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In thermodynamic formalism for dynamical systems, a Hölder continuous potential function $\phi$ on a countable state topological Markov chain $(X,\sigma)$ is either positive recurrent, null recurrent, or transient. These correspond to the three possibilities for equilibrium states (shift-invariant measures maximising the quantity $h(\mu) + \int_X \phi\,d\mu$): existence of a finite equilibrium state is equivalent to positive recurrence; null recurrence is the boundary case where the equilibrium state becomes $\sigma$-finite but not finite, and transience is the case where there is no equilibrium state (all the weight has gone to infinity).

These can be characterised in terms of a particular sequence $a_n>0$: positive recurrence is equivalent to $\limsup a_n > 0$, null recurrence is equivalent to $a_n\to 0$ and $\sum a_n=\infty$, and transience is equivalent to $\sum a_n<\infty$. I imagine this trichotomy for sequences appears in other places as well.

Edit: It's worth mentioning that this trichotomy is also true for random walks on directed graphs (weighted or unweighted) -- historically I believe this is where it was first studied and where the terminology came from, but as a dynamicist I more immediately think of the interpretation above. In this setting the interpretations are as follows:

• Positive recurrent -- with probability 1, a random walk returns to where it started, and the expected return time is finite.
• Null recurrent -- the walk returns to the starting position with probability 1, but the expected return time is infinite.
• Transient -- with probability 1, the walk never returns to its starting position.
In thermodynamic formalism for dynamical systems, a Hölder continuous potential function $\phi$ on a countable state topological Markov chain $(X,\sigma)$ is either positive recurrent, null recurrent, or transient. These correspond to the three possibilities for equilibrium states (shift-invariant measures maximising the quantity $h(\mu) + \int_X \phi\,d\mu$): existence of a finite equilibrium state is equivalent to positive recurrence; null recurrence is the boundary case where the equilibrium state becomes $\sigma$-finite but not finite, and transience is the case where there is no equilibrium state (all the weight has gone to infinity).
These can be characterised in terms of a particular sequence $a_n>0$: positive recurrence is equivalent to $\limsup a_n > 0$, null recurrence is equivalent to $a_n\to 0$ and $\sum a_n=\infty$, and transience is equivalent to $\sum a_n<\infty$. I imagine this trichotomy for sequences appears in other places as well.