TL;DR
This paper provides a precise formula for the H"older exponent of topological entropy in quadratic maps and analyzes the measure and dimension of parameter sets related to entropy values.
Contribution
It introduces a formula for the H"older exponent at almost every quadratic parameter and studies the Hausdorff dimension of parameter sets associated with entropy.
Findings
H"older exponent formula at almost every parameter
Most entropy values come from a set of parameters with Hausdorff dimension less than one
The set of parameters with high entropy has Hausdorff dimension less than one
Abstract
Milnor and Thurston's famous paper proved monotonicity of the topological entropy for the real quadratic family. Guckenheimer showed that it is H\"older continuous. We obtain a precise formula for the H\"older exponent at almost every quadratic parameter. Furthermore, the entropy of most parameters is proven to be in a set of Hausdorff dimension smaller than one, while most values of the entropy arise from a set of parameters of dimension smaller than one.
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