Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spatial Temporal Differentiations   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's ergodic with respect to the homeomorphism $T : X \to
X$. 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 $$\lim_{k \to \infty} \frac{\int_{F_k} \left( \frac{1}{k} \sum_{i = 0}^{k
- 1} T^i f \right) \mathrm{d} \mu}{\mu(F_k)} = \int f \mathrm{d} \mu ? $$

20
The categories of this paper are: acc-phys
19
The categories of this paper are: eess
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
13
The categories of this paper are: quant-ph