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The discrete Markus-Yamabe problem for symmetric planar polynomial maps. (English) Zbl 1284.37015

Summary: We probe deeper into the discrete Markus-Yamabe question for polynomial planar maps and into the normal form for those maps which answer this question in the affirmative. Furthermore, in a symmetric context, we show that the only nonlinear equivariant polynomial maps providing an affirmative answer to the discrete Markus-Yamabe question are those possessing \(\mathbb Z_{2}\) as their group of symmetries. We use this to establish two new tools which give information about the spectrum of a planar polynomial map.

MSC:

37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37C75 Stability theory for smooth dynamical systems
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[1] Chamberland, M., Characterizing two-dimensional maps whose Jacobians have constant eigenvalues, Canadian Mathematical Bulletin, 46, 3, 323-331 (2003) · Zbl 1041.26004
[2] Cima, A.; Gasull, A.; Mañosas, F., The discrete Markus-Yamabe problem, Nonlinear Analysis, 35, 343-354 (1999) · Zbl 0919.34042
[3] Cima, A.; van den Essen, A.; Gasull, A.; Hubbers, E.; Mañosas, F., A polynomial couterexample to the Markus-Yamabe conjecture, Advances in Mathematics, 131, 453-457 (1997) · Zbl 0896.34042
[4] Golubitsky, M.; Stewart, I. N.; Schaeffer, D. G., Singularities and Groups in Bifurcation Theory, Vol. II (1988), Springer-Verlag
[5] van den Essen, A.; Hubbers, E., A new class of invertible polynomial maps, Journal of Algebra, 187, 214-226 (1997) · Zbl 0941.14002
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