Emergence via non-existence of averages

Abstract

Inspired by a recent work by Berger, we introduce the concept of pointwise emergence. This concept provides with a new quantitative perspective into the study of non-existence of averages for dynamical systems. We show that high pointwise emergence on a large set appears for abundant dynamical systems: Any continuous maps on a compact metric space with the specification property have super-polynomial pointwise emergence on a residual subset of the state space. Furthermore, there is a dense subset of any Newhouse open set each element of which has super-polynomial pointwise emergence on a positive Lebesgue measure subset of the state space.

25 pages. Updated introduction, reorganised the paper, strengthened main theorem. Accepted for publication in Advances in Mathematics

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• Historic behavior in non-hyperbolic homoclinic classes
• Infinitely Many Stochastically Stable Attractors
• Complexities of differentiable dynamical systems
• Empirical measures of partially hyperbolic attractors
• Dirac physical measures on saddle-type fixed points
• Non-statistical rational maps

Cited by in corpus (5)

• High pointwise emergence and Katok's conjecture for systems with non-uniform structure
• Abundance of observable Lyapunov irregular sets
• Highly irregular orbits for subshifts of finite type: large intersections and emergence
• Takens' Last Problem and strong pluripotency
• Topological and metric emergence of continuous maps