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Erratum to “Smooth shifts along trajectories of flows” [Topology Appl. 130 (2) (2003) 183–204]
Topology and its Applications 157 (2010) 1721–1722
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Erratum
Erratum to “Smooth shifts along trajectories of flows” [Topology Appl. 130 (2) (2003) 183–204] Sergiy Maksymenko Topology Dept., Institute of Mathematics of NAS of Ukraine, Tereshchenkivs’ka st. 3, Kyiv, 01601 Ukraine
a r t i c l e
i n f o
Article history: Received 6 April 2010 Accepted 6 April 2010
In [2] I studied the group D (Φ) of C ∞ -diffeomorphisms of a manifold M preserving orbits of a C ∞ -flow Φ : M × R → M. Unfortunately that paper contains two mistakes which imply that one should put additional analytical and topological assumptions to the main result (Theorem 1) describing the homotopy type of the identity path component D0 (Φ) of D (Φ). Denote by E (Φ) the subset of C ∞ ( M , M ) consisting of maps f such that (i) f (ω) ⊂ ω for every orbit ω of F , and (ii) f is a local diffeomorphism at every fixed point z ∈ Fix(Φ). Evidently, D (Φ) ⊂ E (Φ). Let also E0 (Φ) ⊂ E (Φ) (resp. D0 (Φ) ⊂ D(Φ)) be the subset of E (Φ) consisting of all maps homotopic (rep. isotopic) to id M in E0 (Φ) (resp. D0 (Φ)). The idea of proof of Theorem 1 was to consider the following shift map
ϕ : C ∞ ( M , R) → C ∞ ( M , M ),
ϕ (α )(x) = Φ x, α (x)
and show that, under additional assumptions on Φ , the map E0 (Φ), and the induced map
ϕ is locally injective, the image ϕ (C ∞ ( M , R)) coincides with
ϕ : C ∞ ( M , R) → E0 (Φ)
(∗)
is a local homeomorphism with respect to every strong topology C rS (r 0). Unfortunately, it turned out that under assumptions of Theorem 1 the map (∗) in not always a local homeomorphism, though ϕ remains locally injective and the relation ϕ (C ∞ ( M , R)) = E0 (Φ) remains true. 1. The first mistake appeared in the “division” Lemma 32 which was wrongly claimed1 and should be formulated as follows: Lemma 32. Let F be either R or C, V ⊂ F be closed disk with smooth boundary such that 0 ∈ Int V , and Z : C ∞ ( V , F) → C ∞ ( V , F) r +1,r be a map defined by the formula: Z (α )(x) = x · α (x). If F = R then the inverse map Z −1 : Z (C ∞ ( V , F)) → C ∞ ( V , F) is C W , W ∞,∞
continuous for all r 0. If F = C, then Z −1 is only C W , W -continuous.
Proof. The case F = R easily follows from the Hadamard lemma, while for F = C this lemma is a particular case of results of [5]. 2
1
DOI of original article: 10.1016/S0166-8641(02)00363-2. E-mail address: [email protected]. In fact, the inequality at the end of the proof of Lemma 32 in [2] proves continuity of Z but not of Z −1 .
0166-8641/$ – see front matter doi:10.1016/j.topol.2010.04.005
© 2002 Elsevier B.V.
All rights reserved.
1722
S. Maksymenko / Topology and its Applications 157 (2010) 1721–1722
This led to incorrect estimations of continuity of the correspondence h → σ given by Eqs. (23)–(26). It now follows r +1,r from the corrected version of Lemma 32 that the correspondence h → σ is C W , W -continuous in Eqs. (23) and (26) and r +2,r
∞,∞
C W , W -continuous in Eq. (24) for each r 0, while it is only C W , W -continuous in Eq. (25). 2. In Definition 15 it should be additionally assumed that the image ϕ V (M) of a neighbourhood M is C rW -open in the r ,r image ϕ V (C ∞ ( V , J )). Equivalently, one may require that the map ϕ V : M → ϕ V (C ∞ ( V , J )) is C W , W -open. Such an assumption was explicitly used in the third paragraph of the proof of Theorem 17: . . . the image Ni = ϕU i (Mi ) is a C rW -neighbourhood of f |U i in C ∞ (U i , M ) for all r ∈ N0 . Unfortunately, this phrase also contains a misprint: instead of C ∞ (U i , M ) I supposed to be written im ϕU i = ϕU i (C ∞ ( V , J )). Openness of ϕ V turns out to be very essential and its violating gives rise to counterexamples to Theorems 17 and 1. For instance such counterexamples are irrational flows on n-dimensional tori and orthogonal flows on R2n (n 2), with orbits being dense on (2n − 1)-spheres centered at the origin. All other results of [2] concerning the kernel and the image of ϕ (§§3, 5) and shift being diffeomorphisms (§6) remain unchanged. For the proof of Theorem 1 it is also necessary to make the following corrections 3–5. 3. For the verification of openness of ϕ V it is necessary to assume that V is not an open subset of M but a compact submanifold with boundary such that dim V = dim M. Then the space C ∞ ( V , R) is well defined and has a series of topologies C rW with good behaviour. Moreover, in order to include the case ∂ M = ∅ it should also be allowed that ∂ V has corners. r ,s
4. Due to 1. Theorem 17 and Lemma 18 should be extended to the case of C W , W -continuous maps (r , s 0). This can be done almost literally the same arguments, however for non-compact M one should assume relative compactness of the set ∞,∞ of singular points for which local shift maps ϕ V are only C W , W . Such a restriction is predicted by the construction of a
in the proof of Lemma 18. Without that assumption U will be open in the so-called very-strong topology, neighbourhood U see [1]. 5. In the proof of Theorem 27 it is necessary to verify for which linear flows the local shift maps ϕ V are open. This leads to additional analytical restrictions on singularities and topological restrictions on the behaviour of orbits of the flow. Corrections to [2] are posed in arXiv [4], and will be published elsewhere. It is proved in [4] that Theorem 1 holds in the following form: Theorem 1. Suppose that a flow Φ satisfies the following conditions. (1) For every singular point z of F there are local coordinates (x1 , . . . , xm ) at z in which F is linear, i.e. F (x) = A z x for some non-zero (m × m)-matrix A z . Moreover, let Q ⊂ M be the set of singular points z of F in which the spectrum of A z is purely imagine and does not coincide with a unique point {0}. Then the closure of Q is compact. (2) Φ has no recurrent non-periodic points. Recall that z is recurrent if there is a sequence {t i } ∈ R such that limi →∞ |t i | = ∞ and limi →∞ Φ( z, t i ) = z. (3) Let z be a periodic point, ωz be its orbit, D be an open (m − 1)-disk transversely intersecting ωz at z, and R : ( D , z) → ( D , z) be a germ of the first return map for ωz . Then either • the tangent map T z R : T z D → T z D has eigen value λ with |λ| = 1, or • the germ R is periodic, i.e. R p = id( D ,z) for some p 1. Then ϕ (C ∞ ( M , R)) = E0 (Φ) and the map ϕ : C ∞ ( M , R) → E0 (Φ) is either a homeomorphism or a Z-covering map with respect to −1 (D (Φ)) is a convex subset of C ∞ ( M , R). strong C ∞ 0 S -topologies. Moreover, ϕ If M is compact, then the inclusion D0 (Φ) ⊂ E0 (Φ) is a homotopy equivalence and these spaces are either contractible or homotopy equivalent to the circle. If Φ has at least one non-closed orbit then they are contractible. Theorem 1 of [2] was essentially used in [3]. It can easily be seen that assumptions (1)–(3) introduced above hold for the situation considered in [3]. Therefore results of [3] remain valid. References [1] [2] [3] [4] [5]
S. Illman, The very-strong C ∞ topology on C ∞ ( M , N ) and K -equivariant maps, Osaka J. Math. 40 (2) (2003) 409–428. S. Maksymenko, Smooth shifts along trajectories of flows, Topology Appl. 130 (2) (2003) 183–204. S. Maksymenko, Homotopy types of stabilizers and orbits of Morse functions on surfaces, Ann. Global Anal. Geom. 29 (3) (2006) 241–285. S. Maksymenko, Local inverses of shift maps along orbits of flows, arXiv:0806.1502 [math.DS], submitted for publication. M.A. Mostow, S. Shnider, Joint continuity of division of smooth functions. I. Uniform Lojasiewicz estimates, Trans. Amer. Math. Soc. 292 (2) (1985) 573–583.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Erratum
Erratum to “Smooth shifts along trajectories of flows” [Topology Appl. 130 (2) (2003) 183–204] Sergiy Maksymenko Topology Dept., Institute of Mathematics of NAS of Ukraine, Tereshchenkivs’ka st. 3, Kyiv, 01601 Ukraine
a r t i c l e
i n f o
Article history: Received 6 April 2010 Accepted 6 April 2010
In [2] I studied the group D (Φ) of C ∞ -diffeomorphisms of a manifold M preserving orbits of a C ∞ -flow Φ : M × R → M. Unfortunately that paper contains two mistakes which imply that one should put additional analytical and topological assumptions to the main result (Theorem 1) describing the homotopy type of the identity path component D0 (Φ) of D (Φ). Denote by E (Φ) the subset of C ∞ ( M , M ) consisting of maps f such that (i) f (ω) ⊂ ω for every orbit ω of F , and (ii) f is a local diffeomorphism at every fixed point z ∈ Fix(Φ). Evidently, D (Φ) ⊂ E (Φ). Let also E0 (Φ) ⊂ E (Φ) (resp. D0 (Φ) ⊂ D(Φ)) be the subset of E (Φ) consisting of all maps homotopic (rep. isotopic) to id M in E0 (Φ) (resp. D0 (Φ)). The idea of proof of Theorem 1 was to consider the following shift map
ϕ : C ∞ ( M , R) → C ∞ ( M , M ),
ϕ (α )(x) = Φ x, α (x)
and show that, under additional assumptions on Φ , the map E0 (Φ), and the induced map
ϕ is locally injective, the image ϕ (C ∞ ( M , R)) coincides with
ϕ : C ∞ ( M , R) → E0 (Φ)
(∗)
is a local homeomorphism with respect to every strong topology C rS (r 0). Unfortunately, it turned out that under assumptions of Theorem 1 the map (∗) in not always a local homeomorphism, though ϕ remains locally injective and the relation ϕ (C ∞ ( M , R)) = E0 (Φ) remains true. 1. The first mistake appeared in the “division” Lemma 32 which was wrongly claimed1 and should be formulated as follows: Lemma 32. Let F be either R or C, V ⊂ F be closed disk with smooth boundary such that 0 ∈ Int V , and Z : C ∞ ( V , F) → C ∞ ( V , F) r +1,r be a map defined by the formula: Z (α )(x) = x · α (x). If F = R then the inverse map Z −1 : Z (C ∞ ( V , F)) → C ∞ ( V , F) is C W , W ∞,∞
continuous for all r 0. If F = C, then Z −1 is only C W , W -continuous.
Proof. The case F = R easily follows from the Hadamard lemma, while for F = C this lemma is a particular case of results of [5]. 2
1
DOI of original article: 10.1016/S0166-8641(02)00363-2. E-mail address: [email protected]. In fact, the inequality at the end of the proof of Lemma 32 in [2] proves continuity of Z but not of Z −1 .
0166-8641/$ – see front matter doi:10.1016/j.topol.2010.04.005
© 2002 Elsevier B.V.
All rights reserved.
1722
S. Maksymenko / Topology and its Applications 157 (2010) 1721–1722
This led to incorrect estimations of continuity of the correspondence h → σ given by Eqs. (23)–(26). It now follows r +1,r from the corrected version of Lemma 32 that the correspondence h → σ is C W , W -continuous in Eqs. (23) and (26) and r +2,r
∞,∞
C W , W -continuous in Eq. (24) for each r 0, while it is only C W , W -continuous in Eq. (25). 2. In Definition 15 it should be additionally assumed that the image ϕ V (M) of a neighbourhood M is C rW -open in the r ,r image ϕ V (C ∞ ( V , J )). Equivalently, one may require that the map ϕ V : M → ϕ V (C ∞ ( V , J )) is C W , W -open. Such an assumption was explicitly used in the third paragraph of the proof of Theorem 17: . . . the image Ni = ϕU i (Mi ) is a C rW -neighbourhood of f |U i in C ∞ (U i , M ) for all r ∈ N0 . Unfortunately, this phrase also contains a misprint: instead of C ∞ (U i , M ) I supposed to be written im ϕU i = ϕU i (C ∞ ( V , J )). Openness of ϕ V turns out to be very essential and its violating gives rise to counterexamples to Theorems 17 and 1. For instance such counterexamples are irrational flows on n-dimensional tori and orthogonal flows on R2n (n 2), with orbits being dense on (2n − 1)-spheres centered at the origin. All other results of [2] concerning the kernel and the image of ϕ (§§3, 5) and shift being diffeomorphisms (§6) remain unchanged. For the proof of Theorem 1 it is also necessary to make the following corrections 3–5. 3. For the verification of openness of ϕ V it is necessary to assume that V is not an open subset of M but a compact submanifold with boundary such that dim V = dim M. Then the space C ∞ ( V , R) is well defined and has a series of topologies C rW with good behaviour. Moreover, in order to include the case ∂ M = ∅ it should also be allowed that ∂ V has corners. r ,s
4. Due to 1. Theorem 17 and Lemma 18 should be extended to the case of C W , W -continuous maps (r , s 0). This can be done almost literally the same arguments, however for non-compact M one should assume relative compactness of the set ∞,∞ of singular points for which local shift maps ϕ V are only C W , W . Such a restriction is predicted by the construction of a
in the proof of Lemma 18. Without that assumption U will be open in the so-called very-strong topology, neighbourhood U see [1]. 5. In the proof of Theorem 27 it is necessary to verify for which linear flows the local shift maps ϕ V are open. This leads to additional analytical restrictions on singularities and topological restrictions on the behaviour of orbits of the flow. Corrections to [2] are posed in arXiv [4], and will be published elsewhere. It is proved in [4] that Theorem 1 holds in the following form: Theorem 1. Suppose that a flow Φ satisfies the following conditions. (1) For every singular point z of F there are local coordinates (x1 , . . . , xm ) at z in which F is linear, i.e. F (x) = A z x for some non-zero (m × m)-matrix A z . Moreover, let Q ⊂ M be the set of singular points z of F in which the spectrum of A z is purely imagine and does not coincide with a unique point {0}. Then the closure of Q is compact. (2) Φ has no recurrent non-periodic points. Recall that z is recurrent if there is a sequence {t i } ∈ R such that limi →∞ |t i | = ∞ and limi →∞ Φ( z, t i ) = z. (3) Let z be a periodic point, ωz be its orbit, D be an open (m − 1)-disk transversely intersecting ωz at z, and R : ( D , z) → ( D , z) be a germ of the first return map for ωz . Then either • the tangent map T z R : T z D → T z D has eigen value λ with |λ| = 1, or • the germ R is periodic, i.e. R p = id( D ,z) for some p 1. Then ϕ (C ∞ ( M , R)) = E0 (Φ) and the map ϕ : C ∞ ( M , R) → E0 (Φ) is either a homeomorphism or a Z-covering map with respect to −1 (D (Φ)) is a convex subset of C ∞ ( M , R). strong C ∞ 0 S -topologies. Moreover, ϕ If M is compact, then the inclusion D0 (Φ) ⊂ E0 (Φ) is a homotopy equivalence and these spaces are either contractible or homotopy equivalent to the circle. If Φ has at least one non-closed orbit then they are contractible. Theorem 1 of [2] was essentially used in [3]. It can easily be seen that assumptions (1)–(3) introduced above hold for the situation considered in [3]. Therefore results of [3] remain valid. References [1] [2] [3] [4] [5]
S. Illman, The very-strong C ∞ topology on C ∞ ( M , N ) and K -equivariant maps, Osaka J. Math. 40 (2) (2003) 409–428. S. Maksymenko, Smooth shifts along trajectories of flows, Topology Appl. 130 (2) (2003) 183–204. S. Maksymenko, Homotopy types of stabilizers and orbits of Morse functions on surfaces, Ann. Global Anal. Geom. 29 (3) (2006) 241–285. S. Maksymenko, Local inverses of shift maps along orbits of flows, arXiv:0806.1502 [math.DS], submitted for publication. M.A. Mostow, S. Shnider, Joint continuity of division of smooth functions. I. Uniform Lojasiewicz estimates, Trans. Amer. Math. Soc. 292 (2) (1985) 573–583.