Optimal Orbits of Hyperbolic Systems

Guo Cheng Yuan and Brian R. Hunt

Nonlinearity 12 (1999), 1207-1224.

Online abstract and download information

Given a dynamical system and a function $f$ from the state space to the real numbers, an optimal orbit for $f$ is an orbit over which the time average of $f$ is maximal. In this paper we consider some basic mathematical properties of optimal orbits: existence, sensitivity to perturbations of $f$, and approximability by periodic orbits with low period. For hyperbolic systems, we conjecture that for (topologically) generic smooth functions, there exists an optimal periodic orbit. In support of this conjecture, we prove that optimal periodic orbits are insensitive to small $C^1$ perturbations of $f$, while the optimality of a non-periodic orbit can be destroyed by arbitrarily small $C^1$ perturbations. In case there is no optimal periodic orbit for a given $f$, we discuss the question of how fast the maximum average over orbits of period at most $p$ must converge to the optimal average, as $p$ increases.

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