Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Corollary 14.5. Suppose that the search directions satisfy (14.5). If the step size is chosen by the exact minimization rule or limited minimization rule, then any limit \( {x}^{ * } \) of the sequence \( {\left\{ {x}_{k}\right\} }_{0}^{\infty } \) is a critical point of \( f \), that is, \( \nabla f\left( {x}^{ * }\ri... | Proof. Suppose that \( \mathop{\lim }\limits_{{i \rightarrow \infty }}{x}_{{k}_{i}} = {x}^{ * } \), but \( \nabla f\left( {x}^{ * }\right) \neq 0 \) . Let \( {\widehat{x}}_{{k}_{i}} = {x}_{{k}_{i}} + {\widehat{\alpha }}_{{k}_{i}}{d}_{{k}_{i}} \) be the point that would be chosen using an Armijo-type rule. Then\n\n\[ f\... | Yes |
Lemma 14.6. (Zoutendijk) Let \( f \) be a function bounded from below on the sublevel set \( M \mathrel{\text{:=}} \left\{ {x : f\left( x\right) \leq f\left( {x}_{0}\right) }\right\} \), with a Lipschitz continuous gradient there, that is, for some \( L > 0 \), \[ \parallel \nabla f\left( y\right) - \nabla f\left( x\ri... | Proof. Recalling \( \begin{Vmatrix}{d}_{k}\end{Vmatrix} = 1 \), we have \[ \left( {{c}_{2} - 1}\right) \left\langle {\nabla f\left( {x}_{k}\right) ,{d}_{k}}\right\rangle \leq \left\langle {\nabla f\left( {x}_{k + 1}\right) - \nabla f\left( {x}_{k}\right) ,{d}_{k}}\right\rangle \leq {t}_{k}L \] where the inequalities fo... | Yes |
Lemma 14.8. (Kantorovich’s inequality) If \( Q \) is a symmetric positive definite \( n \times n \) matrix with eigenvalues \( {\left\{ {\lambda }_{i}\right\} }_{1}^{n} \) in the interval \( \left\lbrack {m, M}\right\rbrack \), then\n\n\[ \frac{\langle {Qx}, x\rangle \cdot \langle {Q}^{-1}x, x\rangle }{\parallel x{\par... | Proof. Since the above inequality remains unchanged if we replace each \( Q \) with \( {\tau Q} \), where \( \tau \) is any positive constant, we assume that \( {mM} = 1 \) . Let \( Q = {U}^{T}{\Lambda U} \) be the spectral decomposition of \( Q \), where \( \Lambda = \operatorname{diag}\left( {{\lambda }_{1},\ldots ,{... | Yes |
Theorem 14.9. In the steepest-descent method for minimizing a strongly convex quadratic function \( q\left( x\right) \) in (14.9), the optimality gap \( E\left( x\right) = q\left( x\right) - \mathop{\min }\limits_{{\mathbb{R}}^{n}}q \) decreases at a geometric rate,\n\n\[ E\left( {x}_{k + 1}\right) \leq {\left( \frac{\... | Proof. Since \( {x}_{k + 1} - {x}_{k} = - {\alpha }_{k}{r}_{k} \), we have\n\n\[ E\left( {x}_{k + 1}\right) = E\left( {x}_{k}\right) - {\alpha }_{k}{\begin{Vmatrix}{r}_{k}\end{Vmatrix}}^{2} + \frac{{\alpha }_{k}^{2}}{2}\left\langle {Q{r}_{k},{r}_{k}}\right\rangle = E\left( {x}_{k}\right) - \frac{1}{2}\frac{{\begin{Vmat... | Yes |
In the steepest-descent method for minimizing a strongly convex quadratic function \( q \) on \( {\mathbb{R}}^{n} \), the optimality gap \( E\left( x\right) = q\left( x\right) - \mathop{\min }\limits_{{\mathbb{R}}^{n}}q \) is halved in every \( O\left( \kappa \right) \) iterations, where \( \kappa = {\lambda }_{\max }/... | Proof. Let \( m \) be the smallest integer satisfying the condition \( {\left( 1 - 2/\left( \kappa + 1\right) \right) }^{m} \leq \) \( 1/2 \) . We have \( \frac{E\left( {x}_{m}\right) }{E\left( {x}_{0}\right) } \leq \frac{1}{2} \), and if \( \kappa \) is large, then\n\n\[ \n- \ln 2 \approx m\ln \left( {1 - \frac{2}{\ka... | Yes |
If \( f \) is a convex function with a Lipschitz continuous derivative satisfying\n\n\[ \parallel \nabla f\left( x\right) - \nabla f\left( y\right) \parallel \leq L\parallel x - y\parallel \;\text{ for all }\;x, y \in {\mathbb{R}}^{n}, \]\n\nthen\n\n\[ f\left( y\right) \leq f\left( x\right) + \langle \nabla f\left( x\r... | Proof. Define the function \( g\left( t\right) \mathrel{\text{:=}} f\left( {x + t\left( {y - x}\right) }\right) \), and note that\n\n\[ g\left( 1\right) - g\left( 0\right) = {\int }_{0}^{1}{g}^{\prime }\left( t\right) {dt} = {g}^{\prime }\left( 0\right) + {\int }_{0}^{1}\left( {{g}^{\prime }\left( t\right) - {g}^{\prim... | Yes |
Lemma 14.12. Let \( f : C \rightarrow \mathbb{R} \) be a differentiable convex function on the convex set \( C \) . The following conditions are equivalent:\n\n\[ \parallel \nabla f\left( y\right) - \nabla f\left( x\right) \parallel \leq L\parallel y - x\parallel \;\text{ for all }\;x, y \in C. \]\n\n(14.10)\n\n\[ \fra... | Proof. \( \left( {14.10}\right) \Rightarrow \left( {14.11}\right) \) : Notice that Lemma 14.11 gives the second inequality in (14.11); to prove the first inequality, define the function\n\n\[ g\left( y\right) \mathrel{\text{:=}} f\left( y\right) - \langle \nabla f\left( x\right), y\rangle \]\n\nwith the gradient \( \na... | Yes |
Lemma 14.15. Let \( C \subseteq {\mathbb{R}}^{n} \) be a closed convex set, and \( f : C \rightarrow \mathbb{R} \) a differentiable function. Let \( s > 0 \). A point \( {x}^{ * } \in C \) satisfies the variational inequality\n\n\[ \left\langle {\nabla f\left( {x}^{ * }\right), x - {x}^{ * }}\right\rangle \geq 0\text{ ... | Proof. By Theorem 6.1, \( {\Pi }_{C}\left( {{x}^{ * } - s\nabla f\left( {x}^{ * }\right) }\right) = {x}^{ * } \) if and only if\n\n\[ \left\langle {\left( {{x}^{ * } - s\nabla f\left( {x}^{ * }\right) }\right) - {x}^{ * }, x - {x}^{ * }}\right\rangle \leq 0\;\text{ for all }\;x \in C, \]\n\nwhich is clearly equivalent ... | Yes |
Lemma 14.19. If \( f \) and \( {x}^{ * } \) are as in the above theorem, then\n\n\[ f\left( x\right) = {\int }_{0}^{1}{Df}\left( {{x}^{ * } + t\left( {x - {x}^{ * }}\right) }\right) \left( {x - {x}^{ * }}\right) {dt}. \] | Proof. For the coordinate function \( {f}_{i} \), define \( \alpha \left( t\right) = {f}_{i}\left( {{x}^{ * } + t\left( {x - {x}^{ * }}\right) }\right) \) . We have \( {\alpha }^{\prime }\left( t\right) = \left\langle {\nabla {f}_{i}\left( {{x}^{ * } + t\left( {x - {x}^{ * }}\right) }\right), x - {x}^{ * }}\right\rangl... | Yes |
Lemma 14.23. A set \( {\left\{ {d}_{i}\right\} }_{i = 1}^{k} \) of \( Q \) -conjugate directions is linearly independent. | Proof. If \( {\alpha }_{1}{d}_{1} + \cdots + {\alpha }_{k}{d}_{k} = 0 \), then\n\n\[ 0 = \left\langle {\mathop{\sum }\limits_{{i = 1}}^{k}{\alpha }_{i}{d}_{i}, Q{d}_{j}}\right\rangle = \mathop{\sum }\limits_{{i = 1}}^{k}{\alpha }_{i}\left\langle {{d}_{i}, Q{d}_{j}}\right\rangle = {\alpha }_{j}\left\langle {{d}_{j}, Q{d... | Yes |
Theorem 14.24. Let \( {\left\{ {d}_{i}\right\} }_{i = 0}^{n - 1} \) be a set of \( Q \) -conjugate directions in \( {\mathbb{R}}^{n} \), and \( {\left\{ {x}_{i}\right\} }_{i = 0}^{n - 1} \) the points generated by the conjugate-direction method using these directions. Then \( {x}_{k} \) is the global minimizer of \( q ... | Proof. Let \( \bar{x} = {x}_{0} + {\gamma }_{0}{d}_{0} + \cdots + {\gamma }_{k - 1}{d}_{k - 1} \) be the minimizer of \( q \) on \( {M}_{k} \). We will show that \( {\gamma }_{i} = {\alpha }_{i}, i = 0,\ldots, k - 1 \), which will imply the first statement of the theorem. By (14.42), we have \[ {\gamma }_{i} = - \frac{... | Yes |
Theorem 14.25. Let the gradient vectors \( {\left\{ {r}_{i}\right\} }_{i = 0}^{k} \) be all nonzero. Then \( {\left\{ {r}_{i}\right\} }_{i = 0}^{k} \) are mutually orthogonal, that is,\n\n\[ \n\left\langle {{r}_{i},{r}_{j}}\right\rangle = 0\text{ for all }i \neq j.\n\]\n\nMoreover,\n\n\[ \n{d}_{k} = - {r}_{k} + {\beta ... | Proof. Note that the vectors \( {\left\{ {d}_{i}\right\} }_{0}^{k - 1} \) are generated from the vectors \( {\left\{ {r}_{i}\right\} }_{0}^{k - 1} \) by the Gram-Schmidt process, so that\n\n\[ \n\operatorname{span}\left\{ {{d}_{0},\ldots ,{d}_{k - 1}}\right\} = \operatorname{span}\left\{ {{r}_{0},\ldots ,{r}_{k - 1}}\r... | Yes |
Corollary 14.27. Let \( {\left\{ {x}_{i}\right\} }_{i = 0}^{k},{\left\{ {r}_{i}\right\} }_{i = 0}^{k} \), and \( {\left\{ {d}_{i}\right\} }_{i = 0}^{k} \) be the iterates, gradients, and conjugate direction vectors, respectively, generated by a conjugate-gradient method.\n\nIf all the gradient vectors \( {r}_{i} \) are... | Proof. The first equality was already proved in Corollary 14.26; to prove the second, we use induction on \( k \) . The equality is trivially true for \( k = 0 \) ; assuming that it is true for \( k - 1 \), let us prove its truth for \( k \) . Let \( {\left\{ {r}_{i}\right\} }_{i = 0}^{k} \) all be nonzero vectors. It ... | Yes |
Theorem 14.29. (Expanding subspace theorem)The point \( {x}_{k} \) generated by the conjugate-gradient method has the variational characterization\n\n\[ E\left( {x}_{k}\right) = \frac{1}{2}\min \left\{ {{\begin{Vmatrix}p\left( Q\right) \left( {x}_{0} - {x}^{ * }\right) \end{Vmatrix}}_{Q}^{2} : p \in {\mathcal{P}}_{k}, ... | Proof. It follows from Theorem 14.24 and Corollary 14.27 that \( {x}_{k} \) is the minimizer \( q \) on the affine subspace \( {M}_{k - 1} = {x}_{0} + \mathcal{K}\left( {Q,{r}_{0}, k - 1}\right) \) . If \( x \in {M}_{k - 1} \) , write\n\n\[ x = {x}_{0} + {\gamma }_{0}{r}_{0} + {\gamma }_{1}Q{r}_{0} + \cdots + {\gamma }... | Yes |
Theorem 14.30. Let \( {x}_{k + 1} \) be the \( \left( {k + 1}\right) \) th point generated by the conjugate-gradient method. If \( {\left\{ {\lambda }_{i}\right\} }_{i = 1}^{n} \) are the eigenvalues of the matrix \( Q \), then\n\n\[ \frac{E\left( {x}_{k + 1}\right) }{E\left( {x}_{0}\right) } \leq \mathop{\min }\limits... | Proof. Let \( Q = {U\Lambda }{U}^{T} \) be the spectral decomposition of \( Q \), where \( \Lambda = \) \( \operatorname{diag}\left( {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right) \), and write \( {x}_{0} - {x}^{ * } = {U\delta } \).\n\nIf \( p \in {\mathcal{P}}_{k} \) is such that \( 1 + {tp}\left( t\right) \) is the ... | Yes |
Corollary 14.31. Let \( {x}_{k} \) be the \( k \) th point generated by the conjugate-gradient method. Let \( \kappa = {\lambda }_{\max }/{\lambda }_{\min } \) be the condition number of \( Q \), where \( {\lambda }_{\max } \) and \( {\lambda }_{\min } \) are the largest and smallest eigenvalues of \( Q \), respectivel... | Proof. Write \( m = {\lambda }_{\min } \) and \( M = {\lambda }_{\max } \) . Theorem 14.30 implies\n\n\[ \frac{E\left( {x}_{k}\right) }{E\left( {x}_{0}\right) } \leq \mathop{\min }\limits_{p}\mathop{\max }\limits_{i}p{\left( {\lambda }_{i}\right) }^{2} \leq \mathop{\min }\limits_{p}\mathop{\max }\limits_{{x \in \left\l... | Yes |
Corollary 14.32. The optimality gap \( E\left( x\right) = q\left( x\right) - q\left( {x}^{ * }\right) \) is halved in every \( O\left( \sqrt{\kappa }\right) \) iterations of the conjugate-gradient method. | Proof. See the proof of Corollary 14.10. | No |
Write \( \mathbb{T} \) for the quotient group \( \mathbb{R}/\mathbb{Z} = \{ x + \mathbb{Z} \mid x \in \mathbb{R}\} \), which can be identified with a circle (as a topological space, this can also be obtained as a quotient space of \( \left\lbrack {0,1}\right\rbrack \) by identifying 0 with 1); there is a natural biject... | To see this, assume first that \( x = \frac{p}{q} \) is rational. In this case the orbit of \( x \) is some subset of \( \left\{ {0,\frac{1}{q},\ldots ,\frac{q - 1}{q}}\right\} \) . Conversely, if the orbit is finite then there must be integers \( m, n \) with \( 1 \leq n < m \) for which \( {T}^{m}\left( x\right) = {T... | Yes |
We show later that the circle rotation \( {R}_{\alpha } : \mathbb{T} \rightarrow \mathbb{T} \) defined by \( {R}_{\alpha }\left( t\right) = t + \alpha \left( {\;\operatorname{mod}\;1}\right) \) is uniquely ergodic if \( \alpha \) is irrational (see Definition 4.9 and Example 4.11). A consequence of this is that for any... | \[ \frac{1}{N}\mathop{\sum }\limits_{{n = 0}}^{{N - 1}}{\chi }_{\lbrack a, b)}\left( {{R}_{\alpha }^{n}\left( t\right) }\right) \rightarrow b - a \] as \( N \rightarrow \infty \) for every \( t \in \mathbb{T} \) (see Theorem 4.10 and Lemma 4.17). | No |
For any \( \alpha \in \mathbb{R} \), define the circle rotation by \( \alpha \) to be the map \[ {R}_{\alpha } : \mathbb{T} \rightarrow \mathbb{T},{R}_{\alpha }\left( t\right) = t + \alpha \;\left( {\;\operatorname{mod}\;1}\right) . \] We claim that \( {R}_{\alpha } \) preserves the Lebesgue measure \( {m}_{\mathbb{T}}... | By Theorem A.8, it is enough to prove it for intervals, where it is clear. Alternatively, we may note that Lebesgue measure is a Haar measure on the compact group \( \mathbb{T} \), which is invariant under any translation by construction (see Sects. 8.3 and C.2). | Yes |
The circle-doubling map is \( {T}_{2} : \mathbb{T} \rightarrow \mathbb{T},{T}_{2}\left( t\right) = {2t}\left( {\;\operatorname{mod}\;1}\right) \). We claim that \( {T}_{2} \) preserves the Lebesgue measure \( {m}_{\mathbb{T}} \) on the circle. | By Theorem A.8, it is sufficient to check this on intervals, so let \( B = \lbrack a, b) \subseteq \lbrack 0,1) \) be any interval. Then it is easy to check that\n\n\[ {T}_{2}^{-1}\left( B\right) = \left\lbrack {\frac{a}{2},\frac{b}{2}}\right) \cup \left\lbrack {\frac{a}{2} + \frac{1}{2},\frac{b}{2} + \frac{1}{2}}\righ... | Yes |
Generalizing Example 2.4, let \( X \) be a compact abelian group and let \( T : X \rightarrow X \) be a surjective endomorphism. Then \( T \) preserves the Haar measure \( {m}_{X} \) on \( X \) by the following argument. | Define a measure \( \mu \) on \( X \) by \( \mu \left( A\right) = {m}_{X}\left( {{T}^{-1}A}\right) \). Then, given any \( x \in X \) pick \( y \) with \( T\left( y\right) = x \) and notice that\n\n\[ \mu \left( {A + x}\right) = {m}_{X}\left( {{T}^{-1}\left( {A + x}\right) }\right) = {m}_{X}\left( {{T}^{-1}A + y}\right)... | Yes |
Lemma 2.6. A measure \( \mu \) on \( X \) is \( T \) -invariant if and only if\n\n\[ \int f\mathrm{\;d}\mu = \int f \circ T\mathrm{\;d}\mu \] \n\n(2.1) \n\nfor all \( f \in {\mathcal{L}}^{\infty } \) . Moreover, if \( \mu \) is \( T \) -invariant, then (2.1) holds for \( f \in {L}_{\mu }^{1} \) . | Proof. If (2.1) holds, then for any measurable set \( B \) we may take \( f = {\chi }_{B} \) to see that\n\n\[ \mu \left( B\right) = \int {\chi }_{B}\mathrm{\;d}\mu = \int {\chi }_{B} \circ T\mathrm{\;d}\mu = \int {\chi }_{{T}^{-1}B}\mathrm{\;d}\mu = \mu \left( {{T}^{-1}B}\right) , \] \n\nso \( T \) preserves \( \mu \)... | Yes |
The shift map in Example 2.8 is an example of a one-sided Bernoulli shift. A more general \( {}^{\left( {13}\right) } \) and natural two-sided definition is the following. Consider an infinitely repeated throw of a loaded \( n \) -sided die. The possible outcomes of each throw are \( \{ 1,2,\ldots, n\} \), and these ap... | \[ X = \{ 1,2,\ldots, n{\} }^{\mathbb{Z}} = \left\{ {x = \left( {\ldots ,{x}_{-1},{x}_{0},{x}_{1},\ldots }\right) \mid {x}_{i} \in \{ 1,2,\ldots, n\} \text{ for all }i \in \mathbb{Z}}\right\} . \] The measure on \( X \) is the infinite product measure \( \mu = \mathop{\prod }\limits_{\mathbb{Z}}{\mu }_{\mathbf{p}} \), ... | Yes |
Consider the 2-to-1 map \( T : \mathbb{R} \rightarrow \mathbb{R} \) defined by\n\n\[ T\left( x\right) = \frac{1}{2}\left( {x - \frac{1}{x}}\right) \]\n\nfor \( x \neq 0 \), and \( T\left( 0\right) = 0 \). For any \( {L}^{1} \) function \( f \), the substitution \( y = T\left( x\right) \) shows that | \[ {\int }_{-\infty }^{\infty }f\left( {T\left( x\right) }\right) \frac{\mathrm{d}x}{\pi \left( {1 + {x}^{2}}\right) } = {\int }_{-\infty }^{\infty }f\left( y\right) \frac{\mathrm{d}y}{\pi \left( {1 + {y}^{2}}\right) } \]\n\n(in this calculation, note that \( T \) is only injective when restricted to \( \left( {0,\inft... | Yes |
Theorem 2.11 (Poincaré Recurrence). Let \( T : X \rightarrow X \) be a measure-preserving transformation on a probability space \( \left( {X,\mathcal{B},\mu }\right) \), and let \( E \subseteq X \) be a measurable set. Then almost every point \( x \in E \) returns to \( E \) infinitely often. That is, there exists a me... | Proof. Let \( B = \left\{ {x \in E \mid {T}^{n}x \notin E}\right. \) for any \( \left. {n \geq 1}\right\} \) . Then\n\n\[ B = E \cap {T}^{-1}\left( {X \smallsetminus E}\right) \cap {T}^{-2}\left( {X \smallsetminus E}\right) \cap \cdots ,\]\n\nso \( B \) is measurable. Now, for any \( n \geq 1 \),\n\n\[ {T}^{-n}B = {T}^... | Yes |
Example 2.12. The map \( T : \mathbb{R} \rightarrow \mathbb{R} \) defined by \( T\left( x\right) = x + 1 \) preserves the Lebesgue measure \( {m}_{\mathbb{R}} \) on \( \mathbb{R} \) . Just as in Definition 2.1, this means that\n\n\[ \n{m}_{\mathbb{R}}\left( {{T}^{-1}A}\right) = {m}_{\mathbb{R}}\left( A\right) \n\] \n\n... | For any bounded set \( E \subseteq \mathbb{R} \) and any \( x \in E \) , the set \n\n\[ \n\left\{ {n \geq 1 \mid {T}^{n}x \in E}\right\} \n\] \n\nis finite. Thus the map \( T \) exhibits no recurrence. | No |
Proposition 2.14. The following are equivalent properties for a measure-preserving transformation \( T \) of \( \left( {X,\mathcal{B},\mu }\right) \) .\n\n(1) \( T \) is ergodic.\n\n(2) For any \( B \in \mathcal{B},\mu \left( {{T}^{-1}B\bigtriangleup B}\right) = 0 \) implies that \( \mu \left( B\right) = 0 \) or \( \mu... | Proof of Proposition 2.14. (1) \( \Rightarrow \) (2): Assume that \( T \) is ergodic, so the implication (2.2) holds, and let \( B \) be an almost invariant measurable set-that is, a measurable set \( B \) with \( \mu \left( {{T}^{-1}B\bigtriangleup B}\right) = 0 \) . We wish to construct an invariant set from \( B \),... | Yes |
Proposition 2.16. The circle rotation \( {R}_{\alpha } : \mathbb{T} \rightarrow \mathbb{T} \) is ergodic with respect to the Lebesgue measure \( {m}_{\mathbb{T}} \) if and only if \( \alpha \) is irrational. | Proof. If \( \alpha \in \mathbb{Q} \), then we may write \( \alpha = \frac{p}{q} \) in lowest terms, so \( {R}_{\alpha }^{q} = {I}_{\mathbb{T}} \) is the identity map. Pick any measurable set \( A \subseteq \mathbb{T} \) with \( 0 < {m}_{\mathbb{T}}\left( A\right) < \frac{1}{q} \) . Then\n\n\[ B = A \cup {R}_{\alpha }A... | Yes |
Proposition 2.17. The circle-doubling map \( {T}_{2} : \mathbb{T} \rightarrow \mathbb{T} \) from Example 2.4 is ergodic (with respect to Lebesgue measure). | Proof. By Example 2.8, \( {T}_{2} \) and the Bernoulli shift \( \sigma \) on \( X = \{ 0,1{\} }^{\mathbb{N}} \) together with the fair coin-toss measure are measurably isomorphic. By Proposition 2.15 the latter is ergodic, and it is clear that measurably isomorphic systems are either both ergodic or both not ergodic. | Yes |
Lemma 2.18. A measure-preserving transformation \( T \) is ergodic if and only if 1 is a simple eigenvalue of the associated operator \( {U}_{T} \) . Hence ergodicity is a unitary property. | Proof. This follows from the proof of the equivalence of (2) and (5) in Proposition 2.14 or via Exercise 2.3.5 applied with \( p = 2 \) : an eigenfunction for the eigenvalue 1 is a \( T \) -invariant function, and ergodicity is characterized by the property that the only \( T \) -invariant functions are the constants. | No |
Theorem 2.19. Let \( T : X \rightarrow X \) be a continuous surjective homomorphism of a compact abelian group \( X \) . Then \( T \) is ergodic with respect to the Haar measure \( {m}_{X} \) if and only if the identity \( \chi \left( {{T}^{n}x}\right) = \chi \left( x\right) \) for some \( n > 0 \) and character \( \ch... | Proof. First assume that there is a non-trivial character \( \chi \) with\n\n\[ \chi \left( {{T}^{n}x}\right) = \chi \left( x\right) \]\n\nfor some \( n > 0 \), chosen to be minimal with this property. Then the function\n\n\[ f\left( x\right) = \chi \left( x\right) + \chi \left( {Tx}\right) + \cdots + \chi \left( {{T}^... | Yes |
Corollary 2.22. \( {}^{\left( {19}\right) } \) Let \( \left( {X,\mathcal{B},\mu, T}\right) \) be a measure-preserving system. Then for any function \( f \in {L}_{\mu }^{1} \) the ergodic averages \( {\mathrm{A}}_{N}^{f} \) converge in \( {L}_{\mu }^{1} \) to a \( T \) - invariant function \( {f}^{\prime } \in {L}_{\mu ... | Proof. By the mean ergodic theorem (Theorem 2.21) we know that for any \( g \in {L}_{\mu }^{\infty } \subseteq {L}_{\mu }^{2} \), the ergodic averages \( {\mathrm{A}}_{N}^{g} \) converge in \( {L}_{\mu }^{2} \) to some \( {g}^{\prime } \in {L}_{\mu }^{2} \) . We claim that \( {g}^{\prime } \in {L}_{\mu }^{\infty } \) .... | Yes |
Let \( \left( {X,{\mathcal{B}}_{X},\mu, T}\right) \) be a measure-preserving system, and fix a small measurable set \( B \in {\mathcal{B}}_{X} \) with \( \mu \left( B\right) = \varepsilon > 0 \) . Consider the ergodic average\n\n\[ \n{\mathrm{A}}_{N}^{{\chi }_{B}} = \frac{1}{N}\mathop{\sum }\limits_{{n = 0}}^{{N - 1}}{... | Since \( T \) preserves \( \mu ,{\int }_{X}{\chi }_{B} \circ {T}^{n}\mathrm{\;d}\mu = \mu \left( B\right) \) for any \( n \geq 0 \), so\n\n\[ \n{\int }_{X}{\mathrm{\;A}}_{N}^{{\chi }_{B}}\mathrm{\;d}\mu = {\int }_{X}{\chi }_{B}\mathrm{\;d}\mu = \mu \left( B\right) = \varepsilon .\n\]\n\nNotice that\n\n\[ \n\sqrt{\varep... | Yes |
Proposition 2.26 (Maximal Inequality). Let \( U : {L}_{\mu }^{1} \rightarrow {L}_{\mu }^{1} \) be a positive linear operator with \( \parallel U\parallel \leq 1 \) . For \( f \in {L}_{\mu }^{1} \) a real-valued function, define inductively the functions\n\n\[ \n{f}_{0} = 0 \n\] \n\n\[ \n{f}_{1} = f \n\] \n\n\[ \n{f}_{2... | Proof. For each \( N \), it is clear that \( {F}_{N} \in {L}_{\mu }^{1} \) . Since \( U \) is positive and linear, and since \n\n\[ \n{F}_{N} \geq {f}_{n} \n\] \n\nfor \( 0 \leq n \leq N \), we have \n\n\[ \nU{F}_{N} + f \geq U{f}_{n} + f = {f}_{n + 1}. \n\] \n\nHence \n\n\[ \nU{F}_{N} + f \geq \mathop{\max }\limits_{{... | Yes |
Lemma 2.27 (Finite Vitali covering lemma). Let \( {B}_{{r}_{1}}\left( {a}_{1}\right) ,\ldots ,{B}_{{r}_{K}}\left( {a}_{K}\right) \) be any collection of balls in a metric space. Then there exists a subcollection \( {B}_{{r}_{j\left( 1\right) }}\left( {a}_{j\left( 1\right) }\right) ,\ldots ,{B}_{{r}_{j\left( k\right) }}... | Proof. By reordering the balls if necessary, we may assume that \[ {r}_{1} \geq {r}_{2} \geq \cdots \geq {r}_{K} \] Let \( j\left( 1\right) = 1 \) . We choose the remaining disjoint balls by induction as follows. Assume that we have chosen \( j\left( 1\right) ,\ldots, j\left( n\right) \) from the indices \( \{ 1,\ldots... | Yes |
For any \( \phi \in {\ell }^{1}\left( \mathbb{Z}\right) \) and \( \alpha > 0 \), define\n\n\[ \n{\phi }^{ * }\left( a\right) = \mathop{\sup }\limits_{{n \geq 1}}\frac{1}{n}\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\phi \left( {a + i}\right)\n\]\n\nand\n\n\[ \n{E}_{\alpha }^{\phi } = \left\{ {a \in \mathbb{Z} \mid {\phi ... | Proof of Lemma 2.29. Let \( {a}_{1},\ldots ,{a}_{K} \) be different elements of \( {E}_{\alpha }^{\phi } \), and let \( \ell \left( j\right) \) for \( j = 1,\ldots, K \) be chosen so that\n\n\[ \n\frac{1}{\ell \left( j\right) }\mathop{\sum }\limits_{{i = 0}}^{{\ell \left( j\right) - 1}}\phi \left( {{a}_{j} + i}\right) ... | Yes |
In Example 1.2 we explained that almost every real number has the property that any block of length \( k \) of digits base 10 appears with asymptotic frequency \( \frac{1}{{10}^{k}} \), thus almost every number is normal base 10 . We now have all the material needed to justify this result: By Corollary 2.20, the map \(... | FIRST Proof of THEOREM 2.30. Recall that \( \left( {X,\mathcal{B},\mu, T}\right) \) is a measure-preserving system, \( \mu \left( X\right) = 1 \), and \( f \in {\mathcal{L}}_{\mu }^{1} \) . It is sufficient to prove the result for a real-valued function \( f \) . Define, for any \( x \in X \) ,\n\n\[ \n{f}^{ * }\left( ... | Yes |
A circle rotation \( {R}_{\alpha } : \mathbb{T} \rightarrow \mathbb{T} \) is not mixing. | There is a sequence \( {n}_{j} \rightarrow \infty \) for which \( {n}_{j}\alpha \left( {\;\operatorname{mod}\;1}\right) \rightarrow 0 \) (if \( \alpha \) is rational we may choose to have \( {n}_{j}\alpha \left( {\;\operatorname{mod}\;1}\right) = 0 \) ). If \( A = B = \left\lbrack {0,\frac{1}{2}}\right\rbrack \) then \... | Yes |
Lemma 2.43. The induced transformation \( {T}_{A} \) is a measure-preserving transformation on the space \( \left( {A,{\left. \mathcal{B}\right| }_{A},{\mu }_{A} = {\left. \frac{1}{\mu \left( A\right) }\mu \right| }_{A},{T}_{A}}\right) \) . If \( T \) is ergodic with respect to \( \mu \) then \( {T}_{A} \) is ergodic w... | Proof of Lemma 2.43. If \( B \subseteq A \) is measurable, then \( B = \mathop{\bigsqcup }\limits_{{n \geq 1}}B \cap {A}_{n} \) is a disjoint union so\n\n\[ \n{\mu }_{A}\left( B\right) = \frac{1}{\mu \left( A\right) }\mathop{\sum }\limits_{{n \geq 1}}\mu \left( {B \cap {A}_{n}}\right) \n\]\n\n(2.37)\n\nNow\n\n\[ \n{T}_... | Yes |
Theorem 2.44 (Kac). Let \( \left( {X,\mathcal{B},\mu, T}\right) \) be an ergodic measure-preserving system and let \( A \in \mathcal{B} \) have \( \mu \left( A\right) > 0 \) . Then the expected return time to \( A \) is \( \frac{1}{\mu \left( A\right) } \) ; equivalently\n\n\[{\int }_{A}{r}_{A}\mathrm{\;d}\mu = 1\] | \( {\text{Proof}}^{\left( {35}\right) } \) . Referring to Fig. 2.2, each column\n\n\[{A}_{n} \sqcup T\left( {A}_{n}\right) \sqcup \cdots \sqcup {T}^{n - 1}\left( {A}_{n}\right)\]\n\ncomprises \( n \) disjoint sets each of measure \( \mu \left( {A}_{n}\right) \), and the entire skyscraper contains almost all of \( X \) ... | Yes |
Lemma 2.45 (Kakutani-Rokhlin). Let \( \\left( {X,\\mathcal{B},\\mu, T}\\right) \) be an invertible ergodic measure-preserving system and assume that \( \\mu \) is non-atomic (that is, \( \\mu \\left( {\\{ x\\} }\\right) = 0 \) for all \( x \\in X \) ). Then for any \( n \\geq 1 \) and \( \\varepsilon > 0 \) there is a ... | Proof of Lemma 2.45. Let \( A \) be a measurable set with \( 0 < \\mu \\left( A\\right) < \\varepsilon /n \) (such a set exists by the assumption that \( \\mu \) is non-atomic) and form the Kakutani skyscraper over \( A \) . Then \( X \) decomposes into a union of disjoint columns of the form\n\n\[ {A}_{k} \\sqcup T\\l... | Yes |
Fix a sequence \( {\left( {a}_{n}\right) }_{n \geq 0} \) with \( {a}_{0} \in {\mathbb{N}}_{0} \) and \( {a}_{n} \in \mathbb{N} \) for \( n \geq 1 \) . Then the rational numbers\n\n\[ \frac{{p}_{n}}{{q}_{n}} = \left\lbrack {{a}_{0};{a}_{1},{a}_{2},\ldots ,{a}_{n}}\right\rbrack \]\n\nfor \( n \geq 0 \) with coprime numer... | Proof. Notice first that the sequence \( {\left( {a}_{n}\right) }_{n \geq 0} \) defines the sequences \( {\left( {p}_{n}\right) }_{n \geq - 1} \) and \( {\left( {q}_{n}\right) }_{n \geq - 1} \) . The claim of the lemma is proved by induction on \( n \) . Assume that (3.3) holds for \( 0 \leq n \leq k - 1 \) and \( {p}_... | Yes |
Lemma 3.2. Let \( {a}_{n} \in \mathbb{N} \) for all \( n \geq 0 \) . Then the limit in (3.10) is irrational. | Proof. Suppose that \( u = \frac{a}{b} \in \mathbb{Q} \) . Then, by (3.14), \[ \left| {{q}_{n}a - b{p}_{n}}\right| < \frac{b}{{a}_{n + 1}{q}_{n}} \leq \frac{b}{{q}_{n}}. \] Since \( {q}_{n} \rightarrow \infty \) by the inequality (3.6) and \( {q}_{n}a - b{p}_{n} \in \mathbb{Z} \) we see that \[ {q}_{n}a - b{p}_{n} = 0 ... | Yes |
Lemma 3.4. The map that sends the sequence\n\n\\[ \n\\left( {{a}_{0},{a}_{1},\\ldots }\\right) \\in {\\mathbb{N}}_{0} \\times {\\mathbb{N}}^{\\mathbb{N}}\n\\]\n\nto the limit in (3.10) is injective. | Proof. Let \\( u = \\left( {{a}_{0},{a}_{1},\\ldots }\\right) \\in {\\mathbb{N}}_{0} \\times {\\mathbb{N}}^{\\mathbb{N}} \\) be given. Then it is clear that\n\n\\[ \nu = \\left\\lbrack {{a}_{0};{a}_{1},\\ldots }\\right\\rbrack \\]\n\nis positive. Applying this to \\( \\left( {{a}_{1},{a}_{2},\\ldots }\\right) \\) and t... | Yes |
Lemma 3.5. The continued fraction map \( T\left( x\right) = \left\{ \frac{1}{x}\right\} \) on \( \left( {0,1}\right) \) preserves the Gauss measure \( \mu \) given by\n\n\[ \mu \left( A\right) = \frac{1}{\log 2}{\int }_{A}\frac{1}{1 + x}\mathrm{\;d}x \] \n\nfor any Borel measurable set \( A \subseteq \left\lbrack {0,1}... | Proof of Lemma 3.5. It is sufficient to show that \( \mu \left( {{T}^{-1}\left\lbrack {0, s}\right\rbrack }\right) = \mu \left( \left\lbrack {0, s}\right\rbrack \right) \) for every \( s > 0 \) . Clearly\n\n\[ {T}^{-1}\left\lbrack {0, s}\right\rbrack = \{ x \mid 0 \leq T\left( x\right) \leq s\} = \mathop{\bigsqcup }\li... | Yes |
For any irrational \( x \in \left\lbrack {0,1}\right\rbrack \smallsetminus \mathbb{Q} \) the sequence \( \left( {{a}_{n}\left( x\right) }\right) \) defined in (3.18) gives the digits of the continued fraction expansion to \( x \) . That is,\n\n\[ x = \left\lbrack {{a}_{1}\left( x\right) ,{a}_{2}\left( x\right) ,\ldots ... | Proof. Define \( {a}_{n} = {a}_{n}\left( x\right) \) and let \( u = \left\lbrack {{a}_{1},{a}_{2},\ldots }\right\rbrack \) be the limit as in (3.10) with \( {a}_{0} = 0 \) . By (3.11) we have\n\n\[ \frac{{p}_{2n}}{{q}_{2n}} < u < \frac{{p}_{{2n} + 1}}{{q}_{{2n} + 1}} \]\n\nand by (3.8) and the inequality (3.6) we have\... | Yes |
Theorem 3.7. The continued fraction map \( T\left( x\right) = \left\{ \frac{1}{x}\right\} \) on \( \left( {0,1}\right) \) is ergodic with respect to the Gauss measure \( \mu \) . | Proof of Theorem 3.7. The description of the continued fraction map as a shift on the space \( {\mathbb{N}}^{\mathbb{N}} \) described above suggests the method of proof: the measure \( \mu \) corresponds to a rather complicated measure on the shift space, but if we can control the measure of cylinder sets (and their in... | Yes |
For almost every real number \( x = \left\lbrack {{a}_{1},{a}_{2},\ldots }\right\rbrack \in \left( {0,1}\right) \), the digit \( j \) appears in the continued fraction with density | Proof. The digit \( j \) appears in the first \( N \) digits with frequency\n\n\[ \frac{1}{N}\left| \left\{ {i \mid i \leq N,{a}_{i} = j}\right\} \right| = \frac{1}{N}\left| \left\{ {i \mid i \leq N,{T}^{i}x \in \left( {\frac{1}{j + 1},\frac{1}{j}}\right) }\right\} \right| \]\n\n\[ \rightarrow \frac{1}{\log 2}{\int }_{... | Yes |
Proposition 3.10. A number \( u \in \left( {0,1}\right) \) is badly approximable if and only if there exists some \( \varepsilon > 0 \) with the property that\n\n\[ \left| {u - \frac{p}{q}}\right| \geq \frac{\varepsilon }{{q}^{2}} \]\n\nfor all rational numbers \( \frac{p}{q} \) . | Proof. If \( u \) is badly approximable, then (3.4) shows that\n\n\[ {q}_{n + 1} \leq \left( {M + 1}\right) {q}_{n} \]\n\nfor all \( n \geq 0 \) . For any \( q \) there is some \( n \) with \( q \in \left( {{q}_{n - 1},{q}_{n}}\right\rbrack \), and by Proposition 3.3 and (3.15) we therefore have\n\n\[ \left| {\frac{p}{... | Yes |
Notice that \( \frac{2}{\sqrt{5} - 1} = \frac{\sqrt{5} + 1}{2} \in \left( {1,2}\right) \) and \( \frac{\sqrt{5} + 1}{2} - 1 = \frac{\sqrt{5} - 1}{2} \). It follows that if \[ \frac{\sqrt{5} - 1}{2} = \left\lbrack {{a}_{1},{a}_{2},\ldots }\right\rbrack \] then \( {a}_{1} + \left\lbrack {{a}_{2},{a}_{3},\ldots }\right\rb... | Indeed, the specific number in Example 3.11 is, in a precise sense, the most badly approximable real number in \( \left( {0,1}\right) \) . In the next section we generalize this example to show that all quadratic irrationals are badly approximable. | Yes |
Theorem 3.13 (Lagrange). Let \( u \) be an irrational positive real number. Then the continued fraction expansion of \( u \) is eventually periodic if and only if \( u \) is a quadratic irrational. | Proof. Assume first that \( u = \left\lbrack \overline{{a}_{0};{a}_{1},\ldots ,{a}_{k}}\right\rbrack \) has a strictly periodic continued fraction expansion, so that \( {u}_{k + 1} = {u}_{0} = u \) . Thus\n\n\[ u = \frac{u{p}_{k} + {p}_{k - 1}}{u{q}_{k} + {q}_{k - 1}} \]\n\nby \( \left( {3.20}\right) \), so\n\n\[ {u}^{... | Yes |
Corollary 3.14. Any quadratic irrational is badly approximable. | Proof. This is an immediate consequence of Theorem 3.13 and Definition 3.9. | No |
Corollary 4.2 (Kryloff-Bogoliouboff). Under the hypotheses of Theorem 4.1, \( {\mathcal{M}}^{T}\left( X\right) \) is non-empty. | Proof. Since \( \mathcal{M}\left( X\right) \) is weak*-compact, the sequence \( \left( {\mu }_{n}\right) \) must have a limit point. | No |
Define the 'North-South' map \( T : X \rightarrow X \) on the circle \( X = \{ z \in \mathbb{C}\left| \right| z - \mathrm{i} \mid = 1\} \) using the stereographic projection \( \pi \) and show its properties. | The 'North-South' map \( T : X \rightarrow X \) is defined by \[ T\left( z\right) = \left\{ \begin{array}{ll} 2\mathrm{i} & \text{ if }z = 2\mathrm{i} \\ {\pi }^{-1}\left( {\pi \left( z\right) /2}\right) & \text{ if }z \neq 2\mathrm{i} \end{array}\right. \] as shown in Fig. 4.1. Using Poincaré recurrence (Theorem 2.11)... | No |
Lemma 4.6. If \( {\mu }_{1},{\mu }_{2} \in {\mathcal{E}}^{T}\left( X\right) \) and \( {\mu }_{1} \neq {\mu }_{2} \) then \( {\mu }_{1} \) and \( {\mu }_{2} \) are mutually singular. | Proof. Let \( f \in C\left( X\right) \) be chosen with \( \int f\mathrm{\;d}{\mu }_{1} \neq \int f\mathrm{\;d}{\mu }_{2} \) (such a function exists by Theorem B.11). Then by the ergodic theorem (Theorem 2.30)\n\n\[{\mathrm{A}}_{n}^{f}\left( x\right) \rightarrow \int f\mathrm{\;d}{\mu }_{1}\]\n\n(4.3)\n\nfor \( {\mu }_{... | Yes |
Theorem 4.8 (Ergodic decomposition). Let \( X \) be a compact metric space and \( T : X \rightarrow X \) a continuous map. Then for any \( \mu \in {\mathcal{M}}^{T}\left( X\right) \) there is a unique probability measure \( \lambda \) defined on the Borel subsets of the compact metric space \( {\mathcal{M}}^{T}\left( X... | Proof. This follows from Choquet's theorem [55] (see also the notes of Phelps [283]). A different proof will be given later (cf. p. 154), and a nontrivial example may be seen in Example 4.13. | No |
The circle rotation \( {R}_{\alpha } : \mathbb{T} \rightarrow \mathbb{T} \) is uniquely ergodic if and only if \( \alpha \) is irrational. The unique invariant measure in this case is the Lebesgue measure \( {m}_{\mathbb{T}} \). | This may be proved using property (5) of Theorem 4.10 (or using property (1); see Theorem 4.14). Assume first that \( \alpha \) is irrational, so \( {\mathrm{e}}^{{2\pi }\mathrm{i}{k\alpha }} = 1 \) only if \( k = 0 \) . If \( f\left( t\right) = {\mathrm{e}}^{{2\pi }\mathrm{i}{kt}} \) for some \( k \in \mathbb{Z} \), t... | Yes |
Let \( X = \{ z \in \mathbb{C}\left| \right| z \mid = 1 \) or \( 2\} \), let \( \alpha \) be an irrational number, and define a continuous map \( T : X \rightarrow X \) by \( T\left( z\right) = {\mathrm{e}}^{{2\pi }\mathrm{i}\alpha }z \). By unique ergodicity on each circle, any invariant measure \( \mu \) takes the fo... | \[ \mu = s{m}_{1} + \left( {1 - s}\right) {m}_{2} \] where \( {m}_{1} \) and \( {m}_{2} \) denote Lebesgue measures on the two circles comprising \( X \) . Thus \( {\mathcal{M}}^{T}\left( X\right) = \left\{ {s{m}_{1} + \left( {1 - s}\right) {m}_{2} \mid s \in \left\lbrack {0,1}\right\rbrack }\right\} \), with the two e... | Yes |
A more sophisticated version of Example 4.12 is a rotation on the disk. Let \( \mathbb{D} = \{ z \in \mathbb{C}\left| \right| z \mid \leq 1\} \), let \( \alpha \) be an irrational number, and define a continuous map \( T : \mathbb{D} \rightarrow \mathbb{D} \) by \( T\left( z\right) = {\mathrm{e}}^{{2\pi }\mathrm{i}\alp... | \[ \mu \left( A\right) = {\int }_{{\mathcal{M}}^{T}\left( X\right) }{m}_{r}\left( A\right) \mathrm{d}\nu \left( {m}_{r}\right) \] | Yes |
Theorem 4.14. Let \( X \) be a compact metrizable group and \( {R}_{g}\left( x\right) = {gx} \) the rotation by a fixed element \( g \in X \) . Then the following are equivalent.\n\n(1) \( {R}_{g} \) is uniquely ergodic (with the unique invariant measure being \( {m}_{X} \), the Haar measure on \( X \) ).\n\n(2) \( {R}... | Proof. (1) \( \Rightarrow \) (2): This is clear.\n\n(2) \( \Rightarrow \) (3): Let \( Y \) denote the closure of the subgroup generated by \( g \) . If \( Y \neq X \) then there is a continuous non-constant function on \( X \) that is constant on each coset of \( Y \) : in fact if \( \mathrm{d} \) is a bi-invariant met... | Yes |
Corollary 4.15. Let \( X = {\mathbb{T}}^{\ell } \), and let \( g = \left( {{\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{\ell }}\right) \in {\mathbb{R}}^{\ell } \) . Then the toral rotation \( {R}_{g} : {\mathbb{T}}^{\ell } \rightarrow {\mathbb{T}}^{\ell } \) given by \( {R}_{g}\left( x\right) = x + g \) is uniquely e... | Exercise 4.3.5. Prove Corollary 4.15\n\n(a) using Theorem 4.14;\n\n(b) using Theorem 4.10(5). | No |
A consequence of Theorem 4.10 and Example 4.11 is that for any irrational number \( \alpha \), and any initial point \( x \in \mathbb{T} \), the orbit \( x,{R}_{\alpha }x,{R}_{\alpha }^{2}x,\ldots \) under the circle rotation is an equidistributed sequence. | Note that this is proved in Example 4.11 by using property (2) of Lemma 4.17. | No |
Corollary 4.20. Let \( X \) be a compact metric space, let \( T : X \rightarrow X \) be a continuous map, and let \( \mu \) be a \( T \) -invariant ergodic probability measure. Then \( \mu \) - almost every point in \( X \) is generic with respect to \( T \) and \( \mu \) . | Proof. Recall that \( C\left( X\right) \) is a separable metric space with respect to the uniform norm\n\n\[ \parallel f{\parallel }_{\infty } = \sup \{ \left| {f\left( x\right) }\right| \mid x \in X\} \]\n\nby Lemma B.8. Let \( {\left( {f}_{n}\right) }_{n \geq 1} \) be a dense sequence in \( C\left( X\right) \) . By a... | Yes |
Theorem 4.21 (Furstenberg). Let \( T : X \rightarrow X \) be a uniquely ergodic homeomorphism of a compact metric space with unique invariant measure \( \mu \) . Let \( G \) be a compact group* with Haar measure \( {m}_{G} \), and let \( c : X \rightarrow G \) be a continuous map. Define the skew-product map \( S \) on... | Proof. To see that \( S \) preserves \( \mu \times {m}_{G} \), let \( f \in C\left( Y\right) \) . Then, by Fubini’s theorem, \[ {\int }_{Y}f \circ S\mathrm{\;d}\left( {\mu \times {m}_{G}}\right) = {\int }_{X}{\int }_{G}f\left( {{Tx}, c\left( x\right) g}\right) \mathrm{d}{m}_{G}\left( g\right) \mathrm{d}\mu \left( x\rig... | Yes |
Corollary 4.22. Let \( \alpha \) be an irrational number. Then the map \( S : {\mathbb{T}}^{k} \rightarrow {\mathbb{T}}^{k} \) defined by\n\n\[ S : \left( \begin{matrix} {x}_{1} \\ {x}_{2} \\ \vdots \\ {x}_{k} \end{matrix}\right) \mapsto \left( \begin{matrix} {x}_{1} + \alpha \\ {x}_{2} + {x}_{1} \\ \vdots \\ {x}_{k} +... | Proof. Notice that the transformation \( S \) is built up from the irrational circle map by taking \( \left( {k - 1}\right) \) skew-product extensions as in Theorem 4.21. By Theorem 4.21, it is sufficient to prove that \( S \) is ergodic with respect to\n\nLebesgue measure on \( {\mathbb{T}}^{k} \) . Let \( f \in {L}^{... | Yes |
If \( \mathcal{A} = \sigma \left( \xi \right) \) is the finite \( \sigma \) -algebra generated by a finite partition \( \xi = \left\{ {{A}_{1},\ldots ,{A}_{n}}\right\} \) of \( X \), then | \[ E\left( {f \mid \mathcal{A}}\right) \left( x\right) = \frac{1}{\mu \left( {A}_{i}\right) }{\int }_{{A}_{i}}f\mathrm{\;d}\mu \] if \( x \in {A}_{i} \) . The \( \sigma \) -algebra being conditioned on is illustrated in Fig. 5.1 for a partition into \( n = 8 \) sets; \( E\left( {f \mid \mathcal{A}}\right) \) is then a ... | Yes |
Lemma 5.6 (Doob’s inequality). Let \( f \in {L}^{1}\left( {X,\mathcal{B},\mu }\right) \), let | Proof. Assume that \( f \geq 0 \) (if necessary replacing \( f \) by \( \left| f\right| \), which makes \( \mu \left( E\right) \) no smaller). Let | No |
Theorem 5.8 (Decreasing martingale theorem). Let \( \left( {X,\mathcal{B},\mu }\right) \) be a probability space. If \( {\mathcal{A}}_{n} \searrow {\mathcal{A}}_{\infty } \) is a decreasing sequence of sub- \( \sigma \) -algebras of \( \mathcal{B} \) then\n\n\[ E\left( {f \mid {\mathcal{A}}_{n}}\right) \rightarrow E\le... | FIRST PART OF PROOF OF THEOREM 5.8, USING \( {L}^{2} \) . Recall from the proof of Theorem 5.1 that in \( {L}^{2}\left( {X,\mathcal{B},\mu }\right) \) the conditional expectation with respect to \( {\mathcal{A}}_{n} \) (or \( {\mathcal{A}}_{\infty } \) ) is precisely the orthogonal projection to \( {L}^{2}\left( {X,{\m... | Yes |
Let \( \xi \) be a countable partition of \( \left( {X,\mathcal{B},\mu }\right) \), with \( \mathcal{A} = \sigma \left( \xi \right) \) the smallest \( \sigma \) -algebra containing \( \xi \) . Then | \[ {\mu }_{x}^{\mathcal{A}} = {\left. \frac{1}{\mu \left( P\right) }\mu \right| }_{P} \] for \( x \in P \in \mathcal{A} \) defines such a measure for almost every \( x \) ; if \( \mu \left( P\right) = 0 \) then \( {\mu }_{x}^{\mathcal{A}} \) is not defined for \( x \in P \) . | No |
Let \( \mathcal{A} = {\mathcal{B}}_{\left\lbrack 0,1\right\rbrack } \times \{ \varnothing ,\left\lbrack {0,1}\right\rbrack \} \subseteq {\mathcal{B}}_{{\left\lbrack 0,1\right\rbrack }^{2}} \). Then | \[ {\mu }_{\left( {x}_{1},{x}_{2}\right) }^{\mathcal{A}} = {\delta }_{{x}_{1}} \times {m}_{\left\lbrack 0,1\right\rbrack } \] where as usual \( m \) denotes Lebesgue (or Haar) measure (see Fig. 5.2). | No |
Let \( X = \{ 0,1{\} }^{\mathbb{R}} \), with the product topology and the \( \sigma \) -algebra of Borel sets. The product measure \( \mu \) of the \( \left( {\frac{1}{2},\frac{1}{2}}\right) \) measure on each of the sets \( \{ 0,1\} \) makes \( X \) into a probability space with the property that there is an uncountab... | \[ \mu \left( {{A}_{{s}_{1}} \cap \cdots \cap {A}_{{s}_{n}}}\right) = \frac{1}{{2}^{n}} \] for any \( n \) distinct reals \( {s}_{1},\ldots ,{s}_{n} \). | Yes |
Lemma 5.17. If \( \left( {X,\mathcal{B},\mu }\right) \) is a Borel probability space and \( \mathcal{A} \subseteq \mathcal{B} \) is a \( \sigma \) - algebra then there is a countably-generated \( \sigma \) -algebra \( \mathcal{A} \) with \( \mathcal{A} = \mathcal{A} \) . | Proof. Recall that \( C\left( \bar{X}\right) \) is separable for any compact metric space \( \bar{X} \) (see Lemma B.8). Since \( C\left( \bar{X}\right) \) is mapped continuously to a dense subspace of \( {L}^{1}\left( {X,\mathcal{B},\mu }\right) \), the same holds for \( {L}^{1}\left( {X,\mathcal{B},\mu }\right) \) . ... | Yes |
Lemma 5.18. Let \( \left( {X,\mathcal{B},\mu }\right) \) be a Borel probability space and let \( \mathcal{A} \subseteq \mathcal{B} \) be a countably-generated \( \sigma \) -algebra. If \( f \in {\mathcal{L}}^{\infty }\left( {X,\mathcal{B}}\right) \) is constant on atoms of \( \mathcal{A} \) , then \( {\left. f\right| }... | Proof of Lemma 5.18. By Theorem 5.14(2), on \( {X}^{\prime } \) we have\n\n\[ \int f\mathrm{\;d}{\mu }_{x}^{\mathcal{A}} = f\left( x\right) \]\n\nsince \( {\mu }_{x}^{\mathcal{A}}\left( {\left\lbrack x\right\rbrack }_{\mathcal{A}}\right) = 1 \) and, by assumption, \( f \) is constant (and equal to \( f\left( x\right) \... | Yes |
Corollary 5.22. Let \( \left( {X,\mathcal{B},\mu }\right) \) be a Borel probability space, and let \( \mathcal{A} \subseteq \mathcal{B} \) be a countably-generated \( \sigma \) -algebra. Then there is a conull set \( {X}^{\prime } = X \smallsetminus N \) in \( \mathcal{A} \), a compact metric space together with its Bo... | This will be proved later; the conclusion described in Corollary 5.22 is depicted in Fig. 5.3. | No |
If \( \bar{X} \) is a compact metric space, and \( f \in {\mathcal{L}}^{\infty }\left( \bar{X}\right) \), then the map\n\n\[ \mathcal{M}\left( X\right) \ni \nu \mapsto \int f\mathrm{\;d}\nu \]\n\nis Borel measurable. In particular, for a Borel subset \( X \) of \( \bar{X} \), we have that \( \mathcal{M}\left( X\right) ... | Proof. Starting with continuous functions, we know that \( \int f\mathrm{\;d}\nu \) depends continuously on \( \nu \) (by definition of the weak*-topology on \( \mathcal{M}\left( X\right) \) ). Arguing just as we did on p. 139, this can be extended to show that \( \int f\mathrm{\;d}\nu \) depends measurably on \( \nu \... | Yes |
Corollary 5.24. Let \( \phi : \left( {X,{\mathcal{B}}_{X},\mu }\right) \rightarrow \left( {Y,{\mathcal{B}}_{Y},\nu }\right) \) be a measure-preserving map between Borel probability spaces, and let \( \mathcal{A} \subseteq {\mathcal{B}}_{Y} \) be a sub-σ-algebra. Then\n\n\[ \n{\phi }_{ * }{\mu }_{x}^{{\phi }^{-1}\mathca... | Proof. First notice that for any \( f \in {L}^{1}\left( {Y,{\mathcal{B}}_{Y},\nu }\right) ,{E}_{\nu }\left( {f \mid \mathcal{A}}\right) \circ \phi \) is \( {\phi }^{-1}\mathcal{A} \) - measurable and\n\n\[ \n{\int }_{{\phi }^{-1}A}{E}_{\nu }\left( {f \mid \mathcal{A}}\right) \circ \phi \mathrm{d}\mu = {\int }_{A}{E}_{\... | Yes |
Lemma 5.25. Let \( X, Y, Z \) be Borel subsets of compact metric spaces \( \bar{X},\bar{Y} \) and \( \bar{Z} \) respectively, and let \( {\phi }_{Z} : X \rightarrow Z \) and \( {\phi }_{Y} : X \rightarrow Y \) be measurable maps. Suppose that \( {\phi }_{Z} \) is \( {\phi }_{Y}^{-1}\left( {\mathcal{B}}_{Y}\right) \) -m... | Proof of Lemma 5.25. Define \( \mathcal{A} = {\phi }_{Y}^{-1}\left( {\mathcal{B}}_{Y}\right) \), which is countably-generated since \( {\mathcal{B}}_{Y} \) is. Since the Borel \( \sigma \) -algebra \( {\mathcal{B}}_{Y} \) of \( Y \) separates points,\n\n\[{\left\lbrack x\right\rbrack }_{\mathcal{A}} = {\phi }_{Y}^{-1}\... | Yes |
Theorem 6.1. Let \( \\left( {X,\\mathcal{B},\\mu, T}\\right) \) be a measure-preserving system and \( f \\in {L}_{\\mu }^{1} \) . Then\n\n\[ \n\\frac{1}{M}\\mathop{\\sum }\\limits_{{n = 0}}^{{M - 1}}f \\circ {T}^{n} \\rightarrow E\\left( {f \\mid \\mathcal{E}}\\right)\n\]\n\nalmost everywhere and in \( {L}^{1} \), wher... | Proof of Theorem 6.1. By Theorem 2.30 the sequence converges to some \( {f}^{ * } \\in {L}^{1} \) both pointwise and in \( {L}^{1} \). Now \( {f}^{ * } \) is \( \\mathcal{E} \) -measurable since \( {f}^{ * } \) is \( T \) -invariant. Moreover, for any set \( A \\in \\mathcal{E} \) with positive measure, we may apply Th... | Yes |
Lemma 6.3. Let \( \left( {X,{\mathcal{B}}_{X},\nu, T}\right) \) be a measure-preserving system on a Borel probability space, and let \( \left\{ {{f}_{1},{f}_{2},\ldots }\right\} \) be dense in \( C\left( \bar{X}\right) \) . Then \( \nu \) is ergodic if and only if\n\n\[ \n\frac{1}{M}\mathop{\sum }\limits_{{n = 0}}^{{M ... | Proof. Ergodicity clearly implies the stated property. For the converse, recall that \n\n\[ \n\frac{1}{M}\mathop{\sum }\limits_{{n = 0}}^{{M - 1}}f \circ {T}^{n}\underset{{L}_{\nu }^{2}}{ \rightarrow }{P}_{T}f \n\] \n\nwhere \( {P}_{T} \) denotes the projection operator onto the space of \( {U}_{T} \) -invariant functi... | Yes |
Theorem 6.5. Let \( \\left( {X,{\\mathcal{B}}_{X},\\mu }\\right) \) be a Borel probability space, and let \( T \) be a measure-preserving transformation on \( X \) . Assume furthermore that there is a T-invariant sub-\\sigma-algebra \( \\mathcal{A} \\subseteq {\\mathcal{B}}_{X} \) . Then there is a measure-preserving s... | Proof. We are going to apply Corollary 5.22 with the choice \( Y = \\mathcal{M}\\left( X\\right) \) appearing in the proof. Let \( S : Y \\rightarrow Y \) be the map defined by \( {S\\nu } = {T}_{ * }\\nu \) for any \( \\nu \\in Y \) . By Lemma 5.23, \( S \) is measurable. Define a map \( \\phi : X \\rightarrow Y \) by... | Yes |
Lemma 6.8. Let \( \mathrm{X} \) and \( \mathrm{Y} \) be ergodic systems with \( \rho \in J\left( {\mathrm{X},\mathrm{Y}}\right) \) . Then almost every ergodic component of \( \rho \) is an ergodic joining of \( \mathrm{X} \) and \( \mathrm{Y} \) . | Proof of Lemma 6.8. Suppose that\n\n\[ \rho = {\int }_{Z}{\rho }_{z}\mathrm{\;d}\tau \left( z\right) \]\n\nis the ergodic decomposition of \( \rho \) from Theorem 6.2, for some probability space \( \left( {Z,{\mathcal{B}}_{Z},\tau }\right) \) . Recall that \( {\pi }_{X} : X \times Y \rightarrow X \) denotes the project... | Yes |
Lemma 6.9. If \( \left( {X,\mathcal{B},\mu, T}\right) \) is ergodic then every eigenvalue of \( {U}_{T} \) is simple, and the set of all eigenvalues of \( {U}_{T} \) is a subgroup of \( {\mathbb{S}}^{1} \) . | Proof. If \( {U}_{T}f = {\lambda f} \) then\n\n\[ \langle f, f\rangle = \left\langle {{U}_{T}f,{U}_{T}f}\right\rangle = \lambda \bar{\lambda }\langle f, f\rangle \]\n\nso \( \lambda \bar{\lambda } = 1 \) . If \( f \) is an eigenfunction corresponding to the eigenvalue \( \lambda \), then\n\n\[ {U}_{T}\left| f\right| = ... | Yes |
Theorem 6.10 (Kronecker factors). Let \( \left( {X,\mathcal{B},\mu, T}\right) \) be an invertible ergodic measure-preserving system on a Borel probability space, and let \( \mathcal{A} \) be the smallest \( \sigma \) -algebra with respect to which all \( {L}_{\mu }^{2} \) eigenfunctions of \( {U}_{T} \) are measurable.... | Proof of Theorem 6.10. Let \( \left\{ {{\chi }_{i} \mid i \in \mathbb{N}}\right\} \subseteq {L}_{\mu }^{2} \) be an enumeration* of the eigenfunctions of \( {U}_{T} \) normalized so that \( \left| {\chi }_{i}\right| = 1 \) almost everywhere, and let \( {U}_{T}{\chi }_{i} = {\lambda }_{i}{\chi }_{i} \) . Define a map \(... | Yes |
Lemma 6.12. Given any countable subgroup \( K \leq {\mathbb{S}}^{1} \) there is an ergodic measure-preserving system \( \left( {X,{\mathcal{B}}_{X},\mu, T}\right) \) on a Borel probability space with the property that \( K \) is the group of eigenvalues of \( {U}_{T} \) . | Proof. Give \( K \) the discrete topology, so that the dual group \( X = \widehat{K} \) is a compact metric abelian group; write \( \mu = {m}_{X} \) for the normalized Haar measure on \( X \) . The map \( \theta : K \rightarrow {\mathbb{S}}^{1} \) defined by \( \theta \left( \kappa \right) = \kappa \) is a character on... | Yes |
Proposition 6.16. Let \( \rho \) be the relatively independent joining of the invertible systems \( \mathrm{X} \) and \( \mathrm{Y} \) over a common factor \( \mathrm{Z} \) as above. Then the following properties hold.\n\n(1) The relatively independent joining is concentrated on the measurable set\n\n\[ \Phi = \left\{ ... | Proof of Proposition 6.16. Property (1) follows easily by substitution:\n\n\[ \rho \left( \Phi \right) = \int {\mu }_{z} \times {\nu }_{z}\left( \Phi \right) \mathrm{d}\lambda \left( z\right) = 1 \]\n\nsince \( {\mu }_{z}\left( {{\left( {\phi }_{X}\right) }^{-1}\left( z\right) }\right) = {\nu }_{z}\left( {{\left( {\phi... | Yes |
Theorem 7.1. Given \( \ell, r \geq 1 \) there is some \( N\left( {\ell, r}\right) \) with the property that if \( N \geq N\left( {\ell, r}\right) \) and\n\n\[ \{ 1,2,\ldots, N\} = {C}_{1} \sqcup \cdots \sqcup {C}_{r} \]\n\nthen for some \( j \) the set \( {C}_{j} \) contains an arithmetic progression of length \( \ell ... | In the proof, it will be convenient to write \( \{ a, a + 1,\ldots, b\} \) as \( \left\lbrack {a, b}\right\rbrack \) and to define a coloring of a set \( \left\lbrack {1, N}\right\rbrack \) into \( r \) colors as a map \( C : \left\lbrack {1, N}\right\rbrack \rightarrow \{ 1,\ldots, r\} \) .\n\nWe define two integer ve... | Yes |
Lemma 7.3. \( \mathbf{V}\left( {\ell, m}\right) \) for all \( m \geq 1 \) implies \( \mathbf{V}\left( {\ell + 1,1}\right) \) . | Proof. Fix \( r \), and let\n\n\[ C : \left\lbrack {1,{2N}\left( {\ell, r, r}\right) }\right\rbrack \rightarrow \{ 1,\ldots, r\} \]\n\nbe given, where \( N\left( {\ell, r, r}\right) \) is as in \( \mathbf{V}\left( {\ell, r}\right) \) . Then there exist \( a,{d}_{1},\ldots ,{d}_{r} > 0 \) such that\n\n\[ a + \mathop{\su... | Yes |
Theorem 7.4 (Furstenberg). For any system \( \left( {X,\mathcal{B},\mu, T}\right) \) and set \( A \in \mathcal{B} \) with \( \mu \left( A\right) > 0 \), and for any \( k \in \mathbb{N} \) ,\n\n\[ \mathop{\liminf }\limits_{{N \rightarrow \infty }}\frac{1}{N}\mathop{\sum }\limits_{{n = 1}}^{N}\mu \left( {A \cap {T}^{-n}A... | In fact Furstenberg proved that\n\n\[ \mathop{\liminf }\limits_{{N - M \rightarrow \infty }}\frac{1}{N - M}\mathop{\sum }\limits_{{n = M}}^{{N - 1}}\mu \left( {A \cap {T}^{-n}A \cap {T}^{-{2n}}A \cap \cdots \cap {T}^{-{kn}}A}\right) > 0, \]\n\nbut the inequality (7.2) is sufficient for Szemerédi's theorem. | Yes |
If \( T = {R}_{\alpha } \) is an ergodic rotation, then the inequality (7.3) is clear: If \( 0,{n\alpha },{2n\alpha },\ldots ,{kn\alpha }\left( {\;\operatorname{mod}\;1}\right) \) are all very close together (which may be arranged for \( \alpha \notin \mathbb{Q} \) and any \( k \geq 1 \) by choice of \( n \) ), then th... | \[ {\chi }_{A},{\chi }_{A - {n\alpha }},\ldots ,{\chi }_{A - {kn\alpha }} \] will be close together in \( {L}_{{m}_{\mathbb{T}}}^{2} \) for any interval (indeed, for any Borel set) \( A \) , so the intersection \[ A \cap {R}_{\alpha }^{-n}A \cap \cdots \cap {R}_{\alpha }^{-{kn}}A \] will have measure close to that of \... | No |
Proposition 7.13. Let \( \left( {X,\mathcal{B},\mu, T}\right) \) be a weak-mixing system. Then for any functions \( {f}_{1},\ldots ,{f}_{k} \in {L}_{\mu }^{\infty } \) , | Proof of Proposition 7.13. Since \( T \) is weak-mixing it is certainly ergodic, so (7.18) holds for \( k = 1 \) by the mean ergodic theorem (Theorem 2.21). We proceed by induction on \( k \) . | No |
Theorem 7.14. Let \( \left( {X,\mathcal{B},\mu, T}\right) \) be a measure-preserving system. Then, for any functions \( {f}_{1},{f}_{2} \in {L}^{\infty }\left( {X,\mathcal{B},\mu }\right) \), \[ \frac{1}{N}\mathop{\sum }\limits_{{n = 1}}^{N}{U}_{T}^{n}{f}_{1}{U}_{T}^{2n}{f}_{2} \] (7.19) converges in \( {L}^{2}\left( {... | Just as in the proof from Sect. 7.4, we will prove this by decomposing the system (more precisely, by decomposing the space of \( {L}^{2} \) -functions on the space) into an orderly and a chaotic part. We will see below that for this result the appropriate splitting will be given by the Kronecker factor. | No |
Proposition 7.15. Let \( T \) be an invertible ergodic measure-preserving transformation on a Borel probability space \( \\left( {X,\\mathcal{B},\\mu, T}\\right) \) . Let \( \\mathcal{K} \) be the \( \\sigma \) -algebra corresponding to the Kronecker factor of \( T \), and let \( {f}_{1},{f}_{2} \\in {L}_{\\mu }^{\\inf... | Proof of Theorem 7.14 assuming Propositions 7.12 and 7.15. By Proposition 7.15, the sequence\n\n\[ \n\\frac{1}{N}\\mathop{\\sum }\\limits_{{n = 1}}^{N}{U}_{T}^{n}{f}_{1}{U}_{T}^{2n}{f}_{2}\n\]\n\nconverges to the same limit as\n\n\[ \n\\frac{1}{N}\\mathop{\\sum }\\limits_{{n = 1}}^{N}{U}_{T}^{n}E\\left( {{f}_{1} \\mid ... | No |
Lemma 7.16. Let \( \left( {X,\mathcal{B},\mu, T}\right) \) be any measure-preserving system on a Borel probability space. Suppose that \( K : {L}_{\mu }^{2} \rightarrow {L}_{\mu }^{2} \) is a compact self-adjoint operator commuting with \( {U}_{T} \) . Then all eigenspaces of \( K \) with non-zero eigenvalue are finite... | Recall that any kernel in \( {L}_{\mu \times \mu }^{2} \) defines a compact operator on \( {L}_{\mu }^{2} \), hence\n\n\[ \n{K}_{H} : g \mapsto \int {F}_{H}\left( {x, y}\right) g\left( y\right) \mathrm{d}\mu \left( y\right) \n\]\n\nand\n\n\[ \nK : g \mapsto \int F\left( {x, y}\right) g\left( y\right) \mathrm{d}\mu \lef... | Yes |
Let \( X = {\mathbb{T}}^{2} \) and define \( {}^{\left( {69}\right) } \) a map \( T : X \rightarrow X \) by\n\n\[ T : x = \left( \begin{array}{l} y \\ z \end{array}\right) \mapsto \left( \begin{array}{l} y + \alpha \\ z + y \end{array}\right) \]\n\nThe iterates of the map \( T \) take the form\n\n\[ {T}^{n} : \left( \b... | For example, the function\n\n\[ f : {\mathbb{T}}^{2} \rightarrow \mathbb{C} \]\n\ndefined by\n\n\[ f\left( \begin{array}{l} y \\ z \end{array}\right) = {\mathrm{e}}^{{2\pi }\mathrm{i}z} \]\n\nhas\n\n\[ {U}_{T}^{n}f\left( \begin{array}{l} y \\ z \end{array}\right) = {\mathrm{e}}^{{2\pi }\mathrm{i}c\left( {n,\alpha }\rig... | Yes |
Example 7.19. The map \( T : {\mathbb{T}}^{2} \rightarrow {\mathbb{T}}^{2} \) constructed in Example 7.17 is a compact extension of the circle rotation \( {R}_{\alpha } : \mathbb{T} \rightarrow \mathbb{T} \) . The character | \[ \left( \begin{array}{l} y \\ z \end{array}\right) \mapsto {\mathrm{e}}^{{2\pi }\mathrm{i}z} \] is mapped to the function \[ \left( \begin{array}{l} y \\ z \end{array}\right) \mapsto \underset{\text{modulus }1}{\underbrace{C\left( {n,\alpha, y}\right) }}{\mathrm{e}}^{{2\pi }\mathrm{i}z} \] in \( {L}_{{\mu }_{y}}^{2} ... | No |
Theorem 7.21. Let \( \left( {X,\mathcal{B},\mu, T}\right) \) be an ergodic measure-preserving system on a Borel probability space and suppose that \n\nis an extension of measure-preserving systems. Then one of the fo... | Proof of Theorem 7.21. Let\n\n\[ \widetilde{\mathrm{X}} = \left( {\widetilde{X} = X \times X,\mathcal{B} \otimes \mathcal{B},\widetilde{\mu } = \mu { \times }_{\mathrm{Y}}\mu ,\widetilde{T} = T \times T}\right) \]\n\nand assume that \( \mathrm{X} \rightarrow \mathrm{Y} \) is not relatively weak-mixing. Then there is a ... | Yes |
Lemma 7.22. There is a function \( \phi \in {L}^{\infty }\left( X\right) \) such that \( H * \phi \notin {L}^{2}\left( Y\right) \) . | Proof. Suppose there is no such function, and choose a sequence \( \left( {\mathcal{P}}_{n}\right) \) of finite partitions of \( X \) with the property that\n\n\[ \sigma \left( {\mathop{\bigcup }\limits_{{n \geq 1}}\sigma \left( {\mathcal{P}}_{n}\right) }\right) = \mathcal{B} \]\n\nThen for \( {x}_{2} \in P \in {\mathc... | Yes |
For any \( y \in \mathop{\bigcap }\limits_{{\ell = 0}}^{k}{T}^{-\ell n}\widetilde{A} \), the set\n\n\[ B = \left\{ {\left( {f,{U}_{T}^{n + m}f,\ldots ,{U}_{T}^{k\left( {n + m}\right) }f}\right) \mid m \in F}\right\} \]\n\nis an \( \varepsilon \) -separated subset of \( {\mathcal{L}}^{ * }\left( y\right) \) . | Proof. Since \( {T}^{\ell \left( {n + m}\right) }y \in A \) for \( \ell = 0,\ldots, k \) and \( m \in F \), the set \( B \) is in \( {\mathcal{L}}^{ * }\left( y\right) \) . Take \( m,{m}^{\prime } \in F, m \neq {m}^{\prime } \) . Since \( y \in \widetilde{A} \) belongs to the set in (7.35), there exists \( \ell \leq k ... | Yes |
Corollary 7.28. If \( \\left( {X,\\mathcal{B},\\mu, T}\\right) \) is a weak-mixing measure-preserving system, then for \( {B}_{0},\\ldots ,{B}_{k} \\in \\mathcal{B} \) , | \[ \\mathop{\\lim }\\limits_{{N \\rightarrow \\infty }}\\frac{1}{N}\\mathop{\\sum }\\limits_{{n = 1}}^{N}\\left\\lbrack {\\mu \\left( {{B}_{0} \\cap {T}^{-n}{B}_{1} \\cap \\cdots \\cap {T}^{-{kn}}{B}_{k}}\\right) }\\right.\n\n\[ {\\left. -\\mu \\left( {B}_{0}\\right) \\mu \\left( {B}_{1}\\right) \\cdots \\mu \\left( {B... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.