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We prove that every two circle diffeomorphisms with break that have the same size of break (i.e., the ratio of the left and right derivatives at the break point) and the same irrational revolution number from a certain class that contains non-Diophantine numbers are C-smoothly conjugate. [ABSTRACT FROM AUTHOR]

We prove that a C2+ α-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class D δ, 0≤ δ< α≤1, α− δ≠1, is C1+ α− δ-smoothly conjugate to a rigid rotation. This is the first sharp result on the smoothness of the conjugacy. We also derive the most precise version of Denjoy’s inequality for such diffeomorphisms. [ABSTRACT FROM AUTHOR]

We prove that under certain regularity conditions imposed on the renormalizations of two circle diffeomorphisms with singularities, their C 1-smooth equivalence follows from exponential convergence of those renormalizations. As an easy corollary, any two analytical critical circle maps with the same order of critical points and the same irrational rotation number are C 1-smoothly conjugate. [ABSTRACT FROM AUTHOR]

It is shown that fractional derivatives of the (integrated) invariant measure of the Feigenbaum map at the onset of chaos have power-law tails in their cumulative distributions, whose exponents can be related to the spectrum of singularities ( f(α). This is a new way of characterizing multifractality in dynamical systems, so far applied only to multifractal random functions [Frisch and Matsumoto, J. Stat. Phys. 108:1181, 2002]. The relation between the thermodynamic approach [Vul, Sinai and Khanin, Russian Math. Surveys 39:1, 1984] and that based on singularities of the invariant measures is also examined. The theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations. [ABSTRACT FROM AUTHOR]

: Circle homeomorphisms with singularities of the break type are considered in the case when rotation numbers have periodic continued fraction expansion. We establish hyperbolicity for renormalizations and then use it in order to prove the following rigidity result. Namely, we show that any two homeomorphisms with a single break point are smoothly conjugate to each other provided they have the same quadratic irrational rotation number and the same ``size'' of a break. [ABSTRACT FROM AUTHOR]

: We consider a spatially periodic inviscid random forced Burgers equation in arbitrary dimension and the random time-dependent Lagrangian system related to it. We construct a unique stationary distribution for ``viscosity'' solutions of the Burgers equation. We also show that with probability 1 there exists a unique minimizing trajectory for the random Lagrangian system which generates a non-trivial ergodic invariant measure for the non-random skew-product extension of the Lagrangian system. [ABSTRACT FROM AUTHOR]

In this paper we study a multidimensional continued fraction algorithm which is related to the Modified Jacobi–Perron algorithm considered by Podsypanin and Schweiger. We demonstrate that this algorithm has many important properties which are natural generalisations of properties of one-dimensional continued fractions. For this reason, we call the transformation associated to the algorithm the d-dimensional Gauss transformation. We construct a coordinate system for the natural extension which reveals its symmetries and allows one to give an explicit formula for the density of its invariant measure. We also discuss the ergodic properties of this invariant measure. [ABSTRACT FROM AUTHOR]

A new proof of the M. Herman theorem on the smooth conjugacy of a circle map is presented here. It is based on the thermodynamic representation of dynamical systems and the study of the ergodic properties for the corresponding radom variables. [ABSTRACT FROM AUTHOR]

We consider a random walk on the d-dimensional lattice ℤ where the transition probabilities p( x,y) are symmetric, p( x,y)= p( y,x), different from zero only if y−x belongs to a finite symmetric set including the origin and are random. We prove the convergence of the finite-dimensional probability distributions of normalized random paths to the finite-dimensional probability distributions of a Wiener process and find our an explicit expression for the diffusion matrix. [ABSTRACT FROM AUTHOR]