Scale-local dimensions of strange attractors
J.G. Reid, T.A. Trainor
TL;DR
This paper compares limit-based and scale-local dimensions of strange attractors, revealing detailed scale-dependent behavior and discussing practical issues in their estimation for complex distributions.
Contribution
It introduces a detailed comparison between limit-based and scale-local dimensions, highlighting their differences and practical considerations for analyzing complex distributions.
Findings
Scale-local dimensions show rich detail across scales.
Limit-based dimensions are averages over semi-infinite scales.
Practical dimension analysis faces definitional challenges on bounded scales.
Abstract
We compare limit-based and scale-local dimensions of complex distributions, particularly for a strange attractor of the Henon map. Scale-local dimensions as distributions on scale are seen to exhibit a wealth of detail. Limit-based dimensions are shown to be averages of scale-local dimensions, in principle over a semi-infinite scale interval. We identify some critical questions of definition for practical dimension analysis of arbitrary distributions on bounded scale intervals.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Geometry and complex manifolds · Stochastic processes and statistical mechanics