When a C*-algebra is a coefficient algebra for a given endomorphism

The paper presents a criterion for a C*-algebra to be a coefficient algebra associated with a given endomorphis

T-entropy and Variational principle for the spectral radius of weighted shift operators

In this paper we introduce a new functional invariant of discrete time dynamical systems -- the so-called t-entropy. The main result is that this t-entropy is the Legendre dual functional to the logarithm of the spectral radius of the weighted shift operator on $L^1(X,m)$ generated by the dynamical system. This result is called the Variational principle and is similar to the classical variational principle for the topological pressure.Comment: 12 pages, v.2: editorial correction

An ergodic support of a dynamical system and a natural representation of Choquet distributions for invariant measures

An ergodic support $X_0$ of a dynamical system $(X,T)$ with metrizable compact phase space $X$ is the set of all points $x\in X$ such that the corresponding sequence of empirical measures $\delta_{x,n} = (\delta_x +\delta_{Tx}+\dots +\delta_{T^{n-1}x})/n$ converges weakly to some ergodic measure. For every invariant probability measure $\mu$ on $X$ it is proven that $\mu(X_0) =1$ and Choquet distribution $\mu^*$ on the set of ergodic measures $\mathop{\mathrm{Erg}} X$ has the natural representation $\mu^*(A) =\mu(\{ x\in X_0 : \lim\delta_{x,n} \in A\})$, where $A\subset \mathop{\mathrm{Erg}} X$.Comment: 4 page

Cramer Asymptotics in the Averaging Method for Systems with Fast Hyperbolic Motions

Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, Vol. 244, 2004, pp. 65–86.A dynamical system $w'=S(w,z,ε)$, $z'=z+εv(w,z,ε)$ is considered. It is assumed that slow motions are determined by the vector field $v(w, z, ε)$ in the Euclidean space and fast motions occur in a neighborhood of a topologically mixing hyperbolic attractor. For the difference between the true and averaged slow motions, a limit theorem is proved and sharp asymptotics for the probabilities of large deviations (that do not exceed $ε^δ$) are calculated; the exponent $δ$ depends on the smoothness of the system and approaches zero as the smoothness increases

Foliated Functions and an Averaged Weighted Shift Operator for Perturbations of Hyperbolic Mappings

Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, Vol. 244, 2004, pp. 35–64.In order to study the perturbations of a family of mappings with a hyperbolic mixing attractor, an apparatus of foliated functions is developed. Foliated functions are analogues of distributions based on smooth measures on leaves (traces), which are embedded manifolds in a neighborhood of the attractor. The dimension of such manifolds must coincide with the dimension of the expanding foliation, and the values of a foliated function on a trace must vary smoothly under smooth transverse deformations of the trace (which include deformations of the measure itself)