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We are interested in looking at the variation in time of the slope of the edge: \( \frac{\partial }{\partial t}{I}_{x} \). If the function \( I\left( \cdot \right) \) is regular enough the order of differentiation may be inverted, | \[ \frac{\partial }{\partial t}{I}_{x} = \frac{\partial }{\partial x}{I}_{t} = \frac{\partial }{\partial x}\left( {\frac{\partial }{\partial x}\phi \left( {I}_{x}\right) }\right) = {\phi }^{\prime \prime }{I}_{xx}^{2} + {\phi }^{\prime }\left( {I}_{x}\right) {I}_{xxx} \] \( \left( {15.45}\right) \)\n\nSuppose the edge ... | Yes |
Theorem 15.4.9. Let \( {T}_{t} \) be a multi-scale analysis satisfying morphology invariance and Euclidean invariance. Then the associated \( F \) satisfies\n\n\[ F\left( {A, p}\right) = F\left( {{Q}_{p}A{Q}_{p}, p}\right) \]\n\nfor all symmetric \( A, p \neq 0 \) . where \( {Q}_{p} \) is the projection matrix given by... | Proof For \( p \neq 0 \), we can select an orthogonal coordinate such that \( p = \left| p\right| \left( {0,\cdots ,0,1}\right) = {p}^{\prime } \) and \( {Q}_{p}A{Q}_{p} = {A}^{\prime } = {\left( {A}_{ij}^{\prime }\right) }_{1 \leq i, j \leq N} \) with \( {A}_{ij}^{\prime } = {A}_{ij} \) for \( 1 \leq i, j \leq N - 1 \... | Yes |
Theorem 15.4.10. Let \( N = 2 \) . Under the same assumptions of the above theorem, the associated \( F \) satisfies\n\n\[ F\left( {A, p}\right) = \left| p\right| \cdot G\left( {\frac{\operatorname{tr}\left( A\right) }{\left| p\right| } - \frac{A\left( {p, p}\right) }{{\left| p\right| }^{3}}}\right) \]\n\nfor all \( A ... | Proof Let \( P \) be the projection \( \frac{p \otimes p}{{\left| p\right| }^{2}}.{P}^{2} = P \) . If we use an orthogonal coordinate system on \( {\mathbf{R}}^{2} \) whose second basis vector is given by \( \frac{p}{\left| p\right| } \), then the matrix of \( {Q}_{p}A{Q}_{p} \) is\n\n\[ {Q}_{p}A{Q}_{p} = \left( \begin... | Yes |
Theorem 1.1. If \( \Lambda \) is invariant, so are \( \bar{\Lambda },\partial \left( \Lambda \right) \), and \( \operatorname{int}\left( \Lambda \right) \) . | Proof. Since \( f \) is a homeomorphism, we have \( f\left( \bar{\Lambda }\right) = \overline{f\left( \Lambda \right) } = \bar{\Lambda } \) . The other two are proved similarly. | No |
Theorem 1.2. For any \( x \in X,\omega \left( x\right) \) is nonempty, compact, and invariant. Moreover, \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}d\left( {{f}^{n}\left( x\right) ,\omega \left( x\right) }\right) = 0. \] | Proof. Since \( X \) is compact, it follows that \( \omega \left( x\right) \) is nonempty and compact. Take \( y \in \omega \left( x\right) \) . There is a subsequence \( {n}_{i} \rightarrow + \infty \) such that \( {f}^{{n}_{i}}\left( x\right) \rightarrow \) \( y \) . Then \( {f}^{{n}_{i} + 1}\left( x\right) \rightarr... | Yes |
Theorem 1.3. Any nonempty compact invariant set contains a minimal set. | Proof. Let \( \Gamma \) be a nonempty compact invariant set of \( f \) . Let \( \mathcal{C} \) be the set of nonempty compact invariant subsets of \( f \) contained in \( \Gamma \) . The inclusion \( \subset \) is a partial order on \( \mathcal{C} \) . Let \( \mathcal{A} \) be a totally ordered subset of \( \mathcal{C}... | Yes |
Theorem 1.4. A compact invariant set \( \Lambda \) is minimal if and only if the orbit of every \( x \in \Lambda \) is dense in \( \Lambda \) . | Proof. Let \( \Lambda \) be minimal. Take any \( x \in \Lambda \) . Since \( \overline{\operatorname{Orb}\left( x\right) } \subset \Lambda \) is nonempty compact invariant, by minimality \( \overline{\operatorname{Orb}\left( x\right) } = \Lambda \) . Conversely, assume \( \Lambda \) is not minimal, then there is a prop... | Yes |
Theorem 1.5. Assume \( X \) is connected. Then any minimal set of \( f \) is either the whole \( X \) or nowhere dense in \( X \) . | Proof. Let \( \Lambda \) be a minimal set of \( f \) . Note that \( \partial \Lambda \) is compact invariant. If \( \partial \Lambda = \varnothing \), then \( \Lambda = \operatorname{int}\left( \Lambda \right) \) . Hence \( \Lambda \) is open. Thus \( \Lambda \) is both open and closed, hence equal to \( X \) since \( ... | Yes |
Theorem 1.6 (Birkhoff). Let \( \Lambda \) be a compact invariant set of \( f \) . The following conditions are equivalent:\n\n(1) \( \Lambda \) is transitive.\n\n(2) For any two open subsets \( U \) and \( V \) of \( \Lambda \), there is \( n \geq 1 \) such that \( {f}^{n}\left( U\right) \cap V \neq \varnothing \) .\n\... | Proof. The proof of \( \left( 1\right) \Rightarrow \left( 2\right) \) is easy, hence omitted. We prove \( \left( 2\right) \Rightarrow \left( 3\right) \) . Take a countable basis \( {V}_{1},{V}_{2},\ldots \) of \( \Lambda \) . For any \( i \geq 1 \), the set \( \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{f}^{-n}{V}_{i}... | No |
Theorem 1.7. Let \( C \) be a chain class of \( f \) . Then\n\n(1) for any \( \epsilon > 0 \) there is \( \delta > 0 \) such that, for any \( x \in C \), any periodic \( \delta \) -chain through \( x \) is contained in the \( \epsilon \) -neighborhood \( B\left( {C,\epsilon }\right) \) of \( C \) ;\n\n(2) \( C \) is in... | Proof. (1) Suppose there is \( {\epsilon }_{0} > 0 \) such that, for every \( n \geq 1 \), there is \( {x}_{0}^{n} \in C \) and a periodic \( 1/n \) -chain\n\n\[ \n{x}_{0}^{n},{x}_{1}^{n},\ldots ,{x}_{{j}_{n}}^{n} \n\]\n\nsuch that \( {x}_{{k}_{n}}^{n} \notin B\left( {C,{\epsilon }_{0}}\right) \) for some \( {k}_{n} \)... | Yes |
Theorem 1.8. Any two orientation-preserving homeomorphisms of \( \left\lbrack {a, b}\right\rbrack \) without fixed points in \( \left( {a, b}\right) \) are topologically conjugate. | Proof. Let \( f \) and \( g \) be two orientation-preserving homeomorphisms of \( \left\lbrack {a, b}\right\rbrack \) without fixed points in \( \left( {a, b}\right) \) . We assume \( f\left( x\right) > x \) and \( g\left( x\right) > x \) for any \( x \in \left( {a, b}\right) \) . See Figure 1.6. For the other cases th... | Yes |
Theorem 1.9. Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \left\lbrack {a, b}\right\rbrack \) be an orientation-preserving diffeomorphism without fixed points in \( \left( {a, b}\right) \) . For any \( r \geq 1, f \) is \( {C}^{r} \) structurally stable if and only if \( {f}^{\prime }\left( a\right) \neq 1 \... | Proof. Assume \( {f}^{\prime }\left( a\right) \neq 1 \) and \( {f}^{\prime }\left( b\right) \neq 1 \) . We prove \( f \) is \( {C}^{r} \) structurally stable for any \( r \geq 1 \) . It suffices to prove it for \( r = 1 \) . There is a \( {C}^{1} \) neighborhood \( {\mathcal{U}}_{1} \) of \( f \) in \( {\operatorname{D... | Yes |
A rigid rotation \( f \) of an irrational angle \( {2\pi \alpha },\alpha \) irrational. It is easy to see that \( f \) is orientation preserving and \( \mathrm{P}\left( f\right) = \varnothing \) . We show the whole \( {S}^{1} \) is a minimal set under \( f \) . Take any \( x \in {S}^{1} \) . We prove \( \overline{\oper... | Take any open interval \( \left( {a, b}\right) \) . It suffices to prove \( \operatorname{Orb}\left( x\right) \) intersects \( \left( {a, b}\right) \) . See Figure 1.9. Since \( x \) is not periodic, \( \operatorname{Orb}\left( x\right) \) consists of infinitely many points. Hence there are two points \( {f}^{n}\left( ... | Yes |
An orientation-reversing homeomorphism as Figure 1.10 shows. \( \operatorname{Fix}\left( f\right) \) consists of two fixed points \( a, b \in {S}^{1} \), and \( f \) interchanges the two cointervals of \( \{ a, b\} \) . Note that \( {f}^{2} \) restricted to \( \left\lbrack {a, b}\right\rbrack \), or \( \left\lbrack {b,... | One can easily prove that \( \Omega \left( f\right) \) consists of the two fixed points \( a \) and \( b \), together with some periodic points of period 2 . | No |
Theorem 1.11. Let \( f : {S}^{1} \rightarrow {S}^{1} \) be an orientation-preserving homeomorphism with \( \mathrm{P}\left( f\right) \neq \varnothing \) . Then all periodic points of \( f \) have the same period, and \( \mathrm{P}\left( f\right) = \Omega \left( f\right) \) . | Proof. Fix any \( x \in \mathrm{P}\left( f\right) \) . Assume the period of \( x \) is \( n \) . Let \( {x}_{1},{x}_{2},\ldots ,{x}_{n} \) be the \( n \) points of \( \operatorname{Orb}\left( x\right) \), ordered counterclockwise. Note that this may not be the order of iteration. Then\n\nLicensed to Peking University. ... | Yes |
Theorem 1.12. Let \( f : {S}^{1} \rightarrow {S}^{1} \) be a homeomorphism with \( \mathrm{P}\left( f\right) = \varnothing \) . Then \( \Omega \left( f\right) \) is a minimal set. Moreover, \( \Omega \left( f\right) \) either coincides with \( {S}^{1} \) or is a Cantor set. | Proof. To prove \( \Omega \left( f\right) \) is minimal, it suffices to prove that every nonempty compact invariant set \( \Lambda \) of \( f \) contains \( \Omega \left( f\right) \) . Take any cointerval \( \left( {a, b}\right) \) of \( \Lambda \) . For any \( n \geq 1 \), either \( {f}^{n}\left( {a, b}\right) = \left... | Yes |
An exceptional homeomorphism of \( {S}^{1} \) . | We informally illustrate the construction of an exceptional homeomorphism of \( {S}^{1} \) . Take a rigid irrational rotation \( f : {S}^{1} \rightarrow {S}^{1} \) and any \( p \in {S}^{1} \) . Replace the points\n\n\[ \ldots ,{f}^{-1}p, p,{fp},{f}^{2}p,\ldots \]\n\ncorrespondingly by countably many closed intervals\n\... | Yes |
Lemma 1.14. The set of attracting sets of \( f \) is countable. | Proof. Choose a countable basis \( \mathcal{B} = {\left\{ {V}_{n}\right\} }_{n = 1}^{\infty } \) for the topology of \( X \) . Let \( A \) be an attracting set of \( f \) with isolating neighborhood \( U \) . Since \( A \) is compact, there are \( {V}_{{i}_{1}},\ldots ,{V}_{{i}_{k}} \) such that \( A \subset {V}_{{i}_{... | Yes |
Lemma 1.15. \( \operatorname{CR}\left( f\right) = \mathop{\bigcap }\limits_{{i = 1}}^{\infty }\left( {{A}_{i} \cup {A}_{i}^{ * }}\right) \) | Proof. Before the proof we fix a basic fact. Fact. For any trapping region \( U \) of \( f \), there is \( {\epsilon }_{0} \) such that no \( {\epsilon }_{0} \) -chain can go from \( {f}^{2}\left( U\right) \) to \( X - f\left( U\right) \) . In fact one can take \( {\epsilon }_{0} = \frac{1}{2}d\left( {X - f\left( U\rig... | No |
Lemma 1.16. If \( x, y \in \mathrm{{CR}}\left( f\right) \), then \( x \) and \( y \) are in the same chain class if and only if, for every \( i, x \) and \( y \) are either both in \( {A}_{i} \) or both in \( {A}_{i}^{ * } \) . | Proof. If for some \( i, x \in {A}_{i} \) but \( y \in {A}_{i}^{ * } \), then there is \( {\epsilon }_{0} > 0 \) such that no \( {\epsilon }_{0} \) -chain can go from \( x \) to \( y \) . Hence \( x \) and \( y \) are not in the same chain class.\n\nConversely, if \( x \) and \( y \) are not in the same chain class, th... | Yes |
Lemma 1.17. Let \( \left( {A,{A}^{ * }}\right) \) be an attracting-repelling pair of \( f \) . There is a continuous function \( \phi : X \rightarrow \left\lbrack {0,1}\right\rbrack \) such that:\n\n(1) \( {\left. \phi \right| }_{{A}^{ * }} = 1,{\left. \;\phi \right| }_{A} = 0 \), and \( \phi \left( x\right) \in \left(... | Proof. Let \( U \) be an isolating neighborhood of \( A \) . Take a continuous function \( \alpha : X \rightarrow \left\lbrack {0,1}\right\rbrack \) such that \( \alpha \left( {X - U}\right) = 1,\alpha \left( {f\left( \bar{U}\right) }\right) = 0 \), and \( \alpha \left( x\right) \in \left( {0,1}\right) \) for every \( ... | Yes |
Theorem 2.2 (Characterization of \( {E}^{s} \) ). Let \( A : E \rightarrow E \) be a hyperbolic linear isomorphism with splitting \( E = {E}^{s} \oplus {E}^{u} \) . Then \( {E}^{s} \) is characterized \( {by} \)\n\n\[ \n{E}^{s} = \left\{ {v \in E \mid {A}^{n}v \rightarrow 0, n \rightarrow + \infty }\right\} \]\n\n\[ \n... | Proof. We prove this for \( {E}^{s} \) . Obviously the first set is contained in the second, and the second is contained in the third. We prove that the third is contained in the fourth. In fact, if\n\n\[ \nv \notin \left\{ {v \in E \mid \text{ there is }\gamma > 0\text{ such that }{A}^{n}v \in {C}_{\gamma }\left( {E}^... | Yes |
Theorem 2.3. Let \( A \) be a hyperbolic linear isomorphism with splitting \( E = \) \( {E}^{s} \oplus {E}^{u} \) . There are a norm \( \parallel \cdot \parallel \) of \( E \) and a constant \( 0 < \tau < 1 \) such that\n\n\[ \n\parallel {Av}\parallel \leq \tau \parallel v\parallel ,\forall v \in {E}^{s},\n\]\n\n\[ \n\... | Proof. Let \( \left| \cdot \right| \) be the original norm of \( E \) . Take \( N \) sufficiently large such that \( C{\lambda }^{N} < 1 \), and define\n\n\[ \n\parallel v\parallel = \mathop{\sum }\limits_{{n = 0}}^{{N - 1}}\left| {{A}^{n}v}\right| ,\forall v \in E.\n\]\n\nThen \( \parallel \cdot \parallel \) is a norm... | Yes |
Lemma 2.4. Let \( f : U \rightarrow E \) be a \( {C}^{1} \) map, and let \( p \in U \) be a point. For any \( \epsilon > 0 \), there are \( \delta > 0 \) and \( r > 0 \) such that, for any \( g \in {\mathcal{B}}^{1}\left( {f,\delta }\right) \) , \( \operatorname{Lip}\left( {g - {Df}\left( p\right) }\right) \leq \epsilo... | Proof. Let \( \epsilon > 0 \) be given. There are \( \delta > 0 \) and \( r > 0 \) such that for any \( g \in {\mathcal{B}}^{1}\left( {f,\delta }\right) \) and any \( x \in B\left( {p, r}\right) \) , \[ \left| {D\left( {g - {Df}\left( p\right) }\right) \left( x\right) }\right| = \left| {{Dg}\left( x\right) - {Df}\left(... | No |
Theorem 2.6 (Persistence of hyperbolic fixed point). Let \( p \in U \) be a hyperbolic fixed point of \( f \) . There are \( {\delta }_{0} > 0 \) and \( {\epsilon }_{0} > 0 \) such that any \( g \in {\mathcal{B}}^{1}\left( {f,{\delta }_{0}}\right) \) has in \( B\left( {p,{\epsilon }_{0}}\right) \) at most one fixed poi... | Proof. If the statement holds for one norm of \( E \), it holds for every norm. It hence suffices to prove the theorem under a norm \( \left| \cdot \right| \) of \( E \) that is adapted to and of box type to \( {Df}\left( p\right) \) . Without loss of generality we assume \( p = 0 \) . Abbreviate \( {Df}\left( 0\right)... | Yes |
Theorem 2.7 (Lipschitz inverse function theorem). Let \( A : E \rightarrow {E}^{\prime } \) be a linear isomorphism, and let \( \phi : E \rightarrow {E}^{\prime } \) be Lipschitz. If\n\n\[ \operatorname{Lip}\phi < m\left( A\right) \]\n\nthen \( A + \phi : E \rightarrow {E}^{\prime } \) is a lipeomorphism and\n\n\[ \ope... | Proof. First we prove \( A + \phi \) is 1-1 and onto. This means that, for any \( z \in {E}^{\prime } \), the equation\n\n\[ \left( {A + \phi }\right) x = z \]\n\nhas a unique solution for \( x \in E \), or\n\n\[ x = {A}^{-1}z - {A}^{-1}\phi \left( x\right) \]\n\nhas a unique solution for \( x \in E \) . In other words... | Yes |
Theorem 2.8. Let \( A \) and \( X \) be two metric spaces with \( X \) complete, and let \( F : A \times X \rightarrow X \) be a map. Assume there is \( 0 < \lambda < 1 \) such that\n\n\[ d\left( {F\left( {a, x}\right), F\left( {a, y}\right) }\right) \leq {\lambda d}\left( {x, y}\right) \]\n\nfor any \( a \in A \) and ... | Proof. Since\n\n\[ d\left( {p\left( a\right), p\left( b\right) }\right) = d\left( {F\left( {a, p\left( a\right) }\right), F\left( {b, p\left( b\right) }\right) }\right) \]\n\n\[ \leq d\left( {F\left( {a, p\left( a\right) }\right), F\left( {a, p\left( b\right) }\right) }\right) + d\left( {F\left( {a, p\left( b\right) }\... | Yes |
Lemma 2.9. Let \( B : E \rightarrow E \) be a linear isomorphism, represented under a direct sum \( E = {E}_{1} \oplus {E}_{2} \) as \[ \left( \begin{array}{ll} {B}_{11} & {B}_{12} \\ {B}_{21} & {B}_{22} \end{array}\right) \] where \( {B}_{ij} = {\left. {\pi }_{i} \circ B\right| }_{{E}_{j}} \) . If there are a norm \( ... | Proof. Let \( P : {E}_{1} \rightarrow {E}_{2} \) be a linear map with \( \left| P\right| \leq 1 \) . For any \( v \in {E}_{1} \) , \[ \left( \begin{matrix} {B}_{11} & {B}_{12} \\ {B}_{21} & {B}_{22} \end{matrix}\right) \left( \begin{matrix} v \\ {Pv} \end{matrix}\right) = \left( \begin{matrix} {B}_{11}v + {B}_{12}{Pv} ... | Yes |
Theorem 2.10 (Persistence of hyperbolicity for a linear map). Let \( A : E \rightarrow \) \( E \) be a hyperbolic linear isomorphism. There is \( {\delta }_{0} > 0 \) such that if a linear map \( B : E \rightarrow E \) satisfies \( \left| {B - A}\right| < {\delta }_{0} \), then \( B \) is hyperbolic. Moreover, the stab... | Proof. If the theorem holds for one norm of \( E \), it will hold for every norm of \( E \) . It hence suffices to prove the theorem under a norm \( \left| \cdot \right| \) of \( E \) that is adapted to and of box type to \( A \) . Let \( E = {E}^{u} \oplus {E}^{s} \) be the hyperbolic splitting of \( A : E \rightarrow... | Yes |
Lemma 2.14 (Characterization of \( {W}_{r}^{s} \), adapted form). Let \( A : E \rightarrow E \) be a hyperbolic linear isomorphism with splitting \( E = {E}^{s} \oplus {E}^{u} \) of skewness \( 0 < \tau < 1 \) with respect to a norm \( \left| \cdot \right| \) of \( E \) that is adapted to and of box type to \( A \). Le... | Proof. First we prove two simple facts. Claim 1. If \( v,{v}^{\prime } \in E\left( r\right) \), then \[ \left| {{\left( A + \phi \right) }_{s}v - {\left( A + \phi \right) }_{s}{v}^{\prime }}\right| \leq \left( {\tau + \operatorname{Lip}\phi }\right) \left| {v - {v}^{\prime }}\right| . \] In fact, \[ \left| {{\left( A +... | Yes |
Theorem 2.15 (Characterizations of \( {W}_{r}^{s} \), general form). Let \( p \in U \) be a hyperbolic fixed point of \( f \) . There are \( r > 0, C \geq 1 \), and \( 0 < \lambda < 1 \) such that\n\n\[ \n{W}_{r}^{s}\left( {p, f}\right) = \left\{ {v \in U\left| \right| {f}^{n}v - p \mid \leq r,\forall n \geq 0}\right\}... | Proof. We prove the theorem for \( {W}_{r}^{s} \) only. If the statement holds for one norm of \( E \), it holds for every norm. It hence suffices to prove the theorem under a norm \( \left| \cdot \right| \) of \( E \) that is adapted to and of box type to \( {Df}\left( p\right) \) . It suffices to prove that there are... | Yes |
Theorem 2.16 (Isolation of a hyperbolic fixed point). Let \( p \in U \) be a hyperbolic fixed point of \( f \) . There is \( r > 0 \) such that if \( w \in U \) satisfies\n\n\[ \left| {{f}^{n}w - p}\right| \leq r,\;\forall n \in \mathbb{Z}, \]\n\nthen \( w = p \) . | Proof. Take \( r > 0, C \geq 1 \), and \( 0 < \lambda < 1 \) such that Theorem 2.15 holds for both \( f \) and \( {f}^{-1} \) . Let \( w \in U \) satisfy \( \left| {{f}^{n}w - p}\right| \leq r \) for all \( n \in \mathbb{Z} \) . By Theorem 2.15, for any \( n \geq 0 \), \n\n\[ \left| {w - p}\right| = \left| {{f}^{-n}\le... | Yes |
Lemma 2.17. Let \( A : E \rightarrow E \) be a hyperbolic linear isomorphism with splitting \( E = {E}^{u} \oplus {E}^{s} \), and let \( \left| \cdot \right| \) be a norm of \( E \) that is adapted to and of box type to \( A \) . Then there is \( \delta > 0 \) such that:\n\n(1) If \( \phi : E \rightarrow E \) is Lipsch... | Proof. We first prove item (1). Let \( 0 < \tau < 1 \) be the skewness of \( A \) with respect to \( \left| \cdot \right| \) . Let\n\n\[ \delta = \min \left\{ {\frac{1 - \tau }{2}, m\left( A\right) }\right\} \]\n\n(Later we will reduce \( \delta \) further.)\n\nLet \( \phi : E \rightarrow E \) be a Lipschitz map such t... | Yes |
Theorem 2.18 (Local stable manifold for a hyperbolic fixed point). Let \( f : U \rightarrow E \) be \( {C}^{k}, k \geq 1 \), and let \( 0 \in U \) be a hyperbolic fixed point of \( f \) with splitting \( E = {E}^{s} \oplus {E}^{u} \) . Then there is \( r > 0 \) such that \( {W}_{r}^{s}\left( {0, f}\right) \) is a \( {C... | The proof of Theorem 2.18 is postponed to Section 4.3, page [91] | No |
Theorem 3.1. \( {\sum }_{2} \) is a Cantor set. | Proof. By the Tychonoff theorem \( {\sum }_{2} \) is compact. Given any \( a \in {\sum }_{2} \) and \( j \geq 1 \), there is \( b \in {C}_{j}\left( a\right) \) that is different from \( a \) . Hence \( {\sum }_{2} \) is perfect. We prove \( {\sum }_{2} \) is totally disconnected. Take any \( a \neq b \) in \( {\sum }_{... | Yes |
Theorem 3.2. Periodic points of \( \sigma \) are dense in \( {\sum }_{2} \), and \( \sigma \) is transitive on \( {\sum }_{2} \) . | Proof. Let \( a \in {\sum }_{2} \) be given. For any \( j \geq 1 \), let \( b \in {\sum }_{2} \) be the bi-sequence that infinitely repeats the \( \left( {{2j} + 1}\right) \) -tuple \( {a}_{-j}\cdots {a}_{j} \) of \( a \) in both directions. Then \( b \) is periodic and \( b \in {C}_{j}\left( a\right) \) . Thus periodi... | Yes |
Theorem 3.3. Let \( X \) be a compact metric space, and let \( f : X \rightarrow X \) be a homeomorphism. Assume \( f \) has periodic points dense in \( X \) and is also transitive on \( X \) . If \( X \) does not reduce to a single periodic orbit, then \( f \) has sensitive dependence on initial conditions. | Proof. Since periodic points are dense in \( X \) and \( X \) does not reduce to a single periodic orbit, there are at least two periodic points \( p, q \in X \) such that\n\n\[ a = d\left( {\operatorname{Orb}\left( p\right) ,\operatorname{Orb}\left( q\right) }\right) > 0. \]\n\nWe prove that \( f \) has sensitive depe... | Yes |
Theorem 3.4 (Smale (1965)). \( f : \Lambda \rightarrow \Lambda \) is topologically conjugate to \( \sigma : {\sum }_{2} \rightarrow {\sum }_{2} \) | Proof. If \( x \notin {H}_{0} \cup {H}_{1} \), then \( {fx} \notin Q \) . Hence\n\n\[ \Lambda = \mathop{\bigcap }\limits_{{n = - \infty }}^{\infty }{f}^{n}\left( {{H}_{0} \cup {H}_{1}}\right) \]\n\nSince \( {H}_{0} \cap {H}_{1} = \varnothing \), for any \( x \in \Lambda \), there is a unique \( a \in {\sum }_{2} \) suc... | Yes |
Corollary 3.5. The horseshoe set \( \Lambda \) is a Cantor set. The horseshoe map \( f : \Lambda \rightarrow \Lambda \) has periodic points dense and is transitive. | We briefly indicate that the horseshoe map is structurally stable. Indeed, the geometric construction that leads to the conjugacy \( h \) is very coarse: stretching and compressing the square \( Q \), bending it over to cross \( Q \) itself. It is intuitively convincing that any diffeomorphism \( g \) that is \( {C}^{1... | No |
Theorem 3.7. Let \( A : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2} \) be an Anosov automorphism. Then the eigenvalues of \( A \) are two irrationals with \( \left| {\lambda }_{1}\right| < 1 < \left| {\lambda }_{2}\right| \), and the slopes of the two eigen-directions are irrational. | Proof. If the two eigenvalues \( {\lambda }_{1} \) and \( {\lambda }_{2} \) of \( A \) are complex conjugate or multiple real roots, they must be of norm 1 as \( \left| {\det A}\right| = 1 \), contradicting that \( A \) is hyperbolic. Thus \( {\lambda }_{1} \) and \( {\lambda }_{2} \) are real and distinct. Since \( \l... | Yes |
Theorem 3.8. Let \( f : {\mathbb{T}}^{2} \rightarrow {\mathbb{T}}^{2} \) be an Anosov toral automorphism induced by \( A : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2} \) .\n\n(1) For any \( a \in {\mathbb{R}}^{2},{W}^{s}\left( {a, A}\right) = a + {E}^{s} \), where \( {\mathbb{R}}^{2} = {E}^{s} \oplus {E}^{u} \) is th... | Proof. (1) Let \( b \in {W}^{s}\left( {a, A}\right) \) . Then \( \left| {{A}^{n}b - {A}^{n}a}\right| \rightarrow 0,\left| {{A}^{n}\left( {b - a}\right) }\right| \rightarrow 0 \) , which means \( b - a \in {E}^{s} \), or \( b \in a + {E}^{s} \) . This proves \( {W}^{s}\left( {a, A}\right) \subset a + {E}^{s} \) . Each s... | Yes |
Theorem 3.9. Let \( f : {\mathbb{T}}^{2} \rightarrow {\mathbb{T}}^{2} \) be an Anosov toral automorphism. Then periodic points of \( f \) are dense in \( {\mathbb{T}}^{2} \), and \( f \) is transitive on \( {\mathbb{T}}^{2} \). | Proof. To prove that periodic points of \( f \) are dense in \( {\mathbb{T}}^{2} \), it suffices to prove that any \ | No |
Lemma 3.10. Periodic points of \( g \) are dense in \( {S}^{1} \) . | Proof. Take any interval \( \left\lbrack {a, b}\right\rbrack \subset {S}^{1} \) . There is \( n \geq 0 \) large such that \( {g}^{n}\left\lbrack {a, b}\right\rbrack \) covers the whole \( {S}^{1} \) . Hence there is a subinterval \( \left\lbrack {{a}^{\prime },{b}^{\prime }}\right\rbrack \subset \left\lbrack {a, b}\rig... | Yes |
Theorem 3.11. The solenoid \( A \) has the following properties:\n\n(1) Periodic points of \( f \) are dense in \( A \) . (2) \( {\left. f\right| }_{A} \) is transitive. | Proof. (1) Let \( x \in A \) . Take any neighborhood \( U \) of \( x \) in \( T \) . We prove there is a periodic point \( p \) of \( f \) in \( U \) . Then \( p \) is in \( A \) since \( T \) is a trapping region.\n\nWe may assume \( U = {f}^{n}\left( {D\left\lbrack {a, b}\right\rbrack }\right) \) for some \( n \) lar... | Yes |
Theorem 4.2 (Characterization of \( {E}^{s} \) ). Let \( \Lambda \subset M \) be a hyperbolic set of \( f \) with splitting \( {T}_{\Lambda }M = {E}^{s} \oplus {E}^{u} \) . For any \( x \in \Lambda ,{E}^{s}\left( x\right) \) is characterized by\n\n\[ \n{E}^{s}\left( x\right) = \left\{ {v \in {T}_{x}M \mid \left| {T{f}^... | Proof. The proof is the same as that of Theorem 2.2, just involving base points of vectors. For instance we prove for \( {E}^{s}\left( x\right) \) that the third set is contained in the fourth. In fact, if\n\n\[ \nv \in {T}_{x}M - \left\{ {v \in {T}_{x}M\mid \exists \gamma > 0\text{ such that }T{f}^{n}v \in {C}_{\gamma... | No |
Theorem 4.3. Let \( \Lambda \subset M \) be a hyperbolic set of \( f \) . Then \( {E}^{s}\left( x\right) \) and \( {E}^{u}\left( x\right) \) vary continuously in \( x \in \Lambda \) . In particular, \( \dim {E}^{s}\left( x\right) \) and \( \dim {E}^{u}\left( x\right) \) are locally constant. Moreover, the closure \( \b... | Proof. Let \( x \in \Lambda \) . We prove \( {E}^{s} \) is continuous at \( x \) . It suffices to prove that whenever a sequence \( {E}^{s}\left( {x}_{k}\right) ,{x}_{k} \in \Lambda \), converges to a linear subspace \( {G}^{s}\left( x\right) \) of \( {T}_{x}M \), then \( {G}^{s}\left( x\right) = {E}^{s}\left( x\right)... | Yes |
Theorem 4.4. Let \( \Lambda \subset M \) be a hyperbolic set of \( f \) with splitting \( {T}_{\Lambda }M = \) \( {E}^{s} \oplus {E}^{u} \) . There are a \( {C}^{\infty } \) Riemannian metric \( \langle \langle \cdot , \cdot \rangle \rangle \) of \( M \) and a constant \( 0 < \tau < 1 \) such that, with respect to the ... | Proof. Let \( \langle \cdot , \cdot \rangle \) be the given Riemannian metric of \( M \) . Take \( N \) sufficiently large such that \( C{\lambda }^{N} < 1 \), and define\n\n\[ \n\langle \langle v, u\rangle \rangle = \mathop{\sum }\limits_{{n = 0}}^{{N - 1}}\left\langle {T{f}^{n}\left( v\right), T{f}^{n}\left( u\right)... | Yes |
Lemma 4.5. Let \( g : M \rightarrow M \) be a diffeomorphism, let \( \Delta \) be an invariant set of \( g \), and let \( B : {T}_{\Delta }M \rightarrow {T}_{\Delta }M \) be a bounded \( {C}^{0} \) bundle isomorphism over \( g \), represented under a \( {C}^{0} \) direct sum \( {T}_{\Delta }M = {E}_{1} \oplus {E}_{2} \... | Proof. Since for fiber-preserving maps things are defined pointwise, the proof will be the same as that of Lemma 2.9, except marking base points \( x \) . It is strongly recommended that the reader compare the two proofs, sentence by sentence. Let \( P : {E}_{1} \rightarrow {E}_{2} \) be a \( {C}^{0} \) bundle homomorp... | Yes |
Theorem 4.6 (Persistence of hyperbolicity for an invariant set). Let \( \Lambda \subset M \) be a compact hyperbolic set of \( f \) . There are a \( {C}^{1} \) neighborhood \( {\mathcal{U}}_{0} \) of \( f \) in \( {\operatorname{Diff}}^{1}\left( M\right) \) and a number \( {a}_{0} > 0 \) such that for any \( g \in {\ma... | Proof. Let \( {T}_{\Lambda }M = {E}^{s} \oplus {E}^{u} \) be the hyperbolic splitting of \( \Lambda \) . We may assume that the given Riemannian norm \( \left| \cdot \right| \) of \( M \) is adapted to \( \Lambda \) . Since \( \Lambda \) is compact, the \( {C}^{0} \) splitting \( {T}_{\Lambda }M = {E}^{s} \oplus {E}^{u... | Yes |
Lemma 4.7. For any \( \delta > 0 \), there is \( K \geq 1 \) such that for any Euclidean space \( E \), any direct sum \( E = {E}_{1} \oplus {E}_{2} \), and any inner product \( \langle \cdot , \cdot \rangle \) of \( E \), if \( \angle \left( {{E}_{1},{E}_{2}}\right) > \delta \), where the angle is with respect to \( \... | The proof is elementary and is left as an exercise. | No |
Lemma 4.8. Let \( \Lambda \subset M \) be a compact hyperbolic set of \( f \), and let \( \left| \cdot \right| \) be a Riemannian norm of \( M \) . Then there are a \( {C}^{1} \) neighborhood \( {\mathcal{U}}_{0} \) of \( f \) in \( {\operatorname{Diff}}^{1}\left( M\right) \) and two numbers \( {a}_{0} > 0 \) and \( K ... | Proof. The proof is standard and we give a sketch only. If \( g \) is \( {C}^{1} \) close to \( f \) and \( \Delta \) is in a small neighborhood of \( \Lambda \), then, by Theorem 4.6, for every \( x \in \Delta \) there is \( y \in \Lambda \) such that \( {E}^{s}\left( {x, g}\right) \oplus {E}^{u}\left( {x, g}\right) \... | Yes |
Lemma 4.10. Let \( f : M \rightarrow M \) be a \( {C}^{1} \) diffeomorphism.\n\n(1) \( {F}_{f}\left( {0}_{x}\right) = {0}_{fx},\;\forall x \in M \) .\n\n(2) \( {D}_{2}\left( {F}_{f}\right) \left( {0}_{x}\right) = {T}_{x}f,\;\forall x \in M \) .\n\n(3) \( {D}_{2}\left( {F}_{f}\right) \) is continuous on \( {TM}\left( {r... | Proof.\n\n\[ \n{F}_{f}\left( {0}_{x}\right) = {\exp }_{fx}^{-1}{fex}{p}_{x}\left( {0}_{x}\right) = {\exp }_{fx}^{-1}\left( {fx}\right) = {0}_{fx}. \]\n\n\[ \n{D}_{2}\left( {F}_{f}\right) \left( {0}_{x}\right) = D\left( {{\exp }_{fx}^{-1}f{\exp }_{x}}\right) \left( {0}_{x}\right) \]\n\n\[ \n= {\left. id\right| }_{{T}_{f... | Yes |
Lemma 4.11. Let \( f : M \rightarrow M \) be a diffeomorphism. Denote \( {\phi }_{g} = {F}_{g} - {Tg} \) . Then for any \( \epsilon > 0 \), there are a \( {C}^{1} \) neighborhood \( \mathcal{U} \) of \( f \) and a number \( r > 0 \) such that, for any \( g \in \mathcal{U},{\operatorname{Lip}}_{2}{\phi }_{g} < \epsilon ... | Proof. Since \( {D}_{2}{\phi }_{f} \) is continuous on \( {TM}\left( {r}_{\rho }\right) \) and since \( {D}_{2}{\phi }_{f}\left( {0}_{x}\right) = 0 \) and \( M \) is compact, for any \( \epsilon > 0 \), there is \( r > 0 \) such that for any \( v \in {TM}\left( r\right) \) one has \( \left| {{D}_{2}{\phi }_{f}\left( v\... | Yes |
Lemma 4.12 (Characterization of \( {W}_{r}^{s} \) on fibers). Let \( \Lambda \subset M \) be a hyperbolic set of \( f \) with splitting \( {T}_{\Lambda }M = {E}^{s} \oplus {E}^{u} \) of skewness \( 0 < \tau < 1 \) with respect to a \( {C}^{0} \) norm \( \left| \cdot \right| \) of \( {T}_{\Lambda }M \) that is adapted t... | Proof. Here \( {T}_{\Lambda }M\left( r\right) ,{Tf},\phi \), and \( {\operatorname{Lip}}_{2}\phi \) correspond to \( E\left( r\right), A,\phi \), and \( \operatorname{Lip}\phi \) of Lemma 2.14. We write the first part of the proof only. Claim 1. If \( x \in \Lambda \) and \( v,{v}^{\prime } \in {T}_{x}M\left( r\right) ... | Yes |
Theorem 4.13 (Characterization of \( {W}_{r}^{s} \) on manifold). Let \( \Lambda \subset M \) be a hyperbolic set of \( f \) . There are \( r > 0, C \geq 1 \), and \( 0 < \lambda < 1 \) such that for any \( x \in \Lambda \) ,\n\n\[ \n{W}_{r}^{s}\left( {x, f}\right) = \left\{ {y \in M \mid d\left( {{f}^{n}y,{f}^{n}x}\ri... | Proof. We give a proof for \( {W}_{r}^{s} \) only. We may assume the Riemannian norm \( \left| \cdot \right| \) of \( M \) is adapted to \( \Lambda \) . It suffices to prove there are \( r > 0, C \geq 1 \), and \( 0 < \lambda < 1 \) such that the second set is contained in the third.\n\nGiven \( x \in \Lambda \), for \... | Yes |
Theorem 4.14 (Uniform expansivity of hyperbolic sets). Let \( \Lambda \subset M \) be a hyperbolic set of \( f \) . Then \( {\left. f\right| }_{\Lambda } \) is expansive. In fact, there are a \( {C}^{1} \) neighborhood \( {\mathcal{U}}_{0} \) of \( f \) and two numbers \( {a}_{0} > 0 \) and \( {r}_{0} > 0 \) such that ... | Proof. We may assume the Riemannian norm \( \left| \cdot \right| \) of \( M \) is adapted to \( \Lambda \) . By definition, \( {\left. g\right| }_{\Delta } \) is \( {r}_{0} \) -expansive means that if \( x, y \in \Delta \) satisfy\n\n\[ d\left( {{g}^{n}x,{g}^{n}y}\right) \leq {r}_{0}\;\forall n \in \mathbb{Z} \]\n\nthe... | Yes |
Lemma 4.15. Let \( \Lambda \subset M \) be a hyperbolic set of \( f \) with splitting \( {T}_{\Lambda }M = \) \( {E}^{u} \oplus {E}^{s} \), and let \( \left| \cdot \right| \) be a \( {C}^{0} \) norm of \( {T}_{\Lambda }M \) that is adapted to and of box type to \( {E}^{u} \oplus {E}^{s} \). Then there is \( \delta > 0 ... | Proof. We first prove item (1). Let\n\n\[ \n\sum \left( {{E}^{u},{E}^{s};0}\right) = \left\{ {\sigma : {E}^{u} \rightarrow {E}^{s} \mid \sigma }\right. \text{is continuous,}\n\]\n\n\[ \n\text{fiber-preserving over}{id},\sigma \left( {0}_{x}\right) = {0}_{x},{\left| \sigma \right| }_{ * } < \infty \} \text{,}\n\]\n\nwhe... | Yes |
Theorem 4.16 (Stable manifolds theorem for a hyperbolic set). Let \( f \) : \( M \rightarrow M \) be a \( {C}^{k} \) diffeomorphism, \( k \geq 1 \), and let \( \Lambda \subset M \) be a hyperbolic set of \( f \) with splitting \( {T}_{\Lambda }M = {E}^{s} \oplus {E}^{u} \) . Then there is \( r > 0 \) such that, for eve... | Proof. First we prove item (1). We first prove there is \( r > 0 \) such that, for every \( x \in \Lambda ,{W}_{r}^{s}\left( x\right) \) is a \( {C}^{k} \) submanifold of \( M \), tangent at \( x \) to \( {E}_{x}^{s} \) . Since\n\n\[ \n{W}_{r}^{s}\left( {x, f}\right) = {\exp }_{x}\left( {{W}_{r}^{s}\left( {{0}_{x},{F}_... | Yes |
Theorem 4.17. Let \( \Lambda \subset M \) be a hyperbolic set of \( f \) . There are \( r > 0 \) and \( \delta > 0 \) such that for any \( x, y \in \Lambda \), if \( d\left( {x, y}\right) \leq \delta \), then \( {W}_{r}^{s}\left( x\right) \cap {W}_{r}^{u}\left( y\right) \neq \varnothing \) transversely. | Proof. Let \( {\Lambda }_{i} = \left\{ {x \in \Lambda : \dim {E}^{s}\left( x\right) = i}\right\} \) . Then \( {\Lambda }_{0},\ldots ,{\Lambda }_{\dim M} \) are finitely many disjoint compact invariant sets. By Theorem 4.1, \( {\Lambda }_{0} \) and \( {\Lambda }_{\dim M} \) are of finite many points. Thus we may assume ... | Yes |
Lemma 4.18. Let \( \Lambda \subset M \) be a hyperbolic set of \( f \) with splitting \( {T}_{\Lambda }M = \) \( {E}^{s} \oplus {E}^{u} \), and let \( \left| \cdot \right| \) be a \( {C}^{0} \) norm of \( {T}_{\Lambda }M \) that is adapted to and of box type to \( {E}^{s} \oplus {E}^{u} \) . Let \( 0 < \tau < 1 \) be t... | Proof. The proof will be like a copy of that of Lemma 2.5. We solve the equation \[ \left( {{Tf} + \phi }\right) \gamma = \gamma \] for \( \gamma \in {\Gamma }^{0}\left( {{T}_{\Lambda }M}\right) \left( r\right) \) . Till the end of the proof of Lemma 4.18, we abbreviate \[ {Tf} = A\text{.} \] It is equivalent to solve ... | Yes |
Theorem 4.20. Anosov diffeomorphisms are \( {C}^{1} \) structurally stable. | Proof. Let \( f : M \rightarrow M \) be Anosov. By Theorem 4.19, there is a \( {C}^{1} \) neighborhood \( \mathcal{U} \) of \( f \) such that for any \( g \in \mathcal{U} \), there is a continuous injective map \( h = {h}_{g} : M \rightarrow M \) such that \( {hf} = {gh} \) . We only need prove \( h \) is onto. By the ... | Yes |
Theorem 4.22. Let \( f : X \rightarrow X \) be a homeomorphism. Let \( \Lambda \subset X \) be an isolated invariant set of \( f \) with an isolating neighborhood \( U \) . For any \( a > 0 \) , there is a \( {C}^{0} \) neighborhood \( \mathcal{U} \) of \( f \) such that, for any \( g \in \mathcal{U} \), the maximal in... | Proof. Since\n\n\[ \mathop{\bigcap }\limits_{{n \in \mathbb{Z}}}{f}^{n}\left( U\right) = \Lambda \]\n\nfor any \( a > 0 \), there is \( N \) large such that\n\n\[ \mathop{\bigcap }\limits_{{n = - N}}^{N}{f}^{n}\left( U\right) \subset B\left( {\Lambda, a/2}\right) \]\n\nTake a sufficiently small \( {C}^{0} \) neighborho... | Yes |
Theorem 4.23 (Structural stability of isolated hyperbolic sets). Let \( \Lambda \) be an isolated hyperbolic set of \( f : M \rightarrow M \) with an isolating neighborhood \( U \) . For any \( \epsilon > 0 \), there is a \( {C}^{1} \) neighborhood \( \mathcal{U} \) of \( f \) such that, for any \( g \in \mathcal{U} \)... | Proof. By Theorem 4.21, there are a \( {C}^{1} \) neighborhood \( {\mathcal{U}}_{0} \) of \( f \) and two numbers \( {a}_{0} > 0 \) and \( {\epsilon }_{0} > 0 \) such that, for any \( g,{g}^{\prime } \in {\mathcal{U}}_{0} \) and any compact invariant set \( \Delta \subset B\left( {\Lambda ,{a}_{0}}\right) \) of \( g \)... | Yes |
Theorem 4.24 (The shadowing lemma). Let \( \Lambda \subset M \) be a hyperbolic set of \( f \) . Then there are \( {\epsilon }_{0} > 0 \) and \( {\delta }_{0} > 0 \) such that every \( {\delta }_{0} \) -pseudo orbit in \( \Lambda \) can be \( {\epsilon }_{0} \) -shadowed by at most one point. Moreover, for any \( 0 < \... | Note that the uniqueness can be derived directly from the expansivity of hyperbolic sets. The shadowing lemma is usually and beautifully proved by transferring it into a fixed point problem of a Banach space; see Katok-Hasselblatt (1995) or Pilyugin (1999). Here we just explain that the shadowing lemma can be regarded ... | Yes |
Theorem 4.26. Let \( \Lambda \subset M \) be an isolated hyperbolic set of \( f \) . Then\n\n\[ \n{W}^{s}\left( \Lambda \right) = \mathop{\bigcup }\limits_{{x \in \Lambda }}{W}^{s}\left( x\right)\n\]\n\n\[ \n{W}^{u}\left( \Lambda \right) = \mathop{\bigcup }\limits_{{x \in \Lambda }}{W}^{u}\left( x\right)\n\] | Proof. It suffices to prove\n\n\[ \n{W}^{s}\left( \Lambda \right) \subset \mathop{\bigcup }\limits_{{x \in \Lambda }}{W}^{s}\left( x\right)\n\]\n\nLet \( y \in {W}^{s}\left( \Lambda \right) \) . Let \( r > 0 \) be the number in Theorem 4.13 such that, for every \( x \in \Lambda \), \n\n\[ \n{W}_{r}^{s}\left( x\right) =... | Yes |
Theorem 4.27 (Improving Theorem 4.24). Let \( \Lambda \subset M \) be a hyperbolic set of \( f \) . For any \( \epsilon > 0 \), there is \( \eta > 0 \) such that every \( \eta \) -pseudo-orbit in the \( \eta \) -neighborhood of \( \Lambda \) is \( \epsilon \) -shadowed by a point. | Proof. For any \( \epsilon > 0 \), by Theorem 4.24, there is \( \delta > 0 \) such that every \( \delta \) -pseudo-orbit in \( \Lambda \) is \( \epsilon /2 \) -shadowed by a point. Take \( 0 < \eta \leq \epsilon /2 \) such that for any \( \eta \) -pseudo-orbit \( \left\{ {x}_{n}\right\} \), if \( d\left( {{y}_{n},{x}_{... | Yes |
Theorem 4.28 (Anosov closing lemma). Let \( \Lambda \subset M \) be a hyperbolic set of \( f \) . For every \( \epsilon > 0 \), there is \( \delta > 0 \) such that every periodic \( \delta \) -pseudo-orbit in \( \Lambda \) is \( \epsilon \) -shadowed by a periodic point. | Proof. By Theorem 4.24, there are \( {\epsilon }_{0} > 0 \) and \( {\delta }_{0} > 0 \) such that every \( {\delta }_{0} \) -pseudo-orbit in \( \Lambda \) can be \( {\epsilon }_{0} \) -shadowed by at most one point.\n\nLet \( 0 < \epsilon \leq {\epsilon }_{0} \) be given. By Theorem 4.24, there is \( 0 < \delta \leq {\... | Yes |
Theorem 4.30. Every transverse homoclinic point is the limit of periodic points. | Proof. We take the case that \( p \) is a hyperbolic fixed point. Let \( x \) be a transverse homoclinic point of \( p \) . Then\n\n\[ \Lambda = \operatorname{Orb}\left( x\right) \cup \{ p\} \]\n\n is a hyperbolic set (Exercise 4.2). For any \( \epsilon > 0 \), let \( \delta > 0 \) be the constant of \( \Lambda \) guar... | No |
Theorem 4.32. If \( \operatorname{CR}\left( f\right) \) is hyperbolic, then \( \operatorname{CR}\left( f\right) = \overline{\mathrm{P}\left( f\right) } \) . | Proof. Let \( x \in \operatorname{CR}\left( f\right) \) . For any \( \epsilon > 0 \), let \( \eta > 0 \) be the constant guaranteed by Theorem 4.29 (treating \( \operatorname{CR}\left( f\right) \) as \( \Lambda \) ). By Theorem 1.7, there is \( \delta > 0 \) such that any periodic \( \delta \) -chain \( {P}_{\delta } \... | Yes |
Theorem 5.1 (The \( \lambda \) -lemma). Let \( p \in M \) be a hyperbolic fixed point of \( f : M \rightarrow M \) . For any \( u \) -disc \( B \) in \( {W}^{u}\left( p\right) \), any point \( x \in {W}^{s}\left( p\right) \), any \( u \) -disc \( D \) transverse to \( {W}^{s}\left( p\right) \) at \( x \), and any \( \e... | The proof can be found in many textbooks, for instance Palis and de Melo (1982). | No |
Consider a 2D diffeomorphism \( f \) that has a homoclinic loop associated with a hyperbolic fixed point \( p \). See Figure 5.2. Using the \( \lambda \) - lemma we conclude that every point \( x \) on the loop is nonwandering. | This is because every small arc \( D \) transverse to the loop at \( x \) will eventually pile on \( {W}^{u}\left( p\right) \), hence to cross \( D \) itself. | Yes |
Let \( p \) and \( q \) be two hyperbolic periodic points of \( f \) such that \( {W}^{s}\left( {\operatorname{Orb}\left( p\right) }\right) \) and \( {W}^{u}\left( {\operatorname{Orb}\left( q\right) }\right) \) have a transverse intersection and \( {W}^{s}\left( {\operatorname{Orb}\left( q\right) }\right) \) and \( {W}... | For instance we verify that \( x \in {W}^{s}\left( {\operatorname{Orb}\left( p\right) }\right) \pitchfork {W}^{u}\left( {\operatorname{Orb}\left( q\right) }\right) \) is nonwandering. Switching to an iterate of \( f \) if necessary, we may assume \( p \) and \( q \) are fixed points of \( f \) . Take a small disc \( D ... | Yes |
Theorem 5.2 (The spectral decomposition theorem). Let \( f : M \rightarrow M \) be a diffeomorphism such that \( \overline{\mathrm{P}\left( f\right) } \) is hyperbolic. Then \( \overline{\mathrm{P}\left( f\right) } \) decomposes in a unique way into finitely many disjoint transitive sets | Proof. First we prove the uniqueness of the decomposition. Assume there is another decomposition\n\n\[ \overline{\mathrm{P}\left( f\right) } = {C}_{1} \cup \cdots \cup {C}_{l} \]\n\nof the same property. For each \( i = 1,\ldots, k \), take \( {x}_{i} \in {B}_{i} \) such that \( \omega \left( {x}_{i}\right) = \) \( {B}... | Yes |
Theorem 5.3. If \( \overline{\mathrm{P}\left( f\right) } \) is hyperbolic, then \( \overline{\mathrm{P}\left( f\right) } \) is isolated. | Proof. Let\n\n\[ \overline{\mathrm{P}\left( f\right) } = {B}_{1} \cup \cdots \cup {B}_{k} \]\n\nbe the spectral decomposition of \( f \) . It suffices to prove that each basic set, say \( {B}_{1} \), is isolated. Take a compact neighborhood \( {U}_{1} \) of \( {B}_{1} \), disjoint from the other \( {B}_{i} \), such tha... | Yes |
Theorem 5.4. Let \( f : X \rightarrow X \) be a homeomorphism, and let \( {\Lambda }_{1},\ldots ,{\Lambda }_{k} \) be finitely many disjoint compact invariant sets of \( f \) with \( {\Lambda }_{1} \cup \cdots \cup {\Lambda }_{k} \supset \mathrm{L}\left( f\right) \). Then\n\n\[ \nX = \mathop{\bigcup }\limits_{{i = 1}}^... | Proof. Take a compact neighborhood \( {U}_{i} \) of \( {\Lambda }_{i} \) in \( X \) such that, for any \( i \neq j \), \n\n\[ \n{U}_{i} \cap {U}_{j} = \varnothing ,\;\left( {f{U}_{i}}\right) \cap {U}_{j} = \varnothing . \n\]\n\nThe second equality means that a point in \( {U}_{i} \) cannot jump positively into a differ... | Yes |
Theorem 5.5. Let \( f : M \rightarrow M \) be a diffeomorphism. If \( \mathrm{L}\left( f\right) \) is hyperbolic, then\n\n\[ M = \mathop{\bigcup }\limits_{{x \in \mathrm{L}\left( f\right) }}{W}^{s}\left( x\right) = \mathop{\bigcup }\limits_{{x \in \mathrm{L}\left( f\right) }}{W}^{u}\left( x\right) . \]\n | Proof. Since \( \mathrm{L}\left( f\right) \) is hyperbolic, by Theorem 4.31, \( \mathrm{L}\left( f\right) = \overline{\mathrm{P}\left( f\right) } \) . By Theorem 5.3, \( \overline{\mathrm{P}\left( f\right) } \), hence \( \mathrm{L}\left( f\right) \), is isolated. Since\n\n\[ M = {W}^{s}\left( {\mathrm{\;L}\left( f\righ... | Yes |
Theorem 5.6. Let \( f : X \rightarrow X \) be a homeomorphism, and let \( \Omega \left( f\right) = \) \( {\Lambda }_{1} \cup \cdots \cup {\Lambda }_{k} \) be a decomposition into finitely many disjoint compact invariant sets. If the \( \left\{ {\Lambda }_{i}\right\} \) have no cycles, then for any neighborhood \( V \) ... | Proof. Suppose to the contrary that there are a compact neighborhood \( {V}_{0} \) of \( \Omega \left( f\right) \) in \( X \) and a sequence \( {g}_{n} \rightarrow f \) in the \( {C}^{0} \) topology, together with \( {a}_{n} \in \Omega \left( {g}_{n}\right) \) such that \( {a}_{n} \notin {V}_{0} \) . We may assume \( {... | Yes |
Lemma 5.7. Let \( x, y \notin \Omega \left( f\right) \) . Let \( \left\lbrack {{x}_{n},{y}_{n}}\right\rbrack \) be an orbit-arc of \( {g}_{n} \) such that \( {x}_{n} \rightarrow x,{y}_{n} \rightarrow y \) . Assume \( x \in {W}^{s}\left( {\Lambda }_{i}\right) \) for some \( i \) . Then there are \( z \in \) \( {W}^{u}\l... | Proof. Take a compact neighborhood \( U \) of \( {\Lambda }_{i} \) such that\n\n\[ x, y \notin U,\;U \cap {\Lambda }_{j} = \varnothing ,\;f\left( U\right) \cap {\Lambda }_{j} = \varnothing ,\;\forall j \neq i.\]\n\nLet \( {p}_{n} \in \left\lbrack {{x}_{n},{y}_{n}}\right\rbrack \) be the point in \( \left\lbrack {{x}_{n... | Yes |
Theorem 5.8 (The \( \Omega \) -stability theorem). Let \( f : M \rightarrow M \) be a diffeomorphism. If \( f \) satisfies Axiom \( A \) and the no-cycle condition, then \( f \) is \( {C}^{r} \) \( \Omega \) -stable for any \( r \geq 1 \) . | Proof. It suffices to prove \( f \) is \( {C}^{1}\Omega \) -stable. In fact we prove \( f \) is \( {C}^{1}\epsilon \) - \( \Omega \) - stable.\n\nBy Theorem 5.3, \( \Omega \left( f\right) \) is isolated. Let \( U \) be an isolating neighborhood of \( \Omega \left( f\right) \) in \( M \) . Let \( \epsilon > 0 \) be give... | Yes |
Theorem 5.9. Let \( f : X \rightarrow X \) be a homeomorphism, and let \( \mathrm{L}\left( f\right) = \) \( {\Lambda }_{1} \cup \cdots \cup {\Lambda }_{k} \) be a decomposition into finitely many disjoint compact invariant sets. If \( \left\{ {\Lambda }_{i}\right\} \) have no cycles, then \( \mathrm{L}\left( f\right) =... | Proof. First we single out a repeatedly used argument below, namely the following lemma. | No |
Lemma 5.10. Let \( x \in \mathrm{{CR}}\left( f\right) - \mathrm{L}\left( f\right) \) . Assume \( x \in {W}^{s}\left( {\Lambda }_{i}\right) \) for some \( i \) . Then there is \( z \in {W}^{u}\left( {\Lambda }_{i}\right) \) such that \( z \in \mathrm{{CR}}\left( f\right) - \mathrm{L}\left( f\right) \) . | Proof. The proof is similar to Lemma 5.7. Take a compact neighborhood \( U \) of \( {\Lambda }_{i} \) such that\n\n\[ x \notin U,\;U \cap {\Lambda }_{j} = \varnothing ,\;f\left( U\right) \cap {\Lambda }_{j} = \varnothing ,\;\forall j \neq i.\]\n\nFor every \( n \geq 1 \), there is a periodic \( 1/n \) -chain\n\n\[ {C}_... | Yes |
Theorem 5.11. Let \( f : M \rightarrow M \) be a diffeomorphism. The following three conditions are equivalent:\n\n(1) \( f \) satisfies Axiom \( A \) and the no-cycle condition.\n\n(2) \( \mathrm{L}\left( f\right) \) is hyperbolic and satisfies the no-cycle condition.\n\n(3) \( \operatorname{CR}\left( f\right) \) is h... | Proof. \( \left( 1\right) \Rightarrow \left( 2\right) \) : Immediate from the definitions.\n\n\( \left( 2\right) \Rightarrow \left( 3\right) \) : By Theorem 4.31, \( \mathrm{L}\left( f\right) = \overline{\mathrm{P}\left( f\right) } \) . By Theorem 5.2, \( \mathrm{L}\left( f\right) \) decomposes into the basic sets \( {... | Yes |
Corollary 5.12. If \( f \) satisfies Axiom A plus the no-cycle condition, then\n\n\[ \overline{\mathrm{P}\left( f\right) } = \mathrm{L}\left( f\right) = \Omega \left( f\right) = \mathrm{{CR}}\left( f\right) \]\n\nand every basic set \( {B}_{i} \) of \( \overline{\mathrm{P}\left( f\right) } \) is a chain class. | Proof. Let \( f \) satisfy Axiom A plus the no-cycle condition. By Theorem 5.11, \( \operatorname{CR}\left( f\right) \) is hyperbolic. By Theorem 4.32, \( \operatorname{CR}\left( f\right) = \overline{\mathrm{P}\left( f\right) } \).\n\nLet the \( \left\{ {B}_{i}\right\} \) be the basic sets of \( \overline{\mathrm{P}\le... | Yes |
Proposition 5.13. Let \( f : X \rightarrow X \) be a homeomorphism. If \( {\left\{ {X}_{i}\right\} }_{i = 0}^{k} \) is a filtration of \( f \), then the maximal invariant set \( {\Gamma }_{i} \) of \( f \) in \( {X}_{i} - {X}_{i - 1} \) is isolated with isolating neighborhood \( {X}_{i} - {X}_{i - 1} \), and the \( \le... | Proof. This proposition follows directly from a simple fact that since \( {X}_{i} \) is a trapping region, if \( x \notin {X}_{i} \) but \( {fx} \in {X}_{i} \), then \( x \notin \operatorname{CR}\left( f\right) \) . See the first half of the proof of Lemma 1.15. We stop the proof here. | No |
Theorem 5.14. Let \( f : X \rightarrow X \) be a homeomorphism. If \( \operatorname{CR}\left( f\right) \) has only finitely many chain classes \( {C}_{1},\ldots ,{C}_{k} \), then rearranging the subscripts if necessary, there is a filtration \( {\left\{ {X}_{i}\right\} }_{i = 0}^{k} \) such that the maximal invariant s... | Proof. We use Theorem 1.13, the fundamental theorem of dynamical systems of Conley. See Shub (1987) for a nice treatment of filtration theory without using Conley theory.\n\nLet \( \phi : X \rightarrow \mathbb{R} \) be a Lyapunov function of \( f \) . Rearranging the subscripts if necessary, we may assume\n\n\[ \phi \l... | Yes |
If \( f \) satisfies Axiom \( A \) and the no-cycle condition, then rearranging the subscripts if necessary, there is a filtration \( \left\{ {M}_{i}\right\} \) such that the maximal invariant set of \( f \) in \( {M}_{i} - {M}_{i - 1} \) is exactly the basic set \( {B}_{i} \) . | The proof is immediate from Corollary 5.12 and Theorem 5.14. | No |
Theorem 6.1. Let \( A : E \rightarrow E \) be a linear isomorphism. The following conditions are equivalent:\n\n(1) \( A \) is hyperbolic.\n\n(2) \( A \) is quasi-hyperbolic.\n\n(3) A satisfies the linear transversality condition. | Proof. Let \( A \) be hyperbolic. By Theorem 2.2, \( {E}^{s} = {B}^{s} = {D}^{s} \), and \( {E}^{u} = \) \( {B}^{u} = {D}^{u} \) . Then \( \left( 1\right) \Rightarrow \left( 2\right) \) and \( \left( 1\right) \Rightarrow \left( 3\right) \) .\n\nWe prove \( \left( 2\right) \Rightarrow \left( 1\right) \) . Suppose \( A \... | Yes |
Theorem 6.2. Let \( \Lambda \subset M \) be a compact invariant set of \( f \), and let \( E \) be a Tf-invariant subbundle of \( {T}_{\Lambda }M \) . The following two conditions are equivalent:\n\n(1) There are \( 0 < \lambda < 1 \) and \( C \geq 1 \) such that \( \left| {T{f}^{n}\left( v\right) }\right| \leq C{\lamb... | Proof. Condition (1) is what we usually mean by a contraction. Clearly \( \left( 1\right) \Rightarrow \left( 2\right) \) . We prove \( \left( 2\right) \Rightarrow \left( 1\right) \) . Since \( \Lambda \) is compact, so is the unit sphere bundle \( E\left( 1\right) = \{ v \in E\left| \right| v \mid = 1\} \) . Then there... | Yes |
Lemma 6.3. \( \Lambda \) is quasi-hyperbolic for \( f \) if and only if there is a positive integer \( N \) such that for any \( 0 \neq v \in {T}_{\Lambda }M \), there is \( - N \leq m \leq N \) such that \( \left| {T{f}^{m}\left( v\right) }\right| > 2\left| v\right| \) . | Proof. Let \( \Lambda \) be quasi-hyperbolic for \( f \) . For any \( v \in {T}_{\Lambda }M \) with \( \left| v\right| = 1 \) , there is an integer \( m = m\left( v\right) \), positive or negative, such that \( \left| {T{f}^{m}\left( v\right) }\right| > 2 \) . By the compactness of the unit sphere bundle of \( {T}_{\La... | Yes |
Lemma 6.4. Let \( \Lambda \) be quasi-hyperbolic for \( f \) . For any \( 0 \neq v \in {T}_{\Lambda }M \), if \( T{f}^{i}\left( v\right) \) is an \( N \) -leftmax and \( T{f}^{j}\left( v\right) \) is an \( N \) -rightmax on the same orbit of \( v \neq 0 \), then \( i < j \) . | Proof. Otherwise, if \( j \leq i \), then there would be \( j \leq k \leq i \) such that \( \left| {T{f}^{k}\left( v\right) }\right| \) assumes the maximum on the whole interval \( \left\lbrack {j - N, i + N}\right\rbrack \), contradicting Lemma 6.3. | Yes |
Lemma 6.5. Let \( \Lambda \) be quasi-hyperbolic for \( f \), and let \( N,\lambda \), and \( C \) be as just determined. If \( u \in {T}_{\Lambda }M \) is an \( N \) -rightmax, then for any \( i \geq 0 \) and any \( n \geq 0,\left| {T{f}^{i + n}\left( u\right) }\right| \geq {C}^{-1}{\lambda }^{-n}\left| {T{f}^{i}\left... | Proof. Let \( u \in {T}_{\Lambda }M \) be an \( N \) -rightmax. By Lemmas 6.3 and 6.4, there is \( 0 \leq {m}_{1} \leq N \) such that \( T{f}^{{m}_{1}}\left( u\right) \) is an \( N \) -rightmax and \( \left| {T{f}^{{m}_{1}}\left( u\right) }\right| \geq 2\left| u\right| \) . Likewise, there is \( 0 \leq {m}_{2} \leq N \... | Yes |
Corollary 6.6. Let \( \Lambda \) be quasi-hyperbolic for \( f \), and let \( N,\lambda \), and \( C \) be determined as right before Lemma 6.5. Then:\n\n(1) For any \( v \in {H}^{s},\left| {T{f}^{n}\left( v\right) }\right| \leq C{\lambda }^{n}\left| v\right| \) for any \( n \geq 0 \).\n\n(2) For any \( v \in {H}^{u},\l... | Proof. For any \( v \in {H}^{u} \), there is an integer \( i \geq 0 \) such that \( T{f}^{-i}\left( v\right) \) is an \( N \) -rightmax. By Lemma 6.5, for any \( n \geq 0 \),\n\n\[ \left| {T{f}^{n}\left( v\right) }\right| \geq {C}^{-1}{\lambda }^{-n}\left| v\right| \]\n\nSince \( {H}^{u} \) is \( {Tf} \) -invariant, th... | No |
Theorem 6.7. Let \( \Lambda \) be quasi-hyperbolic for \( f \), and let \( \lambda \) and \( C \) be determined as right before Lemma 6.5. Then\n\n\[ \left| {T{f}^{n}\left( v\right) }\right| \leq C{\lambda }^{n}\left| v\right| ,\forall v \in {B}^{s}\left( x\right), x \in \Lambda, n \geq 0, \]\n\n\[ \left| {T{f}^{-n}\le... | Proof. We only need to prove that \( {B}^{s} - \{ 0\} \subset {H}^{s} \) . This is obvious because, for any \( 0 \neq v \in {B}^{s} \), there can be no \( N \) -rightmax by Lemma 6.5. The proof for \( {B}^{u} \) is similar.\n\nWe remark that \( {H}^{s} \subset {B}^{s} \) by option (1) of Corollary 6.6. Hence \( {H}^{s}... | No |
Theorem 6.8. A compact invariant set \( \Lambda \) of \( f \) is hyperbolic if and only if \( {B}^{s}\left( x\right) \oplus {B}^{u}\left( x\right) = {T}_{x}M \) for any \( x \in \Lambda \) . | Proof. We only prove the \ | No |
Theorem 6.9. Let \( \Lambda \) be quasi-hyperbolic for \( f \) . For any \( x \in \Lambda ,\omega \left( x\right) \) is hyperbolic of index \( \dim \left( {{B}^{s}\left( x\right) }\right) \) with contraction rates \( \left( {C,\lambda }\right) \) determined as right before Lemma 6.5. Likewise, \( \alpha \left( x\right)... | Proof. Let \( x \in \Lambda \) and \( y \in \omega \left( x\right) \) . Take a subspace \( F\left( x\right) \) of \( E\left( x\right) \) such that\n\n\[ F\left( x\right) \oplus {B}^{s}\left( x\right) = {T}_{x}M. \]\n\nSince \( {B}^{s}\left( x\right) - \{ 0\} = {H}^{s}\left( x\right) \) and since \( {H}^{s}\left( x\righ... | No |
Theorem 6.10. If \( \Lambda \) is quasi-hyperbolic for \( f \), then \( f \) restricted to \( L\left( {\left. f\right| }_{\Lambda }\right) \) is hyperbolic. | Proof. Because the constants \( \left( {C,\lambda }\right) \) determined as right before Lemma 6.5 are independent of \( \omega \left( x\right), x \in \Lambda \), the inequalities carry over to the closure. Then Theorem 6.10 follows from Theorem 6.9. | No |
Theorem 6.11. If \( \Lambda \) is quasi-hyperbolic for \( f \), then \( f \) restricted to \( \operatorname{CR}\left( {\left. f\right| }_{\Lambda }\right) \) is hyperbolic. | Proof. By Theorem 6.10, \( L\left( {\left. f\right| }_{\Lambda }\right) \) is contained in \( \Delta \) . If we prove there is no cycle between the \( \left\{ {\Delta }^{i}\right\} \) with respect to \( {\left. f\right| }_{\Lambda } \), then, by Theorem 5.9, \( \Delta \) will contain \( \operatorname{CR}\left( {\left. ... | Yes |
Theorem 6.12. Let \( \Lambda \subset M \) be a compact invariant set of \( f \) . Then \( \Lambda \) is hyperbolic for \( f \) if and only if \( {T}^{ * }f \) on \( {T}_{\Lambda }^{ * }M \) is hyperbolic. | Proof. Let \( {T}_{\Lambda }M = {E}^{s} \oplus {E}^{u} \) be the hyperbolic splitting of \( f \) with constants \( 0 < \lambda < 1 \) and \( C \geq 1 \) . Then the splitting\n\n\[ \n{\left( {E}^{s}\right) }^{0} \oplus {\left( {E}^{u}\right) }^{0} = {T}_{\Lambda }^{ * }M \n\]\n\nis \( {T}^{ * }f \) -invariant. Since the... | Yes |
Theorem 6.13. Let \( \Lambda \subset M \) be a compact invariant set of \( f \) . Then \( {Tf} \) on \( {T}_{\Lambda }M \) satisfies the linear transversality condition if and only if \( {T}^{ * }f \) on \( {T}_{\Lambda }^{ * }M \) is quasi-hyperbolic. | Proof. First we give the \ | No |
Theorem 6.14. Let \( \Lambda \subset M \) be a compact invariant set of \( f \) . The following conditions are equivalent:\n\n(1) \( \Lambda \) is hyperbolic for \( f \) .\n\n(2) \( {B}^{s}\left( x\right) \oplus {B}^{u}\left( x\right) = {T}_{x}M \) for all \( x \in \Lambda \) .\n\n(3) \( {D}^{s}\left( x\right) \oplus {... | Proof. That \( \left( 1\right) \Leftrightarrow \left( 2\right) \) is just Theorem 6.8. Obviously \( \left( 1\right) \Rightarrow \left( 3\right) \) . It remains to prove \( \left( 3\right) \Rightarrow \left( 1\right) \) .\n\nAssume condition (3). By Theorem 6.13, \( {T}^{ * }f \) on \( {T}_{\Lambda }^{ * }M \) is quasi-... | Yes |
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