1 results
Fast growth of the number of periodic points arising from heterodimensional connections
- Part of
-
- Journal:
- Compositio Mathematica / Volume 157 / Issue 9 / September 2021
- Published online by Cambridge University Press:
- 02 August 2021, pp. 1899-1963
- Print publication:
- September 2021
-
- Article
- Export citation
-
We consider
$C^{r}$-diffeomorphisms (
$1 \leq r \leq +\infty$) of a compact smooth manifold having two pairs of hyperbolic periodic points of different indices which admit transverse heteroclinic points and are connected through a blender. We prove that, by giving an arbitrarily 
$C^{r}$-small perturbation near the periodic points, we can produce a periodic point for which the first return map in the center direction coincides with the identity map up to order 
$r$, provided the transverse heteroclinic points satisfy certain natural conditions involving higher derivatives of their transition maps in the center direction. As a consequence, we prove that 
$C^{r}$-generic diffeomorphisms in a small neighborhood of the diffeomorphism under consideration exhibit super-exponential growth of number of periodic points. We also give examples which show the necessity of the conditions we assume.
