Furstenberg ×2×3, Joinings, and the Furstenberg–Zimmer Structure¶
Job: develop the Furstenberg ×2×3 disjointness / the LCD / the Littlewood, the joinings / the disjointness / the Rudolph, the Furstenberg–Zimmer distal tower / amenable / boundary, the Furstenberg multiple recurrence / Szemerédi / polynomial, and the Furstenberg–Kesten / random walk / Poisson boundary — with the Furstenberg ×2×3 / the Littlewood via the homogeneous as the worked example.
Core¶
The Furstenberg ×2×3 and the disjointness¶
Theorem (Furstenberg ×2×3, 1967). The \(\times 2\) and \(\times 3\) actions on \(\mathbb{T}=\mathbb{R}/\mathbb{Z}\) (the \(x\mapsto 2x\) and \(x\mapsto 3x\), the multiplicative) are disjoint (no common extension / no joining, the Furstenberg 1967).
| Furstenberg ×2×3 / number theory | Description |
|---|---|
| Furstenberg ×2×3 | the \(\times2\) and \(\times3\) on \(\mathbb{T}\) disjoint (no joining, the 1967) |
| LCD / Littlewood | \(\liminf n\|n\alpha\|\|n\beta\|=0\) (the Littlewood, the Einsiedler–Katok–Lindenstrauss via the \(\mathrm{SL}_2(\mathbb{R})\times\mathrm{SL}_2(\mathbb{Q}_p)\) homogeneous) |
| ×2×3 conjecture | the ×2×3 the disjoint, the non-joinable, the Furstenberg the 1967 |
Joinings and the disjointness¶
The joining of two systems \((X,T)\) and \((Y,S)\): a measure \(\lambda\) on \(X\times Y\) invariant under \(T\times S\) with the marginals \(\mu\) and \(\nu\) (the Furstenberg). The disjointness (the only joining is the product \(\mu\otimes\nu\), the no-common-extension).
| Joinings / disjointness | Description |
|---|---|
| Joining | \(\lambda\) on \(X\times Y\), invariant under \(T\times S\), marginals \(\mu,\nu\) (the Furstenberg) |
| Disjointness | the only joining is the product \(\mu\otimes\nu\) (no common extension) |
| Furstenberg disjointness | the ×2×3, the weakly mixing systems disjoint from the Kronecker / the zero-entropy |
| Rudolph–Joinings | the joining theory, the entropy / the relative, the Thouvenot |
The Furstenberg–Zimmer structure theorem and the amenable¶
Theorem (Furstenberg–Zimmer structure). The ergodic system \((X,T)\) has the distal tower (the Furstenberg–Zimmer distal / the proximal, the tower the transfinite, the structure of the ergodic action).
| Furstenberg–Zimmer / amenable | Description |
|---|---|
| Furstenberg–Zimmer structure | the distal tower (the distal / the proximal, the transfinite tower, the ergodic structure) |
| Amenable vs non-amenable | the amenable (the invariant mean, the \(\mathbb{Z}\)), the non-amenable (the free group / the higher rank, the homogeneous rigidity) |
| Furstenberg boundary | the Poisson boundary / the Furstenberg, the non-amenable, the boundary theory |
The Furstenberg recurrence and the multiple recurrence¶
Theorem (Furstenberg multiple recurrence). The \(\{x,T^{n_1}x,\dots,T^{n_k}x\}\) the multiple, the Furstenberg proof of the Szemerédi / the van der Waerden.
| Furstenberg recurrence | Description |
|---|---|
| Multiple recurrence | \(\{x,T^{n_1}x,\dots,T^{n_k}x\}\) (the multiple, the Furstenberg) |
| Szemerédi / van der Waerden | the Furstenberg proof (the ergodic / the multiple recurrence, the Szemerédi) |
| Polynomial (Bergelson–Leibman) | the polynomial recurrence (the polynomial Szemerédi, the Bergelson–Leibman) |
| Topological (Glasner–Weiss) | the topological multiple recurrence (the Glasner, the Weiss) |
The Furstenberg–Kesten and the random products¶
| Furstenberg–Kesten / random walk | Description |
|---|---|
| Furstenberg–Kesten | the random matrix products, the Lyapunov exponent (the Oseledets) |
| Random walk | the Poisson boundary / the Furstenberg, the random walk on the group / the boundary |
| Entropy / escape rate | the entropy / the rate of escape (the Furstenberg, the boundary) |
See it¶
Furstenberg ×2×3 disjointness and the Littlewood conjecture via the homogeneous dynamics. The ×2 and ×3 on \(\mathbb{T}\) (the \(x\mapsto2x\), \(x\mapsto3x\), the multiplicative, the Furstenberg disjointness 1967: the ×2×3 no common extension / the only joining the product). The LCD: the \(\liminf n\|n\alpha\|\|n\beta\|=0\) (the Littlewood, the Einsiedler–Katok–Lindenstrauss via the \(\mathrm{SL}_2(\mathbb{R})\times\mathrm{SL}_2(\mathbb{Q}_p)\) homogeneous, the ×2×3 the Furstenberg). The mechanism: the ×2×3 the Furstenberg disjointness / the invariant measure rigidity (the Einsiedler–Katok–Lindenstrauss: the invariant measures for the ×2×3 are the Haar / the atomic, the rigidity), the LCD the consequence (the \(\liminf=0\) the Littlewood, the Einsiedler–Katok–Lindenstrauss the partial via the homogeneous). The key point: the Furstenberg ×2×3 (the multiplicative disjointness) is the ergodic-theoretic input to the Littlewood conjecture, the Einsiedler–Katok–Lindenstrauss the homogeneous-dynamics approach — the number theory via the ergodic / the homogeneous.
Check¶
Show that the ×2 and ×3 on \(\mathbb{T}\) are disjoint (the Furstenberg 1967), and explain the joining.
The ×2 (\(T:x\mapsto2x\)) and ×3 (\(S:x\mapsto3x\)) on \(\mathbb{T}=\mathbb{R}/\mathbb{Z}\) (the Lebesgue measure \(\mu\), the ergodic — the \(\times 2\) and \(\times 3\) the ergodic for the Lebesgue). The joining: \(\lambda\) on \(\mathbb{T}\times\mathbb{T}\), invariant under \(T\times S\) (\(\lambda\) the \((2x,3y)\)-invariant), with the marginals \(\mu\) (the Lebesgue). The Furstenberg disjointness (1967): the only joining is the product \(\mu\otimes\mu\) (the no-common-extension). The proof: the ×2×3 the coprime (the 2 and 3 the multiplicative independent), the Furstenberg the ×2×3 disjoint (the joining the product, the no common extension). The mechanism: the ×2×3 the Furstenberg the disjointness, the invariant measure rigidity (the Einsiedler–Katok–Lindenstrauss the strengthening, the invariant measures the Haar / the atomic, the rigidity). So the ×2×3 disjoint, the only joining the product.
Explain the Furstenberg multiple recurrence and the Szemerédi theorem, and the polynomial (Bergelson–Leibman).
The Furstenberg multiple recurrence: for the ergodic \((X,T,\mu)\) and \(A\) with \(\mu(A)>0\), the \(\{x,T^{n_1}x,\dots,T^{n_k}x\}\) the multiple — the Furstenberg the multiple recurrence (\(\mu(\bigcap_{i=0}^{k-1}T^{-n_i}A)>0\) for some \(n_1,\dots,n_k\), the multiple recurrence). The Szemerédi: the subset of \(\mathbb{Z}\) with positive density contains arbitrarily long arithmetic progressions (the Szemerédi theorem) — the Furstenberg proof via the multiple recurrence (the ergodic / the Furstenberg, the correspondence the dense subset \(\to\) the ergodic system, the multiple recurrence the AP). The polynomial (Bergelson–Leibman): the polynomial recurrence (the \(\{x,T^{p_1(n)}x,\dots,T^{p_k(n)}x\}\) with \(p_i\) polynomials, the polynomial Szemerédi, the Bergelson–Leibman the polynomial generalization). The topological (Glasner–Weiss): the topological multiple recurrence (the topological / the measurable, the separate).
Pitfalls¶
- The Furstenberg ×2×3 the disjointness (the invariant measure rigidity the EKL the partial, the full disjointness the open) — the EKL the full-measure (the invariant measures the Haar / the atomic for the full-measure set), not the all (the full disjointness the open, the EKL the partial).
- The Furstenberg multiple recurrence / the Szemerédi (the measurable, the topological the Glasner–Weiss the separate) — the measurable the Furstenberg, the topological the Glasner–Weiss (the two related but distinct, the measurable / the topological).
- The Furstenberg–Zimmer distal tower the transfinite (the structure of the ergodic action), the amenable / the non-amenable the distinction (the amenable the invariant mean, the non-amenable the rigidity / the boundary).
- The Furstenberg boundary / the Poisson boundary the non-amenable (the non-amenable the boundary, the amenable the trivial), the boundary theory the Furstenberg.
- The Furstenberg–Kesten / the Kingman the subadditive ergodic (the random matrix products, the Lyapunov), the Oseledets the MET the connection.
Refs¶
- Furstenberg, "Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation" (Math. Systems Theory 1967)
- Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory (Princeton 1981); Bergelson–Leibman, "Polynomial extensions of van der Waerden's and Szemerédi's theorems" (JAMS 1996)
- Furstenberg–Zimmer, "The structure of ergodic systems" (the distal tower); Rudolph, Fundamentals of Measurable Dynamics (the joinings)
- Einsiedler–Katok–Lindenstrauss, "Invariant measures and the set of exceptions to the Littlewood conjecture" (Annals 2006)
Next¶
ET index · Foundations · Homogeneous & Ratner · Oseledets & Pesin · Ornstein & Isomorphism · Analytic number theory · Dynamical systems