Habibulla Akhadkulov

Coupled point results in partially ordered metric spaces without compatibility

AbstractThe existence of fixed points, coupled fixed points, and coupled coincidence points without the assumption of compatibility is established. The results presented in this paper extend, improve, and generalize some well-known results in the literature. Also, an example is given to show that our results are real generalizations of known ones in coupled coincidence fixed point theory.nMSC:54H25, 47H10.

Counting Closed Orbits for the Dyck Shift

The prime orbit theorem and Mertens’ theorem are proved for a shift dynamical system of infinite type called the Dyck shift. Different and more direct methods are used in the proof without any complicated theoretical discussion.

On conjugacies of circle maps with singular points

Let Tf and Tg be P-homeomorphisms of the circle with break point singularities that is, differentiable away from countable many points where the derivative has a jump. Let Tf and Tg have the same irrational rotation number. We describe a formula to find a conjugating map Th between Tf and Tg, i.e., Th○Tfu2009=u2009Tg○Th. Moreover, we provide a sufficient condition for the β - Holder continuity of the conjugating map.

On quasi-symmetry of conjugacies between critical circle maps with break points

We prove that a critical circle homeomorphism with infinite number break points without periodic orbits is conjugated to the linear rotation by a quasi-symmetric map if and only if its rotation number is of bounded type.

On smoothness of conjugations between break equivalent circle maps

Let Tf and Tg be a P-homeomorphisms of the circle with break point singularities that is, differentiable away from countably many points where the derivative has a jump. Let Tf and Tg have the same irrational rotation number. We provide a sufficient and necessary condition for the C1 -smoothness of conjugation map between break equivalent homeomorphisms Tf and Tg.

On invariant measure of the circle maps

Let f be piecewise smooth circle homeomorphisms with break points and the rotation number ρ is irrational. We provide a necessary condition for the absolute continuity of f-invariant measure with respect to Lebesgue measure.