Which orbit types force only finitely many orbit types?

ABSTRACT

For real maps, the question of the title is answered by finding all types of orbits that are available in t-simple systems. Here a t-simple system means a real map which has only finitely many types of orbits.

Keywords:

• Forcing relations
• order-type
• trajectory type
• backward orbit

2010 Mathematics Subject Classifications:

• 54H20
• 26A18

Acknowledgments

The authors thank all the referees for their valuable remarks and suggestions. This work was done during the second author's visit to SRM University-AP.

Disclosure statement

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

Notation

$C^0(\mathbb{R},\mathbb{R})$:= space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$.

$\omega (f,x):=\{y\in X:\mathrm{\exists }(n_k\left)\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}{f}^{n_k}\right(x)\to y\}$.

$Per\left(f\right):=\text{set of all periodic points of }f$.

${\mathcal{F}}_2:=\{f\in C^0(I,I):f\text{ does not have cycles of length greater than 2}\}$

$<a,b>:=$ the closed interval with end points a and b.

Additional information

Funding

The second author acknowledges National Board for Higher Mathematics (NBHM-DAE) (Government of India) for financial support (Ref. No. 2/39(2)/2016/NBHM/R&D-II/11397).