This covers a number of differential equations arising in mathematical physics, and in particular applies to the widely studied example of the Rabi oscillator.
Keywords: cocycles, strong accessibility property., linear differential systems, ergodicity.
Mathematics Subject Classification: Primary: 34F05; Secondary: 37H9.
Citation: Mahesh G. Nerurkar, Héctor J. Sussmann. Construction of ergodic cocycles that are fundamental solutions to linear systems of a special form. Journal of Modern Dynamics, 2007, 1 (2) : 205-253. doi: 10.3934/jmd.2007.1.205
[1]

Boris Kalinin, Victoria Sadovskaya. Linear cocycles over hyperbolic systems and criteria of conformality. Journal of Modern Dynamics, 2010, 4 (3) : 419-441. doi: 10.3934/jmd.2010.4.419
[2]

David Ralston, Serge Troubetzkoy. Ergodicity of certain cocycles over certain interval exchanges. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2523-2529. doi: 10.3934/dcds.2013.33.2523
[3]

Zbigniew Bartosiewicz, Ülle Kotta, Maris Tőnso, Małgorzata Wyrwas. Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales. Mathematical Control & Related Fields, 2016, 6 (2) : 217-250. doi: 10.3934/mcrf.2016002
[4]

Yong He. Switching controls for linear stochastic differential systems. Mathematical Control & Related Fields, 2020, 10 (2) : 443-454. doi: 10.3934/mcrf.2020005
[5]

Mikhail Krastanov, Michael Malisoff, Peter Wolenski. On the strong invariance property for non-Lipschitz dynamics. Communications on Pure & Applied Analysis, 2006, 5 (1) : 107-124. doi: 10.3934/cpaa.2006.5.107
[6]

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
[7]

Priyanjana M. N. Dharmawardane. Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity. Conference Publications, 2013, 2013 (special) : 197-206. doi: 10.3934/proc.2013.2013.197
[8]

Dmitry Dolgopyat. The work of Federico Rodriguez Hertz on ergodicity of dynamical systems. Journal of Modern Dynamics, 2016, 10: 175-189. doi: 10.3934/jmd.2016.10.175
[9]

Jaume Llibre, Lucyjane de A. S. Menezes. On the limit cycles of a class of discontinuous piecewise linear differential systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1835-1858. doi: 10.3934/dcdsb.2020005
[10]

Kai Du, Jianhui Huang, Zhen Wu. Linear quadratic mean-field-game of backward stochastic differential systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 653-678. doi: 10.3934/mcrf.2018028
[11]

Shimin Li, Jaume Llibre. On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5885-5901. doi: 10.3934/dcdsb.2019111
[12]

Elena K. Kostousova. State estimation for linear impulsive differential systems through polyhedral techniques. Conference Publications, 2009, 2009 (Special) : 466-475. doi: 10.3934/proc.2009.2009.466
[13]

Zdzisław Brzeźniak, Paul André Razafimandimby. Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1051-1077. doi: 10.3934/dcdsb.2016.21.1051
[14]

Makoto Mori. Higher order mixing property of piecewise linear transformations. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 915-934. doi: 10.3934/dcds.2000.6.915
[15]

Michael Röckner, Jiyong Shin, Gerald Trutnau. Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3219-3237. doi: 10.3934/dcdsb.2016095
[16]

Jana Rodriguez Hertz, Carlos H. Vásquez. Structure of accessibility classes. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4653-4664. doi: 10.3934/dcds.2020196
[17]

Jinjun Li, Min Wu. Generic property of irregular sets in systems satisfying the specification property. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 635-645. doi: 10.3934/dcds.2014.34.635
[18]

Pablo G. Barrientos, Abbas Fakhari. Ergodicity of non-autonomous discrete systems with non-uniform expansion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1361-1382. doi: 10.3934/dcdsb.2019231
[19]

Victoria Sadovskaya. Fiber bunching and cohomology for Banach cocycles over hyperbolic systems. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4959-4972. doi: 10.3934/dcds.2017213
[20]

Victoria Sadovskaya. Cohomology of $GL(2,\mathbb{R})$-valued cocycles over hyperbolic systems. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2085-2104. doi: 10.3934/dcds.2013.33.2085

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences