Dynamical Systems: Ergodic & Physical Measures¶

Job: develop the measure-theoretic side of smooth dynamics — the Hopf argument for ergodicity of Anosov systems (via absolute continuity of stable/unstable foliations), the Sinai–Ruelle–Bowen (SRB) physical measures for hyperbolic attractors, the Pesin theory (non-uniform hyperbolicity, entropy formula, stable ergodicity), and the Pugh–Shub stable ergodicity conjecture for partially hyperbolic systems.

Core¶

Hopf argument and Anosov ergodicity¶

Theorem (Hopf argument; Anosov–Sinai). For an Anosov diffeomorphism $f$ preserving a smooth (Borel, equivalent to Lebesgue) measure $\mu$, the measure is ergodic and mixing. The proof (Hopf argument): 1. By the Birkhoff ergodic theorem, for any continuous observable $\phi$, the time average $\frac1N\sum_{k=0}^{N-1}\phi\circ f^k$ converges $\mu$-a.e. to a $f$-invariant limit $\bar\phi$. 2. The limit $\bar\phi$ is constant along stable manifolds: $\bar\phi(x)=\bar\phi(y)$ for $y\in W^s(x)$ (since $d(f^n x,f^n y)\to 0$, the averages coincide). Similarly along $W^u$. 3. The absolute continuity of the stable and unstable foliations $W^s,W^u$ (the holonomy maps preserve the measure class of $\mu$) implies $\bar\phi$ is constant on the sets saturated by $W^s$ and $W^u$. 4. For an Anosov system, $W^s$ and $W^u$ are transverse and jointly fill $M$ (the local product structure), so the saturation is the whole $M$, and $\bar\phi$ is constant globally $\Rightarrow$ ergodicity.

The Anosov–Sinai theorem upgrades ergodicity to mixing via the Hopf argument + the decay of correlations (the stable/unstable foliations interact transversally, producing a shearing that mixes). The Bernoulli property (Ornstein–Weiss): geodesic flows on negatively curved manifolds are Bernoulli (isomorphic to a sequence of independent random variables), the strongest mixing property.

SRB measures (Sinai–Ruelle–Bowen)¶

For a $C^{1+\alpha}$ diffeomorphism $f$ with a hyperbolic attractor $\Lambda$ (a hyperbolic set that is an attractor), the SRB measure is the physically relevant invariant measure:

Definition (SRB measure). An invariant measure $\mu$ is an SRB measure if (i) the Lyapunov exponents of $\mu$ are non-zero (positive on unstable, negative on stable), and (ii) the conditional measures of $\mu$ on the unstable manifolds $W^u$ are absolutely continuous with respect to the Riemannian volume on $W^u$.

Theorem (Bowen–Ruelle). For an Axiom A attractor $\Lambda$ of a $C^{1+\alpha}$ diffeomorphism, the SRB measure $\mu_{\mathrm{SRB}}$ exists, is unique, and is the physical measure: for Lebesgue-a.e. $x$ in the basin $B(\Lambda)$, the time averages $\frac1N\sum_{k=0}^{N-1}\phi\circ f^k(x)\to\int\phi\,d\mu_{\mathrm{SRB}}$ for all continuous $\phi$. The SRB measure is the measure observed in physical experiments (typical initial conditions converge to it).

The SRB measure is the smooth-dynamics analog of the Gibbs measure in statistical mechanics: it is the equilibrium state for the potential $\log|\det Df|_{E^u}|$, and it satisfies the Pesin formula (below). The absolute continuity of the unstable foliation is the key hypothesis — it requires $C^{1+\alpha}$ (Hölder derivatives); for $C^1$ maps the unstable foliation may fail to be absolutely continuous, and the SRB / physical measure may not exist.

Pesin theory: non-uniform hyperbolicity¶

The Pesin set $R(\mu)$ (the regular set): the set of points with non-zero Lyapunov exponents (the Oseledets splitting $T_xM=E^s\oplus E^u$ exists with $\chi^s<0<\chi^u$) and the stable/unstable manifolds $W^{s/u}(x)$ exist. For a $C^{1+\alpha}$ diffeomorphism and a smooth invariant measure $\mu$, $\mu(R(\mu))=1$ (by Oseledets).

Theorem (Pesin entropy formula). For a $C^{1+\alpha}$ diffeomorphism $f$ and a smooth invariant measure $\mu$, $$ h_\mu(f)\;=\;\int\sum_{\chi_i(x)>0}\chi_i(x)\,d\mu(x), $$ the sum over positive Lyapunov exponents. The Ruelle inequality $h_\mu\le\int\sum_{\chi_i>0}\chi_i\,d\mu$ holds in general; equality (the Pesin formula) holds iff $\mu$ is an SRB measure.

Theorem (Pesin stable ergodicity). For a $C^{1+\alpha}$ diffeomorphism $f$ with a smooth invariant measure $\mu$, non-uniform hyperbolicity (all Lyapunov exponents non-zero $\mu$-a.e.) + accessibility (the stable and unstable foliations are jointly accessible, i.e. any two points can be connected by a path of stable / unstable leaves) $\Rightarrow$ ergodicity of $\mu$ (Pugh–Shub). This is the non-uniform analog of the Hopf argument.

Property	Uniform hyperbolicity (Anosov)	Non-uniform (Pesin)
Hyperbolic splitting	$T_\Lambda M=E^s\oplus E^u$ on all of $\Lambda$ (continuous, uniform bounds)	$T_xM=E^s\oplus E^u$ on $R(\mu)$, $\mu(R)=1$ (measurable, no uniform bounds)
Stable/unstable manifolds	$W^{s/u}(x)$ for all $x\in\Lambda$, $C^r$ uniform	$W^{s/u}(x)$ for $x\in R(\mu)$, $C^r$ but with non-uniform sizes
Foliations absolute continuity	Yes, uniform (Anosov–Sinai)	Yes, on $R(\mu)$ (Pesin), requires $C^{1+\alpha}$
Ergodicity	Hopf argument (Anosov–Sinai)	Pesin stable ergodicity + accessibility (Pugh–Shub)
Entropy formula	Pesin (equality for smooth $\mu$)	Pesin (equality for smooth / SRB $\mu$)
Examples	Anosov diffeos / flows	Generic partially hyperbolic systems; some non-uniformly hyperbolic maps

Partially hyperbolic systems and stable ergodicity¶

A diffeomorphism $f$ is partially hyperbolic if $TM=E^{ss}\oplus E^c\oplus E^{uu}$ with $E^{ss}$ uniformly contracted, $E^{uu}$ uniformly expanded, and $E^c$ the (possibly non-hyperbolic) center. The Pugh–Shub conjecture on stable ergodicity:

Conjecture (Pugh–Shub). A partially hyperbolic diffeomorphism is stably ergodic (with respect to a smooth measure $\mu$) iff it is accessible (the stable and unstable foliations are jointly accessible) and center bolicity (the center Lyapunov exponents are non-zero a.e.).

Theorem (partial results, Pugh–Shub; Burns–Wilkinson; F. Rodriguez Hita). The conjecture is proven in many cases: (i) time-$t$ maps of Anosov flows are stably ergodic (accessibility + center direction = flow, which is neutral but the system is ergodic); (ii) skew products over Anosov with compact center fibers, under accessibility; (iii) the largely expanding / mostly contracting dichotomy (the center direction is uniformly expanding or contracting $\Rightarrow$ stable ergodicity). The full conjecture (the iff of accessibility + center bolicity) is open in general.

The stable ergodicity is the open condition in the $C^r$ topology: $f$ stably ergodic means $f$ and all $C^r$-small perturbations are ergodic — it is stronger than ergodicity (a property of a single map), and is the smooth-dynamics analog of structural stability (an open condition in the space of dynamics). The frontier is the classification of stably ergodic diffeomorphisms.

See it¶

The Hopf argument for the Arnold cat map + the Pesin formula. For the Arnold cat map $f:(x,y)\mapsto(2x+y,x+y)$ on $\mathbb{T}^2$ (see Hyperbolic Dynamics), the stable and unstable foliations are the linear foliations $W^s=\{(t v_-)\}$ and $W^u=\{(t v_+)\}$ (the eigendirections of the matrix $A$), with $v_+=(\frac{1+\sqrt 5}{2},1)$ and $v_-=(\frac{1-\sqrt 5}{2},1)$ (the golden-ratio slopes).

Hopf argument: the Haar (Lebesgue) measure $\mu$ is invariant. By Birkhoff, the time average $\bar\phi(x)=\lim_{N\to\infty}\frac1N\sum_{k=0}^{N-1}\phi\circ f^k(x)$ exists $\mu$-a.e. Since $d(f^n x,f^n y)\to 0$ for $y\in W^s(x)$, $\bar\phi(x)=\bar\phi(y)$ along $W^s$; similarly along $W^u$. The foliations $W^s,W^u$ are linear with irrational slopes, and the holonomy between transversals is an isometry (a translation in $\mathbb{T}^2$), hence absolutely continuous. The local product structure (any small rectangle in $\mathbb{T}^2$ is spanned by $W^s$ and $W^u$ leaves) then propagates the constant value of $\bar\phi$ to all of $\mathbb{T}^2$, so $\bar\phi$ is constant $\mu$-a.e. $\Rightarrow$ ergodicity. The same argument gives mixing (Anosov–Sinai): the shearing between $W^s$ and $W^u$ under $f$ produces decay of correlations.

Pesin formula: the Haar measure is smooth (Lebesgue), hence an SRB measure (the conditional measures on $W^u$ are the linear Lebesgue measure on the line $W^u$, which is absolutely continuous). The Pesin formula gives

\[ h_\mu(f)=\int\sum_{\chi_i>0}\chi_i\,d\mu=\log\lambda_+=\log\frac{3+\sqrt 5}{2}, \]

the single positive Lyapunov exponent of the cat map (the log of the expanding eigenvalue $\lambda_+$). This equals the topological entropy $h_{\mathrm{top}}(f)=\log\lambda_+$ (computed in Hyperbolic Dynamics via the Markov partition), confirming the variational principle $h_{\mathrm{top}}=\sup_\mu h_\mu$ — the Haar measure is the measure of maximal entropy for the cat map (it attains the supremum). The two computations agree: the Markov partition gives $h_{\mathrm{top}}=\log\rho(A)=\log\lambda_+$, and the Pesin formula gives $h_{\mathrm{Haar}}=\log\lambda_+$; the Haar measure is both SRB and the measure of maximal entropy (for the linear cat map, the two coincide — a special property of algebraic Anosov systems).

Check¶

Explain why the absolute continuity of the stable/unstable foliations is the key input for the Hopf argument, and identify where it fails for general partially hyperbolic systems.

The Hopf argument step 3: the Birkhoff limit $\bar\phi$ is constant along $W^s$ and along $W^u$ (step 2), so it is constant on the sets saturated by these foliations. To conclude the saturation is the whole $M$ (ergodicity), one needs the saturation to be measurable and of full measure, which uses the absolute continuity of the holonomy maps between transversals: a set of positive measure in a transverse disk maps to a set of positive measure, so the saturation of a positive-measure set is positive-measure. Without absolute continuity, the holonomy can map a positive-measure set to a zero-measure set (singular holonomy), and the saturation argument fails — the Birkhoff limit can be constant on the saturated set (a zero-measure "pathological" union) without being globally constant, and ergodicity is not concluded. For Anosov systems the absolute continuity is a theorem (Anosov–Sinai); for general partially hyperbolic systems (with a center direction), the absolute continuity of $W^{ss}$ and $W^{uu}$ holds, but the joint accessibility (whether $W^{ss}$ and $W^{uu}$ together fill $M$) is not automatic — it is the Pugh–Shub accessibility hypothesis. The Hopf argument for partial hyperbolicity needs the accessibility + absolute continuity; without accessibility, the argument fails.

Verify the Pesin formula for the Arnold cat map explicitly, and identify the SRB / measure-of-maximal-entropy coincidence.

The cat map $f$ is linear with $Df=A=\begin{pmatrix}2&1\\1&1\end{pmatrix}$, eigenvalues $\lambda_\pm=\frac{3\pm\sqrt 5}{2}$. The Lyapunov exponents with respect to the Haar measure $\mu$ are $\chi_\pm=\log|\lambda_\pm|$, so $\chi_+=\log\frac{3+\sqrt 5}{2}>0$ (the positive exponent) and $\chi_-=\log\frac{3-\sqrt 5}{2}<0$. The Pesin formula gives $h_\mu=\sum_{\chi_i>0}\chi_i=\chi_+=\log\frac{3+\sqrt 5}{2}$. The topological entropy (from the Markov partition, $h_{\mathrm{top}}=\log\rho(A)=\log\frac{3+\sqrt 5}{2}$, see Hyperbolic Dynamics) equals the metric entropy of the Haar measure, so by the variational principle the Haar measure is the measure of maximal entropy (it attains $h_{\mathrm{top}}$). The Haar measure is also the SRB measure (the conditional measure on $W^u$ is the linear Lebesgue measure, absolutely continuous). Hence for the cat map, the SRB measure and the measure of maximal entropy coincide — this is a special property of linear algebraic Anosov systems (the Haar measure is both); for general Anosov systems (e.g. nonlinear perturbations) the SRB and the measure of maximal entropy may differ. The Pesin formula holds because the Haar measure is SRB, and the equality $h_\mu=h_{\mathrm{top}}$ holds because the Haar measure is the measure of maximal entropy.

Pitfalls¶

The Hopf argument needs absolute continuity of foliations: which is a theorem for Anosov systems (Anosov–Sinai) but requires $C^{1+\alpha}$ (Hölder derivatives); for $C^1$ maps the foliations may be singular, and the Hopf argument fails. Confusing the regularity hypotheses is the standard error.
SRB measures require $C^{1+\alpha}$ (not $C^1$): the absolute continuity of the unstable foliation — the key SRB property — needs Hölder derivatives; for $C^1$ maps the SRB / physical measure may not exist.
Pesin's formula is an equality for SRB / smooth measures; the Ruelle inequality is strict in general: $h_\mu\le\int\sum_{\chi_i>0}\chi_i\,d\mu$ (Ruelle), equality iff $\mu$ is SRB (Pesin). For non-SRB measures, the inequality is strict — the formula is not an identity for all invariant measures.
Stable ergodicity is an open condition in $C^r$, not equivalent to ergodicity: a stably ergodic system remains ergodic under $C^r$-small perturbations; it is stronger than ergodicity (which is a property of a single map). The Pugh–Shub conjecture characterizes stably ergodic partially hyperbolic systems via accessibility + center bolicity.
The Pugh–Shub conjecture is open in general: it is proven in many cases (time-$t$ maps of Anosov flows, skew products, the largely-expanding / mostly-contracting dichotomy), but the full iff (accessibility + center bolicity $\Leftrightarrow$ stable ergodicity) is open. Do not state it as a theorem in full.
Accessibility is not automatic for partially hyperbolic systems: the joint accessibility of $W^{ss}$ and $W^{uu}$ (any two points connected by a stable-unstable path) is a hypothesis, not a theorem; without it, the Hopf / Pesin ergodicity argument fails.
For the cat map, the SRB measure and the measure of maximal entropy coincide (special property of algebraic Anosov): for general Anosov / partially hyperbolic systems, they may differ; do not generalize the coincidence.

Refs¶

Anosov–Sinai, Certain smooth ergodic systems (Russian Math. Surveys 1967); Hopf, Ergodentheorie (1937, the Hopf argument)
Bowen–Ruelle, The ergodic theory of Axiom A flows (Invent. 1975); Sinai, Gibbs measures in ergodic theory (Russian Math. Surveys 1972)
Pesin, Lyapunov characteristic exponents and smooth ergodic theory (Russian Math. Surveys 1977); Ruelle, An inequality for the entropy of differentiable maps (Bol. Soc. Bras. Mat. 1978)
Pugh–Shub, Stably ergodic dynamical systems (Annals 1996) and Ergodic attractors; Burns–Wilkinson, On the ergodicity of partially hyperbolic systems
Ornstein–Weiss, Geodesic flows are Bernoullian (Israel J. Math. 1973)

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