Abstract
Let \((X, \mathcal{B}, \mu, T)\) be a dynamical system where X is a compact metric space with Borel \(\sigma\)-algebra \(\mathcal{B}\), and \(\mu\) is a probability measure that is ergodic with respect to the homeomorphism \(T \colon X \to X\). We study the following differentiation problem: Given \(f \in C(X)\) and \(F_k \in \mathcal{B}\), where \(\mu(F_k) > 0\) and \(\mu(F_k) \to 0\), when can we say that
We establish some sufficient conditions for these sequences to converge.
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Assani, I., Young, A. Spatial-temporal differentiation theorems. Acta Math. Hungar. 168, 301–344 (2022). https://doi.org/10.1007/s10474-022-01276-5
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DOI: https://doi.org/10.1007/s10474-022-01276-5
Key words and phrases
- spatial-temporal differentiation
- dynamical system
- ergodic theorem
- uniquely ergodic system
- strong law of large numbers