## Linearization of generalized interval exchange maps.(English)Zbl 1277.37067

Let $$I$$ be an open bounded interval. A generalized interval exchange map (g.i.e.m.) $$T$$ on $$I$$ is defined by the following data. Let $${\mathcal A}$$ be an alphabet with $$d \geq 2$$ symbols. Consider “top” and “bottom” partitions of $$I$$ into $$d$$ open subintervals indexed by $${\mathcal A}$$. More precisely, the top partition is defined by its singularity points $$u_0^t < u_1^t < \cdots < u_d^t$$, where $$I=(u_0^t, u_d^t)$$, and a bijection $$\pi_t : {\mathcal A} \rightarrow \{ 1,\ldots,d \}$$; then $$I_{\alpha}^t = (u_{\pi_t(\alpha) -1}, u_{\pi_t(\alpha)})$$ for each $$\alpha \in {\mathcal A}$$ and $$I=\sqcup I^t_{\alpha}$$ modulo points $$u_j^t$$. The definition of the bottom partition $$I=\sqcup I^b_{\alpha}$$ is similar, just replace the letter $$t$$ by the letter $$b$$. It is also required that $$\pi_t^{-1}(\{1,\ldots,k \}) \not= \pi_b^{-1}(\{ 1,\ldots,k \})$$ for all $$k=1,\ldots,d-1$$. Consider also a $$(d \times d)$$-antisymmetric matrix $$\Omega$$, whose element $$\Omega_{\alpha\beta}$$ equals $$(+1)$$ if $$\pi_t(\alpha) < \pi_t(\beta)$$ and $$\pi_b(\alpha) > \pi_b(\beta)$$, equals $$(-1)$$ if $$\pi_t(\alpha) > \pi_t(\beta)$$ and $$\pi_b(\alpha) < \pi_b(\beta)$$, and equals $$0$$ otherwise.
Then the g.i.e.m. $$T$$ is defined on $$\sqcup I^t_{\alpha}$$ and its restriction on each $$I^t_{\alpha}$$ is an orientation-preserving homeomorphism onto the corresponding $$I^b_{\alpha}$$. A g.i.e.m. $$T_0$$ is “standard” if $$|I^t_{\alpha}|=|I^b_{\alpha}|$$ for each $$\alpha \in {\mathcal A}$$, and the restriction of $$T_0$$ on each $$I^t_{\alpha}$$ is a translation. We say that a g.i.e.m. $$T$$ is a simple deformation of a standard i.e.m. $$T_0$$ if: (i) $$T$$ and $$T_0$$ have the same discontinuities, (ii) $$T$$ and $$T_0$$ coincide in the neighborhood of each discontinuity and of the endpoints of $$I$$, (iii) $$T$$ is a $$C^r$$-diffeomorphism on each $$I^t_{\alpha}$$. The main result of the paper is a local conjugacy theorem, which, stated in particular generality (for simple deformations), can be summarized as follows.
{Theorem.} For almost all standard i.e.m. $$T_0$$ and for any integer $$r \geq 2$$ amongst the $$C^{r+3}$$-simple deformations of $$T_0$$, those that are $$C^r$$-conjugate to $$T_0$$ by a diffeomorphism $$C^r$$-close to the identity form a $$C^1$$-submanifold of codimension $$d^*=d + (r-\tfrac{1}{2})\cdot \text{rank\,} \Omega - 2r$$.

### MSC:

 37E05 Dynamical systems involving maps of the interval 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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### References:

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