We consider the regularity of measurable solutions χ to the cohomological equation φ = χ ◦ T − χ, where (T,X, μ) is a dynamical system and φ : X → R is a C valued cocycle in the setting in which T : X → X is a piecewise C Gibbs– Markov map, an affine β-transformation of the unit interval or more generally a piecewise C uniformly expanding map of an interval. We show that under mild assumptions,...

zw Let G be a finite abelian group and M an o-minimal expansion of Rexp = (R, +, ·, <, e) admitting the C∞ cell decomposition. Everything is considered in M. We prove that every definable C∞G manifold is affine. Moreover we prove that if X1, . . . , Xn (resp. Y1, . . . , Yn) are definable C ∞G submanifolds of a definable C∞G manifold X (resp. Y ) in general position, then every definable CG map...

Ciphers y = C(x, k) and  = (, ) are isomorphic if there exists invertible computable in both directions map y ↔ , x ↔ , k ↔ . Cipher is vulnerable if and only if isomorphic cipher is vulnerable. Instead of computing the key of a cipher it is sufficient to find suitable isomorphic cipher and compute its key. If φ is arbitrary substitution and T is round substitution, its conjugate  = φTφ...

In this article, we consider the factor complexity of a fixed point of a primitive substitution canonically defined by a β-numeration system. We provide a necessary and sufficient condition on the Rényi expansion of 1 for having an affine factor complexity map C(n), that is, such that C(n) = an+ b for any n ∈ N.

An Anosov diffeomorphism f on a torus T is affine if f lifts to an affine map on R. By a classical result of Franks and Manning, any Anosov diffeomorphism g on T is topologically conjugate to an affine Anosov diffeomorphism. More precisely, there is a homeomorphism φ : T → T such that f = φ◦g◦φ−1 is an affine Anosov diffeomorphism. We call φ the Franks-Manning conjugacy. The linear part of f is...

We classify surjective self-maps (of degree at least two) of affine surfaces according to the log Kodaira dimension. In this paper we are interested in the following question. Question. Classify all smooth affine surfaces X/C which admit a proper morphism f : X → X with degree f > 1. In [5] and [18], a classification of smooth projective surfaces with a self-map of degree > 1 has been given. Th...

Let C be a unital AH-algebra and A be a unital simple C∗-algebra with tracial rank zero. It has been shown that two unital monomorphisms φ, ψ : C → A are approximately unitarily equivalent if and only if [φ] = [ψ] in KL(C,A) and τ ◦ φ = τ ◦ ψ for all τ ∈ T (A), where T (A) is the tracial state space of A. In this paper we prove the following: Given κ ∈ KL(C,A) with κ(K0(C)+\{0}) ⊂ K0(A)+\{0} an...

This paper presents a new texture analysis method based on structural properties. The texture features extracted using this algorithm are invariant to affine transform (including rotation, translation, scaling, and skewing). Affine invariant structural properties are derived based on texel areas. An area-ratio map utilizing these properties is introduced to characterize texture images. Histogra...

Suppose that to every non-degenerate simplex ∆ ⊂ Rn a ‘center’ C(∆) is assigned so that the following assumptions hold: (i) The map ∆ → C(∆) commutes with similarities and is invariant under the permutations of the vertices of the simplex; (ii) The map ∆ → Vol(∆)C(∆) is polynomial in the coordinates of the vertices of the simplex. Then C(∆) is an affine combination of the center of mass CM(∆) a...