In ergodic theory, Kac's lemma, demonstrated by mathematician Mark Kac in 1947, is a lemma stating that in a finite measure space the orbit of almost all the points contained in a set A {\displaystyle A} of such space, whose measure is μ ( A ) {\displaystyle \mu (A)} , return to A {\displaystyle A} within an average time inversely proportional to μ ( A ) {\displaystyle \mu (A)} . The lemma extends what is stated by Poincaré recurrence theorem, in which it is shown that the points return in A {\displaystyle A} infinite times.
Intuition In simple terms, it states that for a measure-preserving system, the expected return time to a given subset of the space is inversely proportional to the measure (or size) of that subset. If you have a space with a total measure of 1 (a probability space), and a subset A {\displaystyle A} , the average number of steps it takes for points starting in A {\displaystyle A} to return to A {\displaystyle A} for the first time is exactly 1 μ ( A ) {\displaystyle {\frac {1}{\mu (A)}}} .
Formal statement
Classic version Sources:
Kakutani-Rokhlin Skyscraper Instead of focusing purely on the geometric/heuristic "shifting of sets" (as in the classic proof), the Rubinstein-Salzedo approach leverages the transfer operator (or pull-back under T {\displaystyle T} ) and indicator functions to keep strict track of the arithmetic of the returns.
Comparison: Rubinstein-Salzedo vs. The Classic Proof While both proofs arrive at the same sum, they approach the bookkeeping of the partition from slightly different angles.
The Rubinstein-Salzedo proof makes the "disjoint skyscraper" structure of the space highly visible. By defining the explicit double-indexed sets B i , j {\displaystyle B_{i,j}} , it creates a physical mental model known as a Kakutani-Rokhlin Skyscraper. In this Skyscraper:
The set A {\displaystyle A} is the "floor" of the building. partition the base by the return time: A k = { x ∈ A : n A ( x ) = k } {\displaystyle A_{k}=\{x\in A:n_{A}(x)=k\}}
Each A i {\displaystyle A_{i}} is a tower base. The sets B i , j = T j ( A i ) {\displaystyle B_{i,j}=T^{j}(A_{i})} are the individual floors stacked above A i {\displaystyle A_{i}} . Applying T {\displaystyle T} simply moves you up one floor in the building, until you hit the top floor B i , i − 1 {\displaystyle B_{i,i-1}} and get mapped back down to the ground floor A {\displaystyle A} . tower k is given by A k ∪ T ( A k ) ∪ ⋯ T ( k − 1 ) ( A k ) {\displaystyle A_{k}\cup T(A_{k})\cup \cdots T^{(k-1)}(A_{k})} and the measure is given by ∑ i = 0 k − 1 μ ( T i ( A k ) ) = k μ ( A k ) {\displaystyle \sum _{i=0}^{k-1}\mu (T^{i}(A_{k}))=k\mu (A_{k})}
each floor of each tower is disjoint T j ( A k ) ∩ T m ( A k ) = ∅ {\displaystyle T^{j}(A_{k})\cap T^{m}(A_{k})=\emptyset } and same is true across towers T i ( A m ) ∩ T j ( A k ) = ∅ {\displaystyle T^{i}(A_{m})\cap T^{j}(A_{k})=\emptyset }
This structural visualization makes this approach highly favored for advanced ergodic topics (like holding-time transformations and suspension flows).
Second version Let ( M , B , μ ) {\displaystyle (M,{\mathcal {B}},\mu )} be a finite measure space. Let f : M → M {\displaystyle f:M\to M} be a measurable transformation preserving μ {\displaystyle \mu } . Let E ∈ B {\displaystyle E\in {\mathcal {B}}} be any measurable set with positive measure. Define the first-return time function ρ E : E → Z + ∪ { ∞ } {\displaystyle \rho _{E}:E\to \mathbb {Z} ^{+}\cup \{\infty \}} by
ρ E ( x ) = min { n ≥ 1 : f n ( x ) ∈ E } {\displaystyle \rho _{E}(x)=\min\{n\geq 1:f^{n}(x)\in E\}}
if this set is nonempty, and otherwise, let it be ρ E ( x ) = ∞ {\displaystyle \rho _{E}(x)=\infty } if no iterate of x {\displaystyle x} is in E {\displaystyle E} . (Note that by the Poincaré recurrence theorem, the set on the right side is nonempty for almost every point.) Then one version of Kac's lemma states that ρ E {\displaystyle \rho _{E}} is integrable (i.e. ρ E ∈ L 1 ( μ | E ) {\displaystyle \rho _{E}\in {\mathcal {L}}^{1}(\mu |_{E})} ), with
∫ E ρ E d μ = μ ( M ) − μ ( E 0 ∗ ) . {\displaystyle \int _{E}\rho _{E}{\text{ d}}\mu =\mu (M)-\mu (E_{0}^{*}).}
This version is provable using elementary measure theory and real analysis. If the system ( f , μ ) {\displaystyle (f,\mu )} is in fact ergodic, then the set E 0 ∗ {\displaystyle E_{0}^{*}} has zero measure, so, dividing both sides by μ ( E ) {\displaystyle \mu (E)} , we indeed get that (almost everywhere in E {\displaystyle E} ) the mean return time is equal to the measure of the whole space divided by the measure of the set E {\displaystyle E} , which is the statement of the lemma.
Application In physics, a dynamical system evolving in time may be described in a phase space, that is by the evolution in time of some variables. If this variables are bounded, that is having a minimum and a maximum, for a theorem due to Liouville, a measure can be defined in the space, having a measure space where the lemma applies. As a consequence, given a configuration of the system (a point in the phase space) the average return period close to this configuration (in the neighbourhood of the point) is inversely proportional to the considered size of volume surrounding the configuration. Normalizing the measure space to 1, it becomes a probability space and the measure P ( A ) {\displaystyle P(A)} of its set A {\displaystyle A} represents the probability of finding the system in the states represented by the points of that set. In this case the lemma implies that the smaller is the probability to be in a certain state (or close to it), the longer is the time of return near that state. In formulas, if A {\displaystyle A} is the region close to the starting point and T R {\displaystyle T_{R}} is the return period, its average value is:
⟨ T R ⟩ = τ / P ( A ) {\displaystyle \langle T_{R}\rangle =\tau /P(A)}
Where τ {\displaystyle \tau } is a characteristic time of the system in question. Note that since the volume of A {\displaystyle A} , therefore P ( A ) {\displaystyle P(A)} , depends exponentially on the n {\displaystyle n} variables in the system ( A = ϵ n {\displaystyle A=\epsilon ^{n}} , with ϵ {\displaystyle \epsilon } infinitesimal side, therefore less than 1, of the volume in n {\displaystyle n} dimensions), P ( A ) {\displaystyle P(A)} decreases very rapidly as the variables of the system increase and consequently the return period increases exponentially. In practice, as the variables needed to describe the system increase, the return period increases rapidly.
References
Further reading Kac, Mark (1947). "On the notion of recurrence in discrete stochastic processes" (PDF). Bulletin of the American Mathematical Society. 53 (10): 1002–1010. doi:10.1090/S0002-9904-1947-08927-8. Walkden, Charles. "MAGIC: 10 lectures course on ergodic theory – Lecture 5". Petersen, Karl E. (1983). Ergodic Theory. Cambridge: Cambridge University Press. p. 46. ISBN 0521236320. Hochman, Michael (2013-01-27). "Notes on ergodic theory" (PDF). Standard Graduate Textbooks
Petersen, K. (1989). Ergodic Theory. Cambridge University Press. Walters, P. (1982). An Introduction to Ergodic Theory. Springer Graduate Texts in Mathematics. Einsiedler, M., & Ward, T. (2011). Ergodic Theory: with a view towards Number Theory. Springer Science & Business Media. Secondary & Survey References
Krengel, U. (1985). Ergodic Theorems. De Gruyter. Shields, P. C. (1996). The Ergodic Theory of Discrete Sample Paths. American Mathematical Society.