Oseledets Multiplicative Ergodic Theorem, Lyapunov Exponents, and Pesin Theory¶

Job: develop the Oseledets MET (the splitting / the Lyapunov exponents), the Pesin formula / Ruelle inequality / non-uniform hyperbolicity, the Pesin stable manifolds / Pesin set / metric, the Furstenberg–Kesten / random matrix products / free probability, and the KZ cocycle / Teichmüller — with the Pesin formula for the SRB / the Anosov / the non-uniform as the worked example.

Core¶

The Oseledets multiplicative ergodic theorem (MET)¶

For a diffeomorphism $T$ on a manifold $M$ preserving a measure $\mu$, and the cocycle $DT^x:TM\to TM$, the Oseledets splitting $T_xM=\bigoplus E_i(x)$ (the Lyapunov decomposition, the subspaces with the exponential growth rates the Lyapunov exponents $\chi_i(x)$).

Theorem (Oseledets MET). For $\mu$-a.e. $x$, the limit $\lim_{n\to\infty}\frac1n\log\|DT^n v\|=\chi_i(x)$ exists for $v\in E_i(x)\setminus\{0\}$ (the Lyapunov exponents), and the splitting is invariant.

Oseledets data	Description
Splitting $T_xM=\bigoplus E_i(x)$	the Lyapunov decomposition (the subspaces with the exponential growth)
Lyapunov exponents $\chi_i(x)$	the exponential growth rates $\lim\frac1n\log\\|DT^n v\\|=\chi_i$ for $v\in E_i$
MET	the limit exists $\mu$-a.e., the splitting invariant (the Oseledets)
Cocycle / regularity	the exponents $\chi_i$ the $T$-invariant functions, the splitting the measurable

The Lyapunov exponents and the hyperbolicity¶

The Lyapunov exponents $\chi_1\ge\cdots\ge\chi_d$ (the sum $\sum\chi_i=\int\log|\det DT|\,d\mu$ the volume growth).

Theorem (Pesin formula). For the SRB measure $\mu_{\mathrm{SRB}}$ on a $C^2$ diffeomorphism: $h_{\mu_{\mathrm{SRB}}}(T)=\sum_{\chi_i>0}\chi_i$ (the entropy = the sum of the positive Lyapunov exponents).

Theorem (Ruelle inequality). The general: $h_\mu(T)\le\sum_{\chi_i>0}\chi_i$ (the upper bound, the equality for the SRB).

Lyapunov / hyperbolicity	Description
Lyapunov exponents $\chi_1\ge\cdots\ge\chi_d$	the sum $\sum\chi_i=\int\log
Pesin formula (SRB)	$h_{\mu_{\mathrm{SRB}}}=\sum_{\chi_i>0}\chi_i$ (the entropy = the positive Lyapunov, the SRB)
Ruelle inequality (general)	$h_\mu\le\sum_{\chi_i>0}\chi_i$ (the upper bound, the equality the SRB)
Non-uniform hyperbolicity (Pesin)	the all $\chi_i\ne0$ $\mu$-a.e. (the Pesin region)

The Pesin theory and the stable manifolds¶

Pesin stable manifold	Description
Stable manifold $W^s(x)$	the $\chi<0$ (the exponentially contracting), the $C^r$ the $\varepsilon$-exponentially close
Unstable manifold $W^u(x)$	the $\chi>0$ (the exponentially expanding)
Pesin set $\mathcal{R}$	the regular set (the points with the Pesin regularity, the $\mu$-full)
Pesin metric	the adapted metric (the Pesin / the Katok–Morse, the Lyapunov-adapted)

The Lyapunov exponents and the random matrices / the free probability¶

Random matrix products / free	Description
Furstenberg–Kesten	the top Lyapunov exponent of the product of i.i.d. random matrices (the subadditive ergodic / the Kingman, the Furstenberg–Kesten formula)
Kingman subadditive ergodic	the subadditive ergodic theorem (the limit of the subadditive sequences, the Kingman)
Free probability	the Lyapunov exponents / the free cumulants (the noncommutative / the free, the free entropy)
Brown measure	the Brown measure / the random matrices / circular (the non-Hermitian / the Brown measure)

The Oseledets for the moduli / the Teichmüller¶

KZ cocycle / Teichmüller	Description
Kontsevich–Zorich cocycle	the Lyapunov exponents on the moduli of abelian differentials (the KZ cocycle)
Teichmüller geodesic flow	the Forni / the Avila–Viana (the Lyapunov exponents, the Teichmüller)
Hodge–Teichmüller	the Lyapunov exponents and the Hodge theory (the higher Teichmüller the relation)

See it¶

Pesin formula for the SRB measure and the non-uniform hyperbolicity. For a $C^2$ Anosov diffeomorphism $T$ (the uniformly hyperbolic, the Pesin special case), the SRB measure $\mu_{\mathrm{SRB}}$ (the Margulis–Sinai / the Bowen, the measure of maximal entropy / the SRB): the Pesin formula $h_{\mu_{\mathrm{SRB}}}(T)=\sum_{\chi_i>0}\chi_i$ (the entropy = the sum of the positive Lyapunov exponents). The Anosov: all $\chi_i\ne0$ (the uniform), the Pesin set $\mathcal{R}=M$ (the full, the uniform). The general (the non-uniformly hyperbolic, the Pesin): the $\chi_i\ne0$ $\mu$-a.e. but not uniformly (the Pesin set $\mu$-full but not all), the stable manifolds exist on the Pesin set (the $C^r$ the $\varepsilon$-exponential), the Pesin formula holds (the SRB). The key point: the Pesin formula the entropy = the Lyapunov (the SRB), the Anosov the uniform special case, the non-uniform the Pesin the general — the dynamical systems / hyperbolic the smooth realization of the ergodic / the Pesin.

Check¶

Show that for an Anosov diffeomorphism, the Pesin set $\mathcal{R}=M$ (the full), and explain the uniform vs the non-uniform.

The Anosov diffeomorphism $T$ on $M$: the tangent bundle $TM=E^s\oplus E^u$ the uniform hyperbolic splitting (the $E^s$ the contracting, the $E^u$ the expanding, the uniform the $\|DT^n|_{E^s}\|\le C\lambda^n$ and $\|DT^{-n}|_{E^u}\|\le C\lambda^n$ for $\lambda<1$ the uniform). The Lyapunov exponents: $\chi_i<0$ on $E^s$, $\chi_i>0$ on $E^u$ (all $\ne0$, the uniform — the bounds uniform over all $x\in M$). The Pesin set $\mathcal{R}$ (the regular set, the points with the Pesin regularity): for the Anosov, $\mathcal{R}=M$ (the full — the uniform hyperbolicity gives the Pesin regularity everywhere, the $\varepsilon$-exponential bounds uniform). The non-uniform: the $\chi_i\ne0$ $\mu$-a.e. but the bounds not uniform (the $C(x)$ the point-dependent, the $\varepsilon(x)$ the point-dependent), the Pesin set $\mathcal{R}$ $\mu$-full but not all (the irregular / the non-Pesin the zero-measure). So the Anosov the uniform ($\mathcal{R}=M$), the non-uniform the Pesin ($\mathcal{R}$ $\mu$-full not all).

Explain the Furstenberg–Kesten formula for the top Lyapunov exponent of the random matrix product, and the Kingman subadditive ergodic theorem.

The random matrix product: $A_n=A_n\cdots A_1$ with $A_i$ i.i.d. random matrices (e.g. the $\mathrm{SL}_d(\mathbb{R})$-valued). The top Lyapunov exponent $\chi_1=\lim_{n\to\infty}\frac1n\log\|A_n\cdots A_1\|$ (the exponential growth of the product norm). The Furstenberg–Kesten: the limit exists a.s. and equals $\inf_n\frac1n\mathbb{E}[\log\|A_n\cdots A_1\|]$ (the subadditivity of $\mathbb{E}[\log\|A_n\cdots A_1\|]$, the limit the infimum). The Kingman subadditive ergodic theorem: the subadditive sequence $a_{n+m}\le a_n+a_m$ (the subadditivity) has the limit $\lim\frac{a_n}{n}=\inf\frac{a_n}{n}$ a.s. (the Kingman, the subadditive ergodic). The Furstenberg–Kesten the application (the $\log\|A_n\cdots A_1\|$ the subadditive, the Kingman the limit, the $\chi_1$ the top Lyapunov). The free probability: the $\chi_1$ / the free cumulants, the noncommutative / the free entropy the connection.

Pitfalls¶

The Oseledets the $\mu$-a.e. (the not all — the irregular / the non-Pesin the zero-measure, the Oseledets the $\mu$-a.e. not the everywhere).
The Pesin formula the SRB (the equality, the Ruelle the inequality for the general — the equality the SRB / the Margulis–Sinai, the non-SRB the strict inequality).
The Pesin stable manifold the non-uniform (the $\varepsilon$-exponential, the $C^r$ the measurable, not the uniform / the smooth — the Pesin the non-uniform, the $C^r$ the $\varepsilon$-exponential the point-dependent, the measurable).
The Furstenberg–Kesten / the Kingman the subadditive ergodic (the limit the infimum, the subadditivity), not the additive (the additive the Birkhoff, the subadditive the Kingman — the two related but distinct).
The KZ cocycle / the Teichmüller the Lyapunov exponents (the Forni / the Avila–Viana), the Hodge–Teichmüller the relation — the Teichmüller the specific (the moduli of abelian differentials), the general the Oseledets.

Refs¶

Oseledets, "A multiplicative ergodic theorem" (Trans. Moscow Math. Soc. 1968)
Pesin, "Characteristic Lyapunov exponents and smooth ergodic theory" (Russian Math. Surveys 1977); Ruelle, "An inequality for the entropy of differentiable maps" (1978)
Furstenberg–Kesten, "Products of random matrices" (Ann. Math. Stat. 1960); Kingman, "Subadditive ergodic theorem" (Ann. Probab. 1973)
Forni, "Deviation of ergodic averages" (JAMS 2002); Avila–Viana, "Simplicity of Lyapunov spectra" (Invent. 2010); Kontsevich–Zorich, "Connected components of the moduli space"

Next¶

ET index · Foundations · Homogeneous & Ratner · Furstenberg & Joinings · Ornstein & Isomorphism · Hyperbolic dynamics · Free probability / entropy · Higher Teichmüller

Pesin stable manifold	Description
Stable manifold \(W^s(x)\)	the \(\chi<0\) (the exponentially contracting), the \(C^r\) the \(\varepsilon\)-exponentially close
Unstable manifold \(W^u(x)\)	the \(\chi>0\) (the exponentially expanding)
Pesin set \(\mathcal{R}\)	the regular set (the points with the Pesin regularity, the \(\mu\)-full)
Pesin metric	the adapted metric (the Pesin / the Katok–Morse, the Lyapunov-adapted)