A persistently singular map of T n that is C 1 robustly transitive. (Une application C 1 robustement transitive dans T n avec singularités persistantes.)

The author finds a C 1 robustly transitive endomorphism displaying critical points on the n-dimensional torus. Some examples for C 1 transitive maps can be found in [P. Berger and A. Rovella, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30, No. 3, 463–475 (2013; Zbl 1373.37096); J. Iglesias et al., Proc. Am. Math. Soc. 144, No. 3, 1235–1250 (2016; Zbl 1354.37036); J. Iglesias and A. Portela, Colloq. Math. 152, No. 2, 285–297 (2018; Zbl 1393.37055); J. C. Morelli, J. Korean Math. Soc. 58, No. 4, 977–1000 (2021; Zbl 1486.37015)]. The main result of the paper is the following: Theorem. Given n ≥ 2, there exists a persistently singular endomorphism supported on T n that is C 1 robustly transitive. The sketch of the construction is the as follows: “Start from an endomorphism induced by a diagonal expanding matrix with integer coefficients, with all but one direction strongly unstable and one central direction. Perturb the map to add a blending region that mixes everything getting the transitivity and then introduce artificially the critical points preserving the transitivity property.”