The Smale horseshoes in a class of non-invertible maps in with two-direction instability

Abstract

We study the existence of Smale horseshoes of new type and the uniformly hyperbolic invariant sets for a class of non-invertible maps in three-dimensional Euclidean spaces with the dimension of instability equal to two. Parameter regions are given, for which the map has a horseshoe and a uniformly hyperbolic invariant set on which the map is topologically conjugate to the two-sided fullshift on four symbols.

Keywords:

• Fullshift
• hyperbolic set
• non-invertible
• Smale horseshoe

AMS Subject Classifications:

• 37D20
• 37D45

Acknowledgements

We would like to appreciate the reviewers for their detailed comments and suggestions on the improvement of the manuscript.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This research was partially supported by the China Postdoctoral Science Foundation [grant number 2016M602126]; the Shandong University Fund [grant number 1090500020314], [grant number 1090516300001], [grant number 1090516300004].