### Abstract

This paper reviews some basic mathematical results on Lyapunov exponents, one of the most fundamental concepts in dynamical systems. The first few sections contain some very general results in nonuniform hyperbolic theory. We consider (f, μ), where f is an arbitrary dynamical system and μ is an arbitrary invariant measure, and discuss relations between Lyapunov exponents and several dynamical quantities of interest, including entropy, fractal dimension and rates of escape. The second half of this review focuses on observable chaos, characterized by positive Lyapunov exponents on positive Lebesgue measure sets. Much attention is given to SRB measures, a very special kind of invariant measures that offer a way to understand observable chaos in dissipative systems. Paradoxical as it may seem, given a concrete system, it is generally impossible to determine with mathematical certainty if it has observable chaos unless strong geometric conditions are satisfied; case studies will be discussed. The final section is on noisy or stochastically perturbed systems, for which we present a dynamical picture simpler than that for purely deterministic systems. In this short review, we have elected to limit ourselves to finite-dimensional systems and to discrete time. The phase space, which is assumed to be or a Riemannian manifold, is denoted by M throughout. The Lebesgue or the Riemannian measure on M is denoted by , and the dynamics are generated by iterating a self-map of M, written f: M. For flows, the reviewed results are applicable to time-t maps and Poincaré return maps to cross-sections. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Lyapunov analysis: from dynamical systems theory to applications'.

Original language | English (US) |
---|---|

Article number | 254001 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 46 |

Issue number | 25 |

DOIs | |

State | Published - Jun 28 2013 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Modeling and Simulation
- Statistics and Probability

### Cite this

**Mathematical theory of Lyapunov exponents.** / Young, Lai-Sang.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 46, no. 25, 254001. https://doi.org/10.1088/1751-8113/46/25/254001

}

TY - JOUR

T1 - Mathematical theory of Lyapunov exponents

AU - Young, Lai-Sang

PY - 2013/6/28

Y1 - 2013/6/28

N2 - This paper reviews some basic mathematical results on Lyapunov exponents, one of the most fundamental concepts in dynamical systems. The first few sections contain some very general results in nonuniform hyperbolic theory. We consider (f, μ), where f is an arbitrary dynamical system and μ is an arbitrary invariant measure, and discuss relations between Lyapunov exponents and several dynamical quantities of interest, including entropy, fractal dimension and rates of escape. The second half of this review focuses on observable chaos, characterized by positive Lyapunov exponents on positive Lebesgue measure sets. Much attention is given to SRB measures, a very special kind of invariant measures that offer a way to understand observable chaos in dissipative systems. Paradoxical as it may seem, given a concrete system, it is generally impossible to determine with mathematical certainty if it has observable chaos unless strong geometric conditions are satisfied; case studies will be discussed. The final section is on noisy or stochastically perturbed systems, for which we present a dynamical picture simpler than that for purely deterministic systems. In this short review, we have elected to limit ourselves to finite-dimensional systems and to discrete time. The phase space, which is assumed to be or a Riemannian manifold, is denoted by M throughout. The Lebesgue or the Riemannian measure on M is denoted by , and the dynamics are generated by iterating a self-map of M, written f: M. For flows, the reviewed results are applicable to time-t maps and Poincaré return maps to cross-sections. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Lyapunov analysis: from dynamical systems theory to applications'.

AB - This paper reviews some basic mathematical results on Lyapunov exponents, one of the most fundamental concepts in dynamical systems. The first few sections contain some very general results in nonuniform hyperbolic theory. We consider (f, μ), where f is an arbitrary dynamical system and μ is an arbitrary invariant measure, and discuss relations between Lyapunov exponents and several dynamical quantities of interest, including entropy, fractal dimension and rates of escape. The second half of this review focuses on observable chaos, characterized by positive Lyapunov exponents on positive Lebesgue measure sets. Much attention is given to SRB measures, a very special kind of invariant measures that offer a way to understand observable chaos in dissipative systems. Paradoxical as it may seem, given a concrete system, it is generally impossible to determine with mathematical certainty if it has observable chaos unless strong geometric conditions are satisfied; case studies will be discussed. The final section is on noisy or stochastically perturbed systems, for which we present a dynamical picture simpler than that for purely deterministic systems. In this short review, we have elected to limit ourselves to finite-dimensional systems and to discrete time. The phase space, which is assumed to be or a Riemannian manifold, is denoted by M throughout. The Lebesgue or the Riemannian measure on M is denoted by , and the dynamics are generated by iterating a self-map of M, written f: M. For flows, the reviewed results are applicable to time-t maps and Poincaré return maps to cross-sections. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Lyapunov analysis: from dynamical systems theory to applications'.

UR - http://www.scopus.com/inward/record.url?scp=84878799013&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878799013&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/46/25/254001

DO - 10.1088/1751-8113/46/25/254001

M3 - Article

VL - 46

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 25

M1 - 254001

ER -