# About new invariants of Kupka–Smale diffeomorphisms on the sphere without sources and sinks

We know a lot of cases, when topological entropy of continious map determine by structure of periodic orbits. The classical result in this field is the Sharkovskiy ordering. Sharkovskiy $\left [ 6 \right ]$ enter mapping, that turn manifold of natural numbers in to ordered set as follows:

$$3 \prec 5 \prec 7 \prec 9 \prec 11 \cdots \prec 2 \cdot 3 \prec 2 \cdot 5 \prec \cdots \prec 2^{2} \cdot 3 \prec 2^{2}\cdot 5 \prec \cdots \prec 2^{3} \prec 2^{2} \prec 2 \prec 1$$

It is valid for continuous interval maps. The existence of a periodic orbit of period three implies the existence of periodic points of any period and topological entropy is positive. We have different situation for orientation-preserving homeomorphism functioning on twodimensional disk $\mathbb{D}^{2}$: the rotation of the disk have topological entropy zero, but such mapping can have point of any period. In J. M. Gambaudo, S. Van Streen and C. Tresser work $\left [ 1 \right ]$ they determine hereditable rotation compatible of periodic orbit and proven, that any periodic orbit of orientation-preserving homeomorphism $f$ on $\mathbb{D}^{2}$ with topological entropy zero are hereditable rotation compatible. Moreover, if f have the smoothness class $C^{1+\varepsilon }$, than the condition on periodic orbit are necessary and sufficient to conclude that topological entropy of f equal zero.

It means, that any diffeomorphism with topological entropy zero on the disk have very simple structure of periodic orbits. Around fixed point located periodic orbit of some period, wich is turning around fixed point and then, around every point of periodic orbit turning another periodic orbit. According that, depending on your choice superstructure, appropriate periodic orbit superstructures flow regular have $\left ( fig.1 \right )$ engagement. Thus, irregular engagement of orbit will be a $C^{1}$ - barrier.

Cascade periodic orbits of orientation-preserving map on disk can be matched with signature, consisting of a sequence $l_{n},n > 0$ rational numbers, that every $l_{n}$ discribe, how orbits of period $q_{n+1}$ associated with orbits of period $q_{n}$. Moreover, with any sequence of signature and periods we can build orientation-preserving homeomorphism of twodimensional disk with specified signature of cascade periodic orbits. However, exist topological barrier for realisation this cascade with orientation-preserving diffeomorphism of twodimensional disk. Thus, in J. M. Gambaudo, D. Sullivan, C. Tresser $\left [ 3 \right ]$ work we can see, that sequence $\lambda _{n} = \frac{l_{n}}{q_{n}}$ is converge for $C^{1}$-map. Limit of this sequence call asymptotic number of rotation.

Classic cascade period dubling on disk have sequence $\lambda _{n} = \frac{\left ( -1 \right )^{n}}{2^{n}}$ and realize by infinitely smooth diffeomorphism $\left [ 3 \right ]$. In R. Bowen and J. Franks works ($\left [ 4 \right ]$и $\left [ 5 \right ]$)built diffeomorfisms, realize sequence $\lambda _{n} = \frac{1}{2^{n}}$ in smoothness class $C^{1}$ and $C^{2}$ respectively. Also in work $\left [ 4 \right ]$ there is a hypothesis about unachievable sequence $\lambda _{n} = \frac{1}{2^{n}}$ for map in smoothness class more than two.

In this work we introduce another invariant, that distinguish diffeomorphisms, wich built using different sequence of signature. We build diffeomorphisms $f_{+},f_{-}: \mathbb{S}^{2} \rightarrow \mathbb{S}^{2}$, which are double applied period doubling bifurcation to diffeomorphism source-sink with turning in one

direction at $f_{+}$ and turning in different directions at $f_{-}$. Rsultant diffeomorphisms have only one sink point $\omega_{+},\omega_{-}$, one sourse orbit $\wp_{\alpha_{+}},\wp_{\alpha_{-}}$ and two saddle orbit $\wp_{\alpha_{+}^{1}},\wp_{\alpha_{+}^{2}},\wp_{\alpha_{-}^{1}},\wp_{\alpha_{-}^{2}}$ respectively. Then we consider orbit spaces of the sink basin

$$\hat{V}_{+}=\left ( W_{\omega _{+}}^{s} \setminus \omega_{+} \right ) / f_{+},\hat{V}_{-}=\left ( W_{\omega _{-}}^{s} \setminus \omega_{-} \right ) / f_{-}$$

and natural projection

$$p_{+}:W_{\omega _{+}}^{s} \setminus \omega _{+} \rightarrow \hat{V_{+}},p_{-}:W_{\omega _{-}}^{s} \setminus \omega _{-} \rightarrow \hat{V_{-}}$$

Since saddle points are hyperbolic, them orbits have invariant neighbourhoods $U_{+}^{1},U_{+}^{2},U_{-}^{1},U_{-}^{2}$

Let

$$\hat{U}_{+}^{1} = p_{+} \left ( U_{+}^{1} \right ),\hat{U}_{+}^{2} = p_{+} \left ( U_{+}^{2} \right ),\hat{U}_{-}^{1} = p_{-} \left ( U_{-}^{1} \right ),\hat{U}_{-}^{2} = p_{-} \left ( U_{-}^{1} \right )$$

Threes

$$S_{+}=\left ( \hat{V}_{+}, \hat{U}_{+}^{1}, \hat{U}_{+}^{2} \right ), S_{-}=\left ( \hat{V}_{-}, \hat{U}_{-}^{1}, \hat{U}_{-}^{2} \right )$$

we call scheme of diffeomorphisms $f_{+},f_{-}$, respectively.

The main result of this paper is the theorem below.

Theorem 1. Scheme $S_{+},S_{-}$ are not equivalent, that is not exist homeomorphism $\varphi : \hat{V}_{+} \rightarrow \hat{V}_{-}$, such that $\varphi \left ( \hat{U}_{+}^{1} \right ) = \hat{U}_{-}^{1},\varphi \left ( \hat{U}_{+}^{2} \right ) = \hat{U}_{-}^{2}$.

Keywords: Kupka–Smale diffeomorphism, cascade of periodic orbits, unstable manifold, stable manifold, orientation-preserving diffeomorphism, topological entropy