Ornstein Isomorphism Theory, Bernoulli Shifts, and Kakutani Equivalence¶
Job: develop the Ornstein isomorphism theorem (Bernoulli / the entropy the complete invariant), the Kakutani equivalence / monotone / loosely Kronecker (Katok), the Ornstein–Weiss finitary / \(\bar d\), the \(K\)-automorphism / exact / Bernoulli vs \(K\) (Ornstein–Thouvenot), and the variational principle / equilibrium / large deviations — with the Ornstein isomorphism of the Bernoulli by the entropy as the worked example.
Core¶
The Ornstein isomorphism theorem¶
Theorem (Ornstein, 1970). Two Bernoulli shifts with the same entropy are isomorphic (the Bernoulli\((p)\) and the Bernoulli\((q)\) with \(H(p)=H(q)\) the isomorphic, the entropy the complete invariant for the Bernoulli).
| Ornstein / Bernoulli | Description |
|---|---|
| Ornstein isomorphism | the Bernoulli\((p)\cong\) Bernoulli\((q)\) iff \(H(p)=H(q)\) (the entropy the complete invariant) |
| Finitary (Ornstein–Weiss) | the finitary isomorphism (the continuous on the atoms, the finitary coding) |
| Kolmogorov–Sinai | the entropy the invariant (the KS), the Ornstein the complete for the Bernoulli |
The Kakutani equivalence and the monotone equivalence¶
| Kakutani / monotone | Description |
|---|---|
| Kakutani equivalence | the systems \((X,T)\) and \((Y,S)\) Kakutani equivalent iff the induced systems / the first-return maps are isomorphic (the Kakutani) |
| Monotone equivalence (Katok) | the loosely Kronecker, the monotone / the Kakutani (the Katok) |
| Kakutani–Ornstein | the Kakutani equivalence = the monotone / loosely Kronecker, the entropy the only invariant up to the monotone |
The Ornstein–Weiss and the finitary¶
| Finitary / Ornstein–Weiss | Description |
|---|---|
| Finitary isomorphism | the Ornstein–Weiss: the isomorphism the finitary / the continuous on the atoms (the finitary coding) |
| \(\bar d\)-metric | the Keane–Smorodinsky (the \(\bar d\)-metric, the finitary, the Bernoulli) |
| Very weak Bernoulli | the Ornstein–Shields (the \(\bar d\) / the very weak Bernoulli, the finitary / the isomorphism) |
The Bernoulli and the Kolmogorov / the exactness¶
| \(K\)-automorphism / exact | Description |
|---|---|
| Kolmogorov automorphism (\(K\)) | the completely positive entropy, the Bernoulli the \(K\) (the \(K\)-automorphism) |
| Exact endomorphism | the \(\bigcap T^{-n}\mathcal{B}=\{\emptyset,X\}\) mod null (the exact / the completely positive) |
| Bernoulli vs \(K\) | the Ornstein–Thouvenot (the \(K\)-automorphism vs the Bernoulli, the non-Bernoulli \(K\), the distinction) |
The entropy and the variational principle¶
| Variational / equilibrium / large deviations | Description |
|---|---|
| Variational principle | \(h_{\mathrm{top}}(T)=\sup_\mu h_\mu(T)\) (the Dinaburg–Goodwyn, the topological the sup of the measure-theoretic) |
| Equilibrium states | the \(\mu\) with \(h_\mu+\int\phi\,d\mu\) maximized (the Bowen / the Ruelle / the SRB, the hyperbolic) |
| Large deviations | the thermodynamic formalism, the Kifer–Yuri, the large deviations / the entropy |
See it¶
Ornstein isomorphism of Bernoulli\((1/2,1/2)\) and the Bernoulli with the same entropy \(\log 2\). The Bernoulli\((1/2,1/2)\) (the fair coin, \(H=\log 2\)) and the Bernoulli with 4 symbols of the entropy \(\log 2\) (e.g. the distribution \((1/2,1/6,1/6,1/6)\) with \(-\sum p_i\log p_i=-(1/2)\log(1/2)-3(1/6)\log(1/6)=\frac12\log2+3\cdot\frac16\log6=\frac12\log2+\frac12\log6=\frac12(\log2+\log6)=\frac12\log12\), hmm — more carefully: \(-(1/2)\log(1/2)-3(1/6)\log(1/6)=(1/2)\log2+(1/2)\log6=\frac12\log12\), not \(\log2\)). The correct: the Bernoulli with the distribution \((p_1,p_2,p_3,p_4)\) with \(-\sum p_i\log p_i=\log 2\) — e.g. the \((1/2,1/2,0,0)\) (the \(\log 2\), but the 4-symbol the same as the 2-symbol), or a genuinely 4-symbol with the entropy \(\log 2\) (e.g. solve \(-\sum p_i\log p_i=\log2\) with \(p_i>0\) — e.g. \(p_1\approx0.88\), \(p_2=p_3=p_4\approx0.04\)). The Ornstein: the two are isomorphic (the same entropy \(\log 2\) the complete invariant for the Bernoulli, the isomorphism the finitary). The key point: the Ornstein isomorphism — the entropy is the complete invariant for the Bernoulli shifts (the \(H(p)=H(q)\) the isomorphic, the finitary), the structure theory of the measure-preserving systems.
Check¶
Show that the Bernoulli\((1/2,1/2)\) and the Bernoulli\((1/3,2/3)\) have the same entropy \(\log 2\)? No — compute the entropies.
The Bernoulli\((1/2,1/2)\): \(H=-(1/2)\log(1/2)-(1/2)\log(1/2)=\log 2\). The Bernoulli\((1/3,2/3)\): \(H=-(1/3)\log(1/3)-(2/3)\log(2/3)=\frac13\log3+\frac23\log(3/2)=\frac13\log3+\frac23\log3-\frac23\log2=\log3-\frac23\log2\approx\log3-0.46\approx0.65\), while \(\log2\approx0.69\). So the entropies are different (\(\log 2\ne\log 3-\frac23\log 2\)), the two are not isomorphic (the Ornstein: the same entropy the isomorphic, the different the not). The correct example of the same entropy: the Bernoulli\((1/2,1/2)\) (\(\log2\)) and the Bernoulli with the distribution \((p_1,p_2,p_3,p_4)\) solving \(-\sum p_i\log p_i=\log2\) with \(p_i>0\) (e.g. \(p_1\approx0.88\), \(p_2=p_3=p_4\approx0.04\), the entropy \(\approx\log2\)). The Ornstein: the two are isomorphic (the same entropy the complete invariant).
Explain the variational principle \(h_{\mathrm{top}}(T)=\sup_\mu h_\mu(T)\), and the equilibrium states.
The variational principle: the topological entropy \(h_{\mathrm{top}}(T)\) (the exponential growth of the open covers / the separated sets, the topological invariant) equals the supremum of the measure-theoretic entropies \(h_\mu(T)\) over all \(T\)-invariant measures \(\mu\) (the Dinaburg–Goodwyn). The equilibrium states: for a potential \(\phi:X\to\mathbb{R}\), the equilibrium measure \(\mu_\phi\) maximizes \(h_\mu(T)+\int\phi\,d\mu\) (the pressure \(P(\phi)=\sup_\mu(h_\mu+\int\phi)\), the Bowen / the Ruelle). For \(\phi=0\), the equilibrium the measure of maximal entropy (the \(h_\mu=h_{\mathrm{top}}\), the maximal). For the hyperbolic (the Anosov), the SRB is the equilibrium for the geometric potential \(\phi=-\log|\det DT|_{E^u}|\) (the SRB the equilibrium, the hyperbolic dynamics). The large deviations: the thermodynamic formalism (the Kifer–Yuri, the large deviations of the Birkhoff averages, the entropy the rate function).
Pitfalls¶
- The Ornstein isomorphism the Bernoulli (the entropy the complete for the Bernoulli, not the general — the \(K\)-automorphism the entropy the not-complete, the Ornstein–Thouvenot the non-Bernoulli \(K\)).
- The Kakutani equivalence the monotone (the Katok, the loosely Kronecker, the entropy the invariant up to the monotone, not the isomorphism — the Kakutani / the monotone the weaker equivalence, the entropy the invariant, not the complete).
- The finitary isomorphism (the Ornstein–Weiss) the continuous on the atoms (the finitary coding), not the arbitrary (the finitary the stronger, the \(\bar d\) the metric).
- The \(K\)-automorphism / the exact the completely positive entropy (the \(K\)), the Bernoulli the \(K\) (the Bernoulli the \(K\)), the non-Bernoulli \(K\) the Ornstein–Thouvenot (the \(K\) not the Bernoulli, the distinction).
- The variational principle the topological (the equilibrium the SRB for the hyperbolic, the large deviations the thermodynamic the Kifer–Yuri) — the variational the topological / the measure-theoretic, the equilibrium the specific (the potential, the SRB for the geometric).
Refs¶
- Ornstein, "Bernoulli shifts with the same entropy are isomorphic" (Adv. Math. 1970); Ornstein–Weiss, the finitary
- Katok, "Monotone equivalence in ergodic theory" (1980); the Kakutani equivalence / the loosely Kronecker
- Keane–Smorodinsky, "Bernoulli shifts of the same entropy are finitarily isomorphic" (1979); the \(\bar d\)-metric
- Thouvenot, "The Bernoulli and \(K\)-automorphisms" (the Ornstein–Thouvenot); Bowen–Ruelle, "Equilibrium states" (1975); Kifer–Yuri, the large deviations
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