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October  2018, 38(10): 5021-5037. doi: 10.3934/dcds.2018220

Moduli of 3-dimensional diffeomorphisms with saddle-foci

Shinobu Hashimoto 1, , Shin Kiriki 2, and  Teruhiko Soma 3,

1. 

Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan
2. 

Department of Mathematics, Tokai University, 4-1-1 Kitakaname, Hiratuka, Kanagawa, 259-1292, Japan
3. 

Department of Mathematical Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan

Received  October 2017 Revised  April 2018 Published  July 2018

Fund Project: This work was partially supported by JSPS KAKENHI Grant Numbers 17K05283 and 26400093.

We consider a space $\mathcal{U}$ of 3-dimensional diffeomorphisms $f$ with hyperbolic fixed points $p$ the stable and unstable manifolds of which have quadratic tangencies and satisfying some open conditions and such that $Df(p)$ has non-real expanding eigenvalues and a real contracting eigenvalue. The aim of this paper is to study moduli of diffeomorphisms in $\mathcal{U}$. We show that, for a generic element $f$ of $\mathcal{U}$, all the eigenvalues of $Df(p)$ are moduli and the restriction of a conjugacy homeomorphism to a local unstable manifold is a uniquely determined linear conformal map.

Keywords: Modulus, homoclinic tangency, saddle-focus.
Mathematics Subject Classification: Primary: 37C15; Secondary: 37C20, 37C05, 37C29.
Citation: Shinobu Hashimoto, Shin Kiriki, Teruhiko Soma. Moduli of 3-dimensional diffeomorphisms with saddle-foci. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5021-5037. doi: 10.3934/dcds.2018220
References:
[1]

C. Bonatti and E. Dufraine, Équivalence topologique de connexions de selles en dimension 3, Ergodic Theory Dynam. Sys., 23 (2003), 1347-1381.  doi: 10.1017/S0143385703000130.  Google Scholar

[2]

M. Carvalho and A. Rodrigues, Complete set of invariants for a Bykov attractor, Regular Chaotic Dynam., 23 (2018), 227-247.  doi: 10.1134/S1560354718030012.  Google Scholar

[3]

W. de Melo, Moduli of stability of two-dimensional diffeomorphisms, Topology, 19 (1980), 9-21.  doi: 10.1016/0040-9383(80)90028-2.  Google Scholar

[4]

W. de Melo and J. Palis, Moduli of stability for diffeomorphisms, Global theory of dynamical systems, (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), 318–339, Lecture Notes in Math., 819, Springer, Berlin, 1980. doi: 10.1016/0040-9383(80)90028-2.  Google Scholar

[5]

W. de Melo and S. J. van Strien, Diffeomorphisms on surfaces with a finite number of moduli, Ergodic Theory Dynam. Sys., 7 (1987), 415-462.  doi: 10.1017/S0143385700004120.  Google Scholar

[6]

E. Dufraine, Some topological invariants for three-dimensional flows, Chaos, 11 (2001), 443-448.  doi: 10.1063/1.1385918.  Google Scholar

[7]

E. Dufraine, Un critére d'existence d'invariant pour la conjugaison de difféomorphismes et de champs de vecteurs, C. R. Math. Acad. Sci. Paris, 334 (2002), 53-58.  doi: 10.1016/S1631-073X(02)02207-0.  Google Scholar

[8]

V. Z. Grines, O. V. Pochinka and S. J. van Strien, On 2-diffeomorphisms with one-dimensional basic sets and a finite number of moduli, Mosc. Math. J., 16 (2016), 727-749.   Google Scholar

[9]

S. Hashimoto, Moduli of surface diffeomorphisms with cubic tangencies, Tokyo J. Math., (to appear). Google Scholar

[10]

T. M. Mitryakova and O. V. Pochinka, On necessary and sufficient conditions for the topological conjugacy of surface diffeomorphisms with a finite number of orbits of heteroclinic tangency, Proc. Steklov Inst. Math., 270 (2010), 194-215.  doi: 10.1134/S0081543810030156.  Google Scholar

[11]

T. M Mitryakova and O. V. Pochinka, Necessary and sufficient conditions for the topological conjugacy of 3-diffeomorphisms with heteroclinic tangencies, Trans, Moscow Math. Soc., (2016), 69-86.   Google Scholar

[12]

S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 57 (1983), 5-71.   Google Scholar

[13]

Y. Nishizawa, Existence of horseshoe sets with nondegenerate one-sided homoclinic tangencies in $\mathbb{R}^3$, Hokkaido Math. J., 37 (2008), 133-145.  doi: 10.14492/hokmj/1253539582.  Google Scholar

[14]

J. Palis, A differentiable invariant of topological conjugacies and moduli of stability, Dynamical Systems, Vol. Ⅲ-Warsaw, Astérisque, Soc. Math. France, Paris, 51 (1978), 335-346.  Google Scholar

[15]

R. A. Posthumus, Homoclinic points and moduli, Ergodic Theory Dynam. Sys., 9 (1989), 389-398.  doi: 10.1017/S0143385700005022.  Google Scholar

[16]

R. A. Posthumus and F. Takens, Homoclinic tangencies: Moduli and topology of separatrices, Ergodic Theory Dynam. Sys., 13 (1993), 369-385.  doi: 10.1017/S0143385700007422.  Google Scholar

[17]

A. Rodrigues, Moduli for heteroclinic connections involving saddle-foci and periodic solutions, Discrete Cont. Dyn. Sys., 35 (2015), 3155-3182.  doi: 10.3934/dcds.2015.35.3155.  Google Scholar

[18]

F. Takens, Heteroclinic attractors: Time averages and moduli of topological conjugacy, Bol. Soc. Brasil. Mat., 25 (1994), 107-120.  doi: 10.1007/BF01232938.  Google Scholar

[19]

Y. Togawa, A modulus of $3$-dimensional vector fields, Ergodic Theory Dynam. Sys., 7 (1987), 295-301.  doi: 10.1017/S0143385700004028.  Google Scholar

show all references

References:
[1]

C. Bonatti and E. Dufraine, Équivalence topologique de connexions de selles en dimension 3, Ergodic Theory Dynam. Sys., 23 (2003), 1347-1381.  doi: 10.1017/S0143385703000130.  Google Scholar

[2]

M. Carvalho and A. Rodrigues, Complete set of invariants for a Bykov attractor, Regular Chaotic Dynam., 23 (2018), 227-247.  doi: 10.1134/S1560354718030012.  Google Scholar

[3]

W. de Melo, Moduli of stability of two-dimensional diffeomorphisms, Topology, 19 (1980), 9-21.  doi: 10.1016/0040-9383(80)90028-2.  Google Scholar

[4]

W. de Melo and J. Palis, Moduli of stability for diffeomorphisms, Global theory of dynamical systems, (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), 318–339, Lecture Notes in Math., 819, Springer, Berlin, 1980. doi: 10.1016/0040-9383(80)90028-2.  Google Scholar

[5]

W. de Melo and S. J. van Strien, Diffeomorphisms on surfaces with a finite number of moduli, Ergodic Theory Dynam. Sys., 7 (1987), 415-462.  doi: 10.1017/S0143385700004120.  Google Scholar

[6]

E. Dufraine, Some topological invariants for three-dimensional flows, Chaos, 11 (2001), 443-448.  doi: 10.1063/1.1385918.  Google Scholar

[7]

E. Dufraine, Un critére d'existence d'invariant pour la conjugaison de difféomorphismes et de champs de vecteurs, C. R. Math. Acad. Sci. Paris, 334 (2002), 53-58.  doi: 10.1016/S1631-073X(02)02207-0.  Google Scholar

[8]

V. Z. Grines, O. V. Pochinka and S. J. van Strien, On 2-diffeomorphisms with one-dimensional basic sets and a finite number of moduli, Mosc. Math. J., 16 (2016), 727-749.   Google Scholar

[9]

S. Hashimoto, Moduli of surface diffeomorphisms with cubic tangencies, Tokyo J. Math., (to appear). Google Scholar

[10]

T. M. Mitryakova and O. V. Pochinka, On necessary and sufficient conditions for the topological conjugacy of surface diffeomorphisms with a finite number of orbits of heteroclinic tangency, Proc. Steklov Inst. Math., 270 (2010), 194-215.  doi: 10.1134/S0081543810030156.  Google Scholar

[11]

T. M Mitryakova and O. V. Pochinka, Necessary and sufficient conditions for the topological conjugacy of 3-diffeomorphisms with heteroclinic tangencies, Trans, Moscow Math. Soc., (2016), 69-86.   Google Scholar

[12]

S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 57 (1983), 5-71.   Google Scholar

[13]

Y. Nishizawa, Existence of horseshoe sets with nondegenerate one-sided homoclinic tangencies in $\mathbb{R}^3$, Hokkaido Math. J., 37 (2008), 133-145.  doi: 10.14492/hokmj/1253539582.  Google Scholar

[14]

J. Palis, A differentiable invariant of topological conjugacies and moduli of stability, Dynamical Systems, Vol. Ⅲ-Warsaw, Astérisque, Soc. Math. France, Paris, 51 (1978), 335-346.  Google Scholar

[15]

R. A. Posthumus, Homoclinic points and moduli, Ergodic Theory Dynam. Sys., 9 (1989), 389-398.  doi: 10.1017/S0143385700005022.  Google Scholar

[16]

R. A. Posthumus and F. Takens, Homoclinic tangencies: Moduli and topology of separatrices, Ergodic Theory Dynam. Sys., 13 (1993), 369-385.  doi: 10.1017/S0143385700007422.  Google Scholar

[17]

A. Rodrigues, Moduli for heteroclinic connections involving saddle-foci and periodic solutions, Discrete Cont. Dyn. Sys., 35 (2015), 3155-3182.  doi: 10.3934/dcds.2015.35.3155.  Google Scholar

[18]

F. Takens, Heteroclinic attractors: Time averages and moduli of topological conjugacy, Bol. Soc. Brasil. Mat., 25 (1994), 107-120.  doi: 10.1007/BF01232938.  Google Scholar

[19]

Y. Togawa, A modulus of $3$-dimensional vector fields, Ergodic Theory Dynam. Sys., 7 (1987), 295-301.  doi: 10.1017/S0143385700004028.  Google Scholar

Figure 3.1.  The images of the parallel straight segments $\gamma_k^\natural$ in $D_a(p)$ by $h$
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Figure 1.1.  A saddle-focus $p$ and a homoclinic quadratic tangency $q$ in $D_a(p)$
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Figure 1.2.  The front curve $\widetilde \gamma_0$ divides $\widetilde H_0$ into the two sheets $\widetilde H_0^+$ and $\widetilde H_0^-$. The folding curve $\gamma_0$ of $H_0$ is the orthogonal image of $\widetilde\gamma_0$
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Figure 1.3.  The half disk $\widetilde H_{0;u}^-$ meets $W^s(p)$ transversely at two points near $q$, one of which is $\widehat z$
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Figure 1.4.  Trip from $\widetilde H_0^-$ to $\widetilde H_m$: $f^{u+v}(\widetilde H_0^-)\supset D$, $f^{m_0}(D)\supset D_0$, $f^m(D_0)\supset D_m$ and $f^{N+n_0}(D_m)\supset \widetilde H_m$, where $N$, $n_0$ are the positive integers with $f^N(q) = \widetilde q$ and $f^{n_0}(\widetilde q) = q_0$. The dotted line passing through $q$ represents a straight segment tangent to $\widetilde \rho$ at $q$
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Figure 2.1.  The image $h(\widetilde H_{(j)})$ is contained in $\widehat H_{(j)}'$, but $h(\widetilde H_{(j)}^\pm)$ is not necessarily contained in $\widehat H_{(j)}'^\pm$
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Figure 2.2.  The case of $\boldsymbol{x}, \boldsymbol{y}\in \widetilde H_{(j)}^+$, $\boldsymbol{x}'\in \widetilde H_{(j)}'^+$ and $\boldsymbol{y}'\in \widetilde H_{(j)}'^-$
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Figure 2.3.  The shaded region represents $\mathcal{N}_\varepsilon(\widetilde\gamma_{m_j, n_j}', \widetilde H_{(j)}')$
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Figure 2.4.  The situation which does not actually occur. $d_1: = {\rm dist}(\boldsymbol{x}', \boldsymbol{y}')<\nu(\varepsilon)$, $d_2: = {\rm dist}_{\widetilde H_{(j)}}(\boldsymbol{x}, \boldsymbol{y})<\delta(\varepsilon)$ and $d_3: = {\rm dist}_{\widetilde H_{(j)}'}(\boldsymbol{x}', \boldsymbol{y}')<\varepsilon$
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Figure 4.1.  Correspondence of straight segments via $h$
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Figure 4.2.  Correspondence via $h$ with respect to the new coordinate on $D_{a'}(p')$
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