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Abstract Chaotic dynamical systems may be characterised by a positive Lyapunov exponent, which measures the exponential rate of separation of nearby trajectories. However in a wide range of so-called weakly chaotic systems, the separation of nearby trajectories is sub-exponential, for example stretched exponential, in time; and therefore in such cases the Lyapunov exponent vanishes. When a hole is introduced in chaotic systems, the Lyapunov exponent on the system’s fractal repeller can be related to the generation of entropy and the escape rate from the system via the escape rate formalism, but no suitable generalisation exists to weakly chaotic systems. In this work we show that in a paradigmatic one-dimensional weakly chaotic iterated map, the Pomeau-Manneville map, the generation of generalised Lyapunov stretching is completely suppressed in the presence of a hole. These results are shown based on numerical evidence, and explained with a fully analytic stochastic model.
Brevitt et al. (Sun,) studied this question.