Ergodic Theorems, Recurrence, and the Pointwise Convergence¶

Job: develop the Birkhoff / pointwise (maximal / Hopf–Dunford–Schwartz), the Poincaré recurrence / conservative-dissipative (Hopf), the subadditive ergodic (Kingman / Furstenberg–Kesten), the martingale convergence (Doob), and the Besicovitch / density / differentiation — with the Birkhoff via the maximal + the Besicovitch density as the worked example.

Core¶

Birkhoff and the pointwise convergence¶

Birkhoff / maximal / pointwise	Description
Birkhoff ergodic theorem	$\frac1N\sum_{k=0}^{N-1}f(T^k x)\to f^*$ a.e. + $L^1$ (the recalled from ergodic theory, made measure-theoretic)
Maximal / Hopf–Dunford–Schwartz	the maximal $f^*(x)=\sup_N
Pointwise convergence	the Birkhoff the pointwise (the maximal + the density, the convergence)

Poincaré recurrence and the conservative / dissipative¶

Recurrence / conservative-dissipative	Description
Poincaré recurrence	$\mu(A)>0\Rightarrow$ a.e. $x\in A$ returns to $A$ infinitely often (the finite-measure)
Conservative / dissipative (Hopf)	the conservative / the dissipative decomposition (the infinite-measure, the Hopf)
Recurrence vs ergodicity	the recurrence the weaker, the ergodicity the stronger, the mixing / the weak / the strong

The subadditive ergodic and the Kingman¶

Subadditive / Kingman / Furstenberg–Kesten	Description
Subadditive ergodic (Kingman)	the subadditive $a_{n+m}\le a_n+a_m\circ T^n$ the $\lim a_n/n$ a.e. (the subadditive the Birkhoff)
Furstenberg–Kesten	the random matrix products the top Lyapunov (the ergodic theory / Oseledets the Lyapunov, the random matrices)

The martingale convergence and the Doob¶

Martingale / Doob / conditional	Description
Martingale convergence (Doob)	the $L^1$-bounded martingale the a.e. convergence (the $L^p$ for $p>1$)
Doob maximal / upcrossing	the Doob maximal the weak (1,1), the upcrossing the inequality, the convergence
Conditional expectation	$\mathbb{E}[\cdot

The Besicovitch / the density theorem and the differentiation¶

Besicovitch / density / differentiation / RN	Description
Besicovitch covering	the Besicovitch covering theorem (the fine cover, the bounded overlap, the $\mathbb{R}^n$ the specific)
Density theorem	the Lebesgue density $\lim_{r\to0}\mu(B_r\cap E)/\mu(B_r)=1_E(x)$ a.e. (the Besicovitch / the Vitali)
Differentiation of measures	$\lim\nu(B_r)/\mu(B_r)=d\nu/d\mu$ a.e. (the Radon–Nikodym the pointwise, the foundations)

See it¶

Birkhoff via the maximal ergodic (Hopf–Dunford–Schwartz) and the Besicovitch density. The Birkhoff the pointwise convergence $\frac1N\sum f(T^k x)\to\int f$ (the ergodic). The maximal: the Hopf–Dunford–Schwartz the maximal $f^*$ the weak (1,1) (the $\mu\{f^*>\lambda\}\le\|f\|_1/\lambda$, the maximal the bound), the Birkhoff the pointwise (the maximal + the density, the convergence). The Besicovitch: the density $\lim_{r\to0}\mu(B_r\cap E)/\mu(B_r)=1_E(x)$ a.e. (the density the pointwise, the Besicovitch the covering), the differentiation $\lim\nu(B_r)/\mu(B_r)=d\nu/d\mu$ (the RN the pointwise). The key point: the Birkhoff the maximal (Hopf–Dunford–Schwartz the weak (1,1)) + the density (Besicovitch) — the pointwise ergodic the measure-theoretic / the covering.

Check¶

Show that the Hopf–Dunford–Schwartz maximal inequality (the weak (1,1)) implies the Birkhoff pointwise convergence, and explain the density.

The Hopf–Dunford–Schwartz maximal: $f^*(x)=\sup_N|\frac1N\sum_{k=0}^{N-1}f(T^k x)|$, the $\mu\{f^*>\lambda\}\le\|f\|_1/\lambda$ (the weak (1,1), the maximal the bound). The Birkhoff: the $\frac1N\sum f(T^k x)\to f^*$ a.e. The mechanism: the maximal the weak (1,1) $\Rightarrow$ the $L^p$ the bounded (the interpolation the $L^p$), the $L^2$ the convergence (the von Neumann the $L^2$), the maximal the weak (1,1) + the $L^2$ the convergence $\Rightarrow$ the pointwise the convergence (the density: the $L^2$ the dense in $L^1$, the maximal the weak (1,1) the extension). The density: the Besicovitch $\lim\mu(B_r\cap E)/\mu(B_r)=1_E$ a.e. (the Lebesgue density, the Besicovitch the covering the bounded overlap), the differentiation $\lim\nu(B_r)/\mu(B_r)=d\nu/d\mu$ (the RN the pointwise, the density the convergence). So the maximal the weak (1,1) + the density $\Rightarrow$ the Birkhoff the pointwise.

Explain the subadditive ergodic (Kingman) and the Furstenberg–Kesten, and the relation to the Lyapunov exponents.

The subadditive ergodic (Kingman): the subadditive sequence $a_{n+m}\le a_n+a_m\circ T^n$ (the subadditive, the $a_n$ the $T$-dependent) has the limit $\lim a_n/n=\inf a_n/n$ a.e. (the Kingman, the subadditive the Birkhoff — the additive the Birkhoff, the subadditive the Kingman). The Furstenberg–Kesten: the random matrix products $A_n\cdots A_1$ (the i.i.d. matrices), the $\log\|A_n\cdots A_1\|$ the subadditive (the $\log\|A_{n+m}\cdots A_1\|\le\log\|A_{n+m}\cdots A_{n+1}\|+\log\|A_n\cdots A_1\|$ the subadditivity), the Kingman the limit $\lim\frac1n\log\|A_n\cdots A_1\|=\chi_1$ a.e. (the top Lyapunov exponent, the Furstenberg–Kesten). The relation: the $\chi_1$ the top Lyapunov (the ergodic theory / Oseledets the MET, the random matrices the Furstenberg–Kesten), the Kingman the subadditive the mechanism. The additive the Birkhoff, the subadditive the Kingman — the two related but distinct (the additive the linear, the subadditive the non-linear).

Pitfalls¶

The Birkhoff the pointwise (the maximal the weak (1,1) the Hopf, the pointwise the convergence) — the maximal the weak (1,1) the key, the pointwise the consequence.
The subadditive the Kingman (the subadditive not the additive, the Birkhoff the additive) — the subadditive the non-linear, the additive the Birkhoff, the two the different.
The martingale the Doob (the $L^1$ the a.e. not the $L^1$ the norm, the $p>1$ the $L^p$) — the $L^1$-bounded the a.e. convergence (not the $L^1$ the norm), the $p>1$ the $L^p$ the convergence.
The Besicovitch / the density the covering (the fine cover, the bounded overlap), the density the pointwise — the Besicovitch the $\mathbb{R}^n$ the specific (the Vitali the general, the Besicovitch the $\mathbb{R}^n$).
The Poincaré recurrence the finite-measure (the $\mu(A)>0\Rightarrow$ the return), the infinite the Hopf (the conservative / the dissipative) — the finite the Poincaré, the infinite the Hopf.

Refs¶

Birkhoff, "Proof of the ergodic theorem" (PNAS 1931); Hopf–Dunford–Schwartz, the maximal ergodic
Kingman, "Subadditive ergodic theorem" (Ann. Probab. 1973); Furstenberg–Kesten, "Products of random matrices" (1960)
Doob, Stochastic Processes (1953); the martingale convergence / the maximal
Besicovitch, "On the fundamental geometric properties of linearly measurable plane sets" (1939); the covering / the density

Next¶

MT index · Foundations · Geometric Measure Theory · Probability & Stochastic · Harmonic & Singular Integrals · Ergodic theory · Stochastic processes / martingales