In the present paper we give an improvement of a previous result of the paper [M. M. Skriganov. Ergodic theory on $SL(n)$, diophantine approximations and anomalies in the lattice point problem. Inv. Math.132(1), (1998), 1–72, Theorem 2.2] on logarithmically small errors in the lattice point problem for polyhedra. This improvement is based on an analysis of hidden symmetries of the problem generated by the Weyl group for $SL(n,\mathbb{B})$. Let $UP$ denote a rotation of a given compact polyhedron $P\subset\mathbb{B}^n$ by an orthogonal matrix $U\in SO(n)$, $tUP$ a dilation of $UP$ by a parameter $t>0$ and $N(tUP)$ the number of integer points $\gamma\in\mathbb{Z}^n$ which fall into the polyhedron $tUP$. We show that for almost all rotations $U$ (in the sense of the Haar measure on $SO(n)$) the following asymptotic formula \[ N(t\UP)=t^n{\rm vol} P+ O((\log t)^{n-1+\varepsilon}),\quad t\to\infty, \] holds with arbitrarily small $\varepsilon>0$.

We describe a strategy which allows one to produce computer assisted proofs of almost everywhere strong convergence of Jacobi–Perron type algorithms in arbitrary dimension. Numerical work is carried out in dimension three to illustrate our method. To the best of our knowledge this is the first result on almost everywhere strong convergence in dimension greater than two.

We study one-dimensional dynamical systems with indifferent fixed points $p$ and $q$. The dynamical systems we study in this paper have ergodic, infinite, invariant measures. We consider the limit of the ratio of the sojourn time of the trajectory in a small neighborhood of $p$ to that in a small neighborhood of $q$. We show that the limit does not exist under some conditions, which has been announced in a previous paper. In fact we prove that the limit supreme of the ratio is $\infty$ and that the limit infimum is 0.

Let $G$ be a finitely generated nilpotent group of unitary operators on a Hilbert space ${\cal H}$. We prove that ${\cal H}$ is decomposable into a direct sum ${\cal H}=\bigoplus_{\alpha\in A}{\cal L}_{\alpha}$ of pairwise orthogonal closed subspaces so that elements of $G$ permute the subspaces ${\cal L}_{\alpha}$, and if $T({\cal L}_{\alpha})={\cal L}_{\alpha}$, then the action of $T$ on ${\cal L}_{\alpha}$ is either scalar or has continuous spectrum. We also provide examples showing that analogous results do not hold for solvable non-nilpotent groups.

In this paper we show that a $C^r$ diffeomorphism having an invariant Floquet torus which is non-normally hyperbolic can be approximated, in the $C^{r-3}$ topology, by another one which exhibits a homoclinic tangency.

A partially ordered set that is invariantly associated to a subshift is constructed. A property of subshifts, also an invariant of topological conjugacy, is described. If this property is present in a subshift then the constructed partially ordered set is a partially ordered semigroup (with zero). In the description of these invariants the notion of context is instrumental.

It is shown that under certain conditions a hyperbolic set will possess an invariant structure in addition to the invariant submanifolds. This invariant structure is in the form of a ‘jet of a foliation’ along the leaves.

We construct stabilized $C^*$-algebras from subshifts by using the dynamical property of the symbolic dynamical systems. We prove that the construction is dynamical and the $C^*$-algebras are isomorphic to the tensor product $C^*$-algebras between the algebra of all compact operators on a separable Hilbert space and the $C^*$-algebras constructed from creation operators on sub-Fock spaces associated with the subshifts. We also prove that the gauge actions on the stabilized $C^*$-algebras are invariant for topological conjugacy as two-sided subshifts under some conditions. Hence, if two subshifts are topologically conjugate as two-sided subshifts, the associated stabilized $C^*$-algebras are isomorphic so that their K-groups are isomorphic.

We study the link between variations of entropy on a compact hyperbolic manifold $M$ and infinitesimal flat conformal deformations of $M$. We remark that the entropy of the Liouville measure increases in the direction of these deformations. We then explain a new construction of infinitesimal flat conformal deformations by bending along totally geodesic hypersurfaces of $M$ which allows us to extend a theorem of L. Flaminio.

To each function we associate a period, a polytope, a group $\Gamma$, a subgroup $\Delta \subset\Gamma$ and a coset of $\Gamma/\Delta$, and show these objects to be universal for the class of Markov chains with that beta function. One consequence is that polynomial matrices employed in coding problems of Markov chains can be chosen canonically.

Suppose that $X^1,X^2,\dotsc$ is a stationary stochastic process of positive $k\times k$ matrices, and let ${}^nY^1=X^nX^{n-1}\dots X^1$ be the corresponding product matrices. For a special case, Bellman showed that the elements $[{}^nY^1]_{ij}$ converge in the sense that $n^{-1}\mathrm{E}\{\log[{}^nY^1]_{ij}\}\rightarrow a$ as $n\rightarrow\infty$. The constant $a$ is independent of $i$ and $j$. Bellman also conjectured that, asymptotically, the $n^{-1/2}\{\log[{}^nY^1]_{ij}-na\}$ terms are distributed according to a normal distribution with a common variance, independent of $ij$. Later Furstenberg and Kesten generalized and strengthened Bellman's result and established the validity of his conjecture.

This paper extends these results to the case of nonlinear mappings that are monotonic and homogeneous of degree one on $R^k_+$. Specifically, given a stationary process $H^1,H^2,\dots$ of such mappings, we define the composite mappings ${}^nF^1(\cdot)=H^n(H^{n-1}(\dots (H^1(\cdot))\dots)$. Under appropriate conditions, the components $[{}^nF^1(x^0)]_i$ have the property that, almost surely, $n^{-1}\log[{}^nF^1(x^0)]_i\rightarrow a$ independent of $x^0$ and $i$. Furthermore the components $n^{-1/2}\{\log[{}^nF^1(x^0)]_i-na\}$ are asymptotically distributed according to a normal distribution with a common variance.

We develop a new technique for calculating the first cohomology of certain classes of actions of higher-rank abelian groups (${\mathbb Z}^k$ and ${\mathbb R}^k$, $k\ge 2$) with values in a linear Lie group. In this paper we consider the discrete-time case. Our results apply to cocycles of different regularity, from Hölder to smooth and real-analytic. The main conclusion is that the corresponding cohomology trivializes, i.e. that any cocycle from a given class is cohomologous to a constant cocycle. The principal novel feature of our method is its geometric character; no global information about the action based on harmonic analysis is used. The method can be developed to apply to cocycles with values in certain infinite dimensional groups and to rigidity problems.

We study multifractal spectra of a simple class of interval transformations with a neutral fixed point. The analyticity of their spectral curves depends on whether the absolutely continuous invariant measure is infinite or not.

We show that if two aperiodic automorphisms of a separable nuclear unital purely infinite simple $C^{\ast}$-algebra are asymptotically unitarily equivalent, then they are outer conjugate with respect to an automorphism which is isotopic to the identity automorphism. Thus, by Kirchberg and Phillips, they have the same KK-class if and only if they are outer conjugate with respect to an automorphism which is in the KK-class of the identity automorphism.

We study a weak Gibbs property of equilibrium states for potentials of weak bounded variation and for maps admitting indifferent periodic points. We further establish statistical properties of the weak Gibbs measures and bounds of their pointwise dimension. We apply our results to higher-dimensional maps (which are not necessarily conformal) with indifferent periodic points and show that their absolutely continuous finite invariant measures are weak Gibbs measures.

A proof for a non-singular version of the inverse Vitali lemma is given. The result is used to describe non-singular orbit equivalence within the framework of Rudolph's restricted orbit equivalence and in the construction of an alternative proof of the Hurewicz ergodic theorem.

Let $T$ be a tree, ${\mathop{\rm End}\nolimits}(T)$ be the number of ends of $T$ and let $L(T)$ be the infimum of topological entropies of transitive maps of $T$. We give an elementary approach to the estimate that $L(T)\ge (1/{\mathop{\rm End}\nolimits} (T))\log 2$. We also divide the set of all trees (up to homeomorphisms) into pairwise disjoint subsets ${\cal P}(i)$, $i\in \{0\}\cup\N$ and prove that $L(T)=(1/({\mathop{\rm End}\nolimits} (T)-i))\log 2$ if $T\in {\cal P}(i)$ with $i=0,1$, and $L(T)\le (\text{respectively} =) (1/({\mathop{\rm End}\nolimits} (T)-i))\log 2$ if $T\in {\cal P}(i)$ (respectively $T\in {\cal P}'(i))$ with $i\ge 2$, where ${\cal P}'(i)$ is an infinite subset of ${\cal P}(i)$. Furthermore, we show that there is a tree $T$ such that the topological entropy of each transitive map of $T$ is larger than $L(T)$, and hence disprove a conjecture of Alseda et al (1997).

We analyse the convergence conditions of the Eckmann and Ruelle algorithm (ERA) used to estimate the Liapunov exponents, for the tangent map, of an ergodic measure, invariant under a smooth dynamical system. We find sufficient conditions for this convergence that are related to those ensuring the convergence to the tangent map of the best linear $L^p$-fittings of the action of a mapping $f$ on small balls. Under such conditions, we show how to use ERA to obtain estimates of the Liapunov exponents, up to an arbitrary degree of accuracy. We propose an adaptation of ERA for the computation of Liapunov exponents in smooth manifolds, which allows us to avoid the problem of detecting the spurious exponents.

We prove, for a Borel measurable dynamics $f$, the existence of Liapunov exponents for the function $S_r(x)$, mapping each point $x$ to the matrix of the best linear $L^p$-fitting of the action of $f$ on the closed ball of radius $r$ centred at $x$, and we show how to use ERA to get reliable estimates of the Liapunov exponents of $S_r$. We also propose a test for checking the differentiability of an empirically observed dynamics.

Let $(M,g)$ be a compact, differentiable Riemannian manifold without conjugate points and bounded asymptote. We show that, if the geodesic flow of $(M,g)$ is either topologically stable, or satisfies the $\epsilon$-shadowing property for some appropriate $\epsilon > 0$, then every abelian subgroup of the fundamental group of $M$ is infinite cyclic. The proof is based on the existence of homoclinic geodesics in perturbations of $(M,g)$, whenever there is a subgroup of the fundamental group of $M$ isomorphic to $\mathbb{Z}\times \mathbb{Z}$.

We show that every shift of finite type $X$ with positive entropy has proper subshifts of finite type with entropy strictly smaller than the entropy of $X$, but with entropy arbitrarily close to the entropy of $X$. Consequently, $X$ contains an infinite chain of subshifts of finite type which is strictly decreasing in entropy.