The geometry of Baire spaces

We introduce the concept of Baire embeddings and we classify them up to C1+epsilon conjugacies. We show that two such embeddings are C1+epsilon-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 Holder equivalent IFS-like Baire embeddings are C1+epsilon 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.