The geometry of Baire spaces.

BAIRE spaces; GEOMETRY; MATHEMATICAL analysis; MATHEMATICS; ALGEBRA; EMBEDDINGS (Mathematics); GEOMETRIC rigidity; GIBBS phenomenon

We introduce the concept of Baire embeddings and we classify them up to C 1+ϵ conjugacies. We show that two such embeddings are C 1+ϵ-equivalent if and only if they have exponentially equivalent geometries. Next, we introduce the class of iterated function system (IFS)-like Baire embeddings and we also show that two Hölder equivalent IFS-like Baire embeddings are C 1+ϵ conjugate if and only if their scaling functions are the same. In the remaining sections, we introduce metric scaling functions and we show that the logarithm of such a metric scaling function and the logarithm of Sullivan's scaling function multiplied by the Hausdorff dimension of the Baire embedding are cohomologous up to a constant. This permits us to conclude that if the Bowen measures coincide for two IFS-like Baire embeddings, then the embeddings are bi-Lipschitz conjugate. [ABSTRACT FROM AUTHOR]

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