On C1 structural stability of maps on the two-torus

Jorge Iglesias; Aldo Portela; Álvaro Rovella; Juliana Xavier

doi:10.3934/dcds.2025135

Article Contents

2026, Volume 48: 48-63. Doi: 10.3934/dcds.2025135

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On C¹ structural stability of maps on the two-torus

1.
Facultad de Ingeniería, Uruguay
2.
Facultad de Ciencias, Uruguay

^*Corresponding author: Juliana Xavier
^*Corresponding author: Juliana Xavier

Received: July 2025

Early access: September 8, 2025

Published online: September 8, 2025

All of the authours are supported by [SNI].

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Abstract

It has long been known since the work of Franks, Shub, Smale, and Sullivan that within the homotopy class of any linear isomorphism of the torus, there existed many non-equivalent $ C^1 $ structurally stable diffeomorphisms with prescribed basic pieces. However, it remained unknown whether there existed at least one $ C^1 $ stable map in certain classes of non-invertible automorphisms of the two torus. To shed light on the scope of this problem, we conducted a brief survey of known results, stated and proved some new necessary conditions for stability, and provided several examples of $ C^1 $ stable maps that were neither invertible nor expanding.

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Mathematics Subject Classification: Primary: 37C75, 37C20.

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References

[1]	R. Bamón, J. Kiwi, J. Rivera-Letelier and R. Urzúa, On the topology of solenoidal attractors of the cylinder, Ann. I. H. Poincaré, 23 (2006), 209-236. doi: 10.1016/j.anihpc.2005.03.002.
[2]	P. Berger, Structural stability of attractor-repellor endomorphisms with singularities, Ergodic Theory Dynam. Systems, 32 (2012), 1-33. doi: 10.1017/S0143385710000878.
[3]	J. Delgado, N. Romero, A. Rovella and F. Vilamajó, $\Omega$ -stability for maps with nonwandering critical points, Fund. Math., 193 (2007), 23-35. doi: 10.4064/fm193-1-3.
[4]	J. M. Franks, Homology and Dynamical Systems, CBMS Regional Conference Series in Mathematics., 49 (1982).
[5]	J. Iglesias, A. Portela and A. Rovella, $C^1$ stable maps: Examples without saddles, Fund. Math., 208 (2010), 23-33. doi: 10.4064/fm208-1-2.
[6]	J. Iglesias, A. Portela and A. Rovella, $C^1$ stability of endomorphisms on two dimensional manifolds, Fund. Math., 219 (2012), 37-58. doi: 10.4064/fm219-1-3.
[7]	J. Iglesias, A. Portela, A. Rovella and J. Xavier, Attracting sets on surfaces, Proc. Amer. Math. Soc., 143 (2015), 765-779. doi: 10.1090/S0002-9939-2014-12274-X.
[8]	J. Iglesias, A. Portela, A. Rovella and J. Xavier, Branched coverings of the sphere having a completely invariant continuum with infinitely many Wada Lakes, Topology and its Applications., 339 (2023), Paper No. 108686, 8 pp. doi: 10.1016/j.topol.2023.108686.
[9]	R. Mañé, Axiom A for endomorphisms, Lecture Notes in Math., 597 (1977), 379-388. doi: 10.1007/BFb0085367.
[10]	R. Mañé and C. Pugh, Stability of endomorphisms, in: Dynamical Systems–Warwick 1974, Lecture Notes in Math., 468 (1975), 175-184. doi: 10.1007/BFb0082622.
[11]	F. Przytycki, On $\Omega$-stability and structural stability of endomorphisms satisfying Axiom A, Studia Mathematica, 60 (1977), 61-77. doi: 10.4064/sm-60-1-61-77.
[12]	F. Przytycki, Chaos after bifurcation of a Morse Smale diffeomorphism after a one-cycle saddle-node and iterations of multivalued mappings of an interval and a circle, Bol. Soc. Bras. Mat., 18 (1987), 29-79. doi: 10.1007/BF02584831.
[13]	M. Shub, Endomorphisms of Compact Differentiable Manifolds, Thesis (Ph.D.)–University of California, Berkeley ProQuest LLC, Ann Arbor, MI, (1967).
[14]	M. Tsujii, Fat solenoidal attractors, Nonlinearity, 14 (2001), 1011-1027. doi: 10.1088/0951-7715/14/5/306.