New classes of C¹-robustly transitive maps with persistent critical points

Abstract

Recently, the authors proved in [C. Lizana and W. Ranter, Topological obstructions for robustly transitive endomorphisms on surfaces, Adv. Math. 390 (2021), pp. 107901] that every $C^1$-robustly transitive toral endomorphism displaying critical points must be homotopic to a linear endomorphism having at least one eigenvalue with modulus greater than one. Here, we exhibit some examples of $C^1$-robustly transitive surface endomorphisms displaying critical points in certain homotopy classes.

Keywords:

• Robustly transitive endomorphisms
• critical points
• partially hyperbolic endomorphisms
• blender set

Acknowledgments

The authors are grateful to E. Pujals and R. Potrie for insightful comments to improve this work. The authors would like to thank to UFBA, UFAL and ICTP for the nice enviroment and support during the preparation of this work. The authors are grateful to the anonymous referees for helpful comments to improve our paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 ${\mathcal{W}}_{\mathrm{loc}}^u(p_G,G)$ is the local unstable manifold defined by the set of points $y\in {\mathbb{T}}^2$ such that there exists a full orbit $\tilde{y}=(y_i{)}_{i\in \mathbb{Z}}$ with $y_0=y$ and $d(y_{-k},x_{-k})<R$ for all $k\ge 0$, where $(x_i{)}_{i\in \mathbb{Z}}$ is a full orbit of $p_G$ and R>0 is locally constant for G. For further details see [13].

2 The graph and conditions of ${\phi }^{\prime }$ are quite different from the one in [8], since the one used in [8] does not satisfy the properties needed to preserve the partial hyperbolic structure.

Additional information

Funding

This W. Ranter was partially supported by CNPq/MCTI/FNDCT [project number 409198/2021-8], Brazil and C. Lizana was partially supported by CNPq/MCTI/FNDCT [project number 406750/2021-1], Brazil