Teplins’kyi O. Yu.

We prove the existence of two real-analytic diffeomorphisms of the circle with break of the same size and an irrational rotation number of semibounded type that are not $C^{1+γ}$-smoothly conjugate for any $γ > 0$. In this way, we show that the previous result concerning the $C^1$-smoothness of conjugacy for these mappings is the exact estimate of smoothness for this conjugacy.

For a discontinuous dynamical system with discrete time on a two-dimensional cylinder generated by a quasiperiodically driven mapping of the shift of intervals with overlapping, we prove the existence and uniqueness of a limiting semiinvariant absorbing belt whose width lies within the same limits as the width of overlapping. In the case of overlapping of constant width, this belt is invariant, and the dynamics inside the belt is equivalent to a skew shift on a two-dimensional torus.

We prove that any C^3+β -smooth orientation-preserving circle diffeomorphism with rotation number from the Diophantine class D_δ , 0 < β < δ < 1, is C ^2+β-δ -smoothly conjugate to the rigid rotation of the circle by appropriate angle.

We suggest a method for the construction of complete asymptotic expansions for eigenvalues and eigen-functions of spectral boundary-value problems for differential equations with rapidly varying coefficients in the case of multiple spectra of the averaged problem. The effect of splitting of multiple eigen-values is illustrated by an example of a special fourth-order problem.

For linear difference equations in the space of bounded number sequences, we prove an analog of the Erugin theorem on reducibility and present sufficient conditions for the reducibility of countable linear systems of difference equations with periodic coefficients.