Akhtam Dzhalilov

Circle homeomorphisms with two break points

Let f be a circle class P homeomorphism with two break points 0 and c. If the rotation number of f is of bounded type and f is C2(S1{0, c}) then the unique f-invariant probability measure is absolutely continuous with respect to the Lebesgue measure if and only if 0 and c are on the same orbit and the product of their f-jumps is 1. We indicate how this result extends to class P homeomorphisms of rotation number of bounded type and with a finite number of break points such that f admits at least two break points 0 and c not on the same orbit and that the jump of f at c is not the product of some f-jumps at breaks points not belonging to the orbits of 0 and c.

On conjugations of circle homeomorphisms with two break points

Let f i ∈ C 2+ α ( S 1 ∖{ a i , b i }), α >0, i =1,2, be circle homeomorphisms with two break points a i , b i , that is, discontinuities in the derivative Df i , with identical irrational rotation number ρ and μ 1 ([ a 1 , b 1 ])= μ 2 ([ a 2 , b 2 ]), where μ i are the invariant measures of f i , i =1,2. Suppose that the products of the jump ratios of Df 1 and Df 2 do not coincide, that is, Df 1 ( a 1 −0)/ Df 1 ( a 1 +0)⋅ Df 1 ( b 1 −0)/ Df 1 ( b 1 +0)≠ Df 2 ( a 2 −0)/ Df 2 ( a 2 +0)⋅ Df 2 ( b 2 −0)/ Df 2 ( b 2 +0) . Then the map ψ conjugating f 1 and f 2 is a singular function, that is, it is continuous on S 1 , but Dψ ( x )=0 almost everywhere with respect to Lebesgue measure.

On the conjugation of piecewise smooth circle homeomorphisms with a finite number of break points

Let fi, i = 1, 2 be orientation preserving circle homeomorphisms with a finite number of break points, at which the first derivatives Dfi have jumps, and with identical irrational rotation number ρ = ρf1 = ρf2 . The jump ratio of fi at the break point b is denoted by σfi(b), i.e. σfi (b) := Dfi(b−0) Dfi(b+0) . Denote by σfi , i = 1, 2, the total jump ratio given by the product over all break points b of the jump ratios σfi(b) of fi. We prove, that for circle homeomorphisms fi, i = 1, 2, which are C , ε > 0, on each interval of continuity of Dfi and whose total jump ratios σf1 and σf2 do not coincide, the congugacy between f1 and f2 is a singular function.Let fi, i = 1, 2, be orientation preserving circle homeomorphisms with a finite number of break points , at which the first derivatives Dfi have jumps, and whose irrational rotation numbers coincide. Denote by the jump ratio of fi at a break point and by its total jump ratio, given by the product of the jump ratios over all break points .We prove that, for two such circle homeomorphisms fi, i = 1, 2, which on each interval of continuity of Dfi are C2+e, e > 0, and whose total jump ratios and do not coincide, the map h conjugating f1 and f2 is a singular function.

On Critical Circle Homeomorphisms with Infinite Number of Break Points

We prove that a critical circle homeomorphism with infinite number of break points without periodic orbits is conjugated to the linear rotation by a quasisymmetric map if and only if its rotation number is of bounded type. And we also prove that any two adjacent atoms of dynamical partition of a unit circle are comparable.

On Herman’s Theorem for Piecewise Smooth Circle Maps with Two Breaks

In this paper we consider general orientation preserving circle homeomorphisms \(f\in C^{2+\varepsilon } (S ^{1}\setminus \{ a^{(0)}, c^{(0)} \} )\;, \varepsilon >0\), with an irrational rotation number \(\rho _{f}\) and two break points \(a^{(0)}, c^{(0)} \). Denote by \(\sigma _f(x_b):=\frac{Df_{-}(x_b)}{Df_{+}(x_b)},\, x_b=a^{(0)},c^{(0)}\), the jump ratios of f at the two break points and by \(\sigma _f:= \sigma _f(a^{(0)})\cdot \sigma _f(c^{(0)}) \) its total jump ratio. Let h be a piecewise-linear (PL) circle homeomorphism with two break points \(a_{0}\), \(c_{0}\), irrational rotation number \(\rho _{h}\) and total jump ratio \(\sigma _h=1\). Denote by \(\mathbf {B}_{n}(h)\) the partition determined by the break points of \(h^{q_{n}}\) and by \(\mu _{h}\) the unique h-invariant probability measure. It is shown that the derivative \(Dh^{q_{n}}\) is constant on every element of \(\mathbf {B}_{n}(h)\) and takes either two or three values. Furthermore we prove, that \( \log Dh^{q_{n}}\) can be expressed in terms of the \(\mu _{h}-\) measures of some intervals of the partition \(\mathbf {B}_{n}(h)\) multiplied by the logarithm of the jump ratio \( \sigma _{h}(a_{0})\) of h at the break point \(a_{0}\). M. Herman showed, that the invariant measure \(\mu _h\) is absolutely continuous iff the two break points belong to the same orbit. We complement Herman’s result for the above class of piecewise \( C^{2+\varepsilon } \)-circle maps f with irrational rotation number \(\rho _f\) and two break points \( a^{(0)}, c^{(0)}\) not lying on the same orbit with total jump ratio \(\sigma _f=1\) as follows: if \(\mu _f\) denotes the invariant measure of the P-homeomorphism f, then for almost all values of \(\mu _f([a^{(0)}, c^{(0)}])\) the measure \(\mu _f\) is singular with respect to Lebesgue measure.

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 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.