Abstract
We give two general constructions of braid equivalences which exist between certain deformations of the 2-branched Horsehoe map. We then give numerical evidence suggesting that these constructions of braid equivalences are always realised in the Hénon family.
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Notes
- 1.
Hénon considered a different parametrisation of this family,
$$\begin{aligned} H_{a,b}(x,y)=(-y+1-ax^2,bx). \end{aligned}$$ - 2.
Note that we will use underlinings to distinguish representations, so \(\underline{3}\) denotes the third point in the orbit of \(\underline{1}=p-1\), but 3 denotes the third point from the left.
- 3.
In the literature such braids are called permutation braids—however, it will be useful to have an adjective to describe this property, as in [8].
- 4.
Formally, these are not braids as they do not have the same set of initial and terminal endpoints. To avoid confusion, let us call these objects almost-braids.
- 5.
Although Holmes only used the term cabling for iterated torus knots (not iterated horseshoe knots) we will use the term for both cases.
References
J.S. Birman, Braids, Links, and Mapping Class Groups, in Annals of Mathematics Studies (Princeton University Press, Princeton, N.J., 1974) (No. 82. MR 0375281 (51 #11477))
P. Boyland, Braid Types and a Topological Method of Proving Positive Entropy (Boston University, 1984)
P. Boyland, Topological methods in surface dynamics. Topol. Appl. 58(3), 223–298 (1994) (MR 1288300 (95h:57016))
A. de Carvalho, T. Hall, The forcing relation for horseshoe braid types. Exp. Math. 11(2), 271–288 (2002)
A. de Carvalho, T. Hall, P. Hazard, Braid Equivalence in the Hénon Family II (2015) (in preparation)
R. Devaney, Z. Nitecki, Shift automorphisms in the Hénon mapping. Commun. Math. Phys. 67(2), 137–146 (1979) (MR 539548 (80f:58035))
H. El Hamouly, C. Mira, Singularités dues au feuilletage du plan des bifurcations d’un difféomorphisme bi-dimensionnel, C. R. Acad. Sci. Paris Sér. I Math. 294(12), 387–390 (1982) (MR 659728 (83m:58056))
T. Hall, The creation of horseshoes. Nonlinearity 7, 861–924 (1994)
V.L. Hansen, in Braids and Coverings: Selected Topics, ed. by L. Gæde, H.R. Morton. London Mathematical Society Student Texts, vol. 18 (Cambridge University Press, Cambridge, 1989, With appendices. MR 1247697 (94g:57004))
P. Holmes, Knotted periodic orbits in suspensions of Smale’s horseshoe: period multiplying and cabled knots. Phys. D 21(1), 7–41 (1986) (MR 860006 (88b:58112))
I. Kan, H. Koçak, J.A. Yorke, Antimonotonicity: concurrent creation and annihilation of periodic orbits. Ann. Math. (2) 136(2), 219–252 (1992) (MR 1185119 (94c:58135))
M. Lyubich, The quadratic family as a qualitatively solvable model of chaos. Not. Am. Math. Soc. 47(9), 1042–1052 (2000) (MR 1777885 (2001g:37063))
J. Milnor, Remarks on iterated cubic maps. Exp. Math. 1(1), 5–24 (1992) (MR 1181083 (94c:58096))
J. Palis Jr., F. Taken, Hyperbolicity & Sensitive Chaotic Dynamics at Homoclinic Bifurcations Cambridge Studies in Advanced Mathematics, vol. 35 (Cambridge University Press, 1993)
A. Sannami, A topological classification of the periodic orbits of the Henon family. Technical Report (Hokkaido University, 1988)
Acknowledgements
A. de Carvalho was partially supported by FAPESP grant 2011/16265-8. T. Hall was partially supported by FAPESP grant 2011/17581-0. P. Hazard was supported by FAPESP grant 2008/10659-1 and Leverhulme Trust grant RPG-279. We would like to thank the institutions IME-USP, IMPA and the IMS at Stony Brook for their hospitality during the time in which parts of this work was done. Thanks also to A. Hammerlindl for his help with numerical computations. Finally, we would like to thank C. Tresser for useful discussions on braids and the Hénon family.
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de Carvalho, A., Hall, T., Hazard, P. (2019). Braid Equivalence in the Hénon Family I. In: Pacifico, M., Guarino, P. (eds) New Trends in One-Dimensional Dynamics. Springer Proceedings in Mathematics & Statistics, vol 285. Springer, Cham. https://doi.org/10.1007/978-3-030-16833-9_6
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