##
**Topological invariants as numbers of translation.**
*(English)*
Zbl 0575.58026

Dynamical systems and bifurcations, Proc. Workshop, Groningen/Neth. 1984, Lect. Notes Math. 1125, 64-86 (1985).

[For the entire collection see Zbl 0552.00007.]

The authors interpret as numbers of translation two topological invariants [see S. Newhouse, the first author and F. Takens, Publ. Math., Inst. Hautes Étud. Sci. 57, 5-71 (1983; Zbl 0518.58031)]. Let a family \(F_{\mu}(x)\) (x,\(\mu\in R)\) of \(C^{\infty}\) diffeomorphisms be a saddle-node bifurcation, in \(\mu =0\), then the time t(x,y) to go from x to y, \(x,y<0\), by a flow \(X_ 0(t)\) is a topological invariant of the family \(F_{\mu}\) \((X_ 0(t)\) is a flow of \(X_ 0\); here, \(X_ 0\) is the unique field in which \(F_ 0\) imbeds).

Let F be a diffeomorphism of class \(C^ 2\) of some 2-dimensional manifold \(M^ 2\) preserving orientation, with two saddle fixed points p,q having common separatrix \(\gamma \subset W^ u(p)\cap W^ s(q)\). Let \(S_ p\subset W^ s(p)\) and \(U_ q\subset W^ u(q)\) be separatrices arranging on the same side of \(\gamma\). Let finally \(\lambda\),\(\mu\), \(0<\lambda <1<\mu\) be the stable and unstable eigenvalues in p and q, respectively. Then the ratio log \(\mu\) /log \(\lambda\) is a topological invariant of conjugacy.

To interpret these topological invariants, the authors introduce a relative asymptotic number of translation \(\rho\) (g;f), a pair of adapted foliations (\({\mathfrak J}_ L,{\mathfrak J}_ R)\) for \(F_{\mu}\) and a map \(h_{xy}: R^+\hookleftarrow (R^+=\{u| u\geq 0\}\), \(x,y\in R=\{(0,\xi)\subset R^+\times R| \xi >0\})\), also an adapted system \(S=\{{\mathfrak J}_ L,{\mathfrak J}_ R,\sigma \}\) for F.

We give two propositions: Proposition 1. Let \(F_{\mu}=X_{\mu}(1)\) be a family of flows. Let (\({\mathfrak J}_ L,{\mathfrak J}_ R)\) be any pair of adapted foliations for \(F_{\mu}\) and consider a point \(x_ 0\). Then the relative asymptotic number of translation \(\rho (h_{xy};T_{x_ 0})\) is independent of the choice of (\({\mathfrak J}_ L,{\mathfrak J}_ R)\), \(x_ 0\) and \(\rho (h_{xy};T_ x)=t(x,y)\) \((T_{x_ 0}=h_{xF_ 0(x)}\), t(x,y) is the time described above).

Proposition 2. Let F be a diffeomorphism of manifold \(M^ 2\) as described above. Then log \(\mu\) /log \(\lambda\) \(=\inf \{\rho (g_ S;f)|\) S adapted system for \(F\}\) ; here g is the conjugate of \(F| U_ q\) by the transport along the leaves of \({\mathfrak J}_ R\) from \(U_ q\) to \(\sigma\), and next along the leaves of \({\mathfrak J}_ L\) from \(\sigma\) to \(S_ p\).

The authors interpret as numbers of translation two topological invariants [see S. Newhouse, the first author and F. Takens, Publ. Math., Inst. Hautes Étud. Sci. 57, 5-71 (1983; Zbl 0518.58031)]. Let a family \(F_{\mu}(x)\) (x,\(\mu\in R)\) of \(C^{\infty}\) diffeomorphisms be a saddle-node bifurcation, in \(\mu =0\), then the time t(x,y) to go from x to y, \(x,y<0\), by a flow \(X_ 0(t)\) is a topological invariant of the family \(F_{\mu}\) \((X_ 0(t)\) is a flow of \(X_ 0\); here, \(X_ 0\) is the unique field in which \(F_ 0\) imbeds).

Let F be a diffeomorphism of class \(C^ 2\) of some 2-dimensional manifold \(M^ 2\) preserving orientation, with two saddle fixed points p,q having common separatrix \(\gamma \subset W^ u(p)\cap W^ s(q)\). Let \(S_ p\subset W^ s(p)\) and \(U_ q\subset W^ u(q)\) be separatrices arranging on the same side of \(\gamma\). Let finally \(\lambda\),\(\mu\), \(0<\lambda <1<\mu\) be the stable and unstable eigenvalues in p and q, respectively. Then the ratio log \(\mu\) /log \(\lambda\) is a topological invariant of conjugacy.

To interpret these topological invariants, the authors introduce a relative asymptotic number of translation \(\rho\) (g;f), a pair of adapted foliations (\({\mathfrak J}_ L,{\mathfrak J}_ R)\) for \(F_{\mu}\) and a map \(h_{xy}: R^+\hookleftarrow (R^+=\{u| u\geq 0\}\), \(x,y\in R=\{(0,\xi)\subset R^+\times R| \xi >0\})\), also an adapted system \(S=\{{\mathfrak J}_ L,{\mathfrak J}_ R,\sigma \}\) for F.

We give two propositions: Proposition 1. Let \(F_{\mu}=X_{\mu}(1)\) be a family of flows. Let (\({\mathfrak J}_ L,{\mathfrak J}_ R)\) be any pair of adapted foliations for \(F_{\mu}\) and consider a point \(x_ 0\). Then the relative asymptotic number of translation \(\rho (h_{xy};T_{x_ 0})\) is independent of the choice of (\({\mathfrak J}_ L,{\mathfrak J}_ R)\), \(x_ 0\) and \(\rho (h_{xy};T_ x)=t(x,y)\) \((T_{x_ 0}=h_{xF_ 0(x)}\), t(x,y) is the time described above).

Proposition 2. Let F be a diffeomorphism of manifold \(M^ 2\) as described above. Then log \(\mu\) /log \(\lambda\) \(=\inf \{\rho (g_ S;f)|\) S adapted system for \(F\}\) ; here g is the conjugate of \(F| U_ q\) by the transport along the leaves of \({\mathfrak J}_ R\) from \(U_ q\) to \(\sigma\), and next along the leaves of \({\mathfrak J}_ L\) from \(\sigma\) to \(S_ p\).

Reviewer: S.Baizaev

### MSC:

37G99 | Local and nonlocal bifurcation theory for dynamical systems |

37C10 | Dynamics induced by flows and semiflows |

57R30 | Foliations in differential topology; geometric theory |