This paper contains a generalization to the $d$-torus $\mathbb{T}^d$ of earlier results (B. Host. Nombres normaux, entropie, translations. Israel J. Math. 91 (1995), 419–428). Given a probability measure $\mu$ on and an endomorphism $T$ of $\mathbb{T}^d$, we explore the relations between three properties: the uniform distribution of the sequence $(T^n\mathbf{t})$ for $\mu$-almost all $\mathbf{t}$; the behaviour of $\mu$ relative to the translations by some rational subgroups of $\mathbb{T}^d$; and the entropy of $\mu$ for another endomorphism $S$ of $\mathbb{T}^d$.

It is shown that a polynomial with a Cremer periodic orbit has a non-accessible critical point in its Julia set provided that the Cremer periodic orbit is approximated by small cycles. Also, this paper contains a new proof of the Douady–Shishikura inequality for the number of non-repelling cycles of a complex polynomial.

We consider the dynamics of piecewise smooth interval maps $f$ with a nowhere vanishing derivative. We show that if $f$ is not infinitely renormalizable, then all its periodic orbits of sufficiently high period are hyperbolic repelling. If, in addition all periodic orbits of $f$ are hyperbolic, then $f$ has at most finitely many periodic attractors and there is a hyperbolic expansion outside the basins of these periodic attractors. In particular, if $f$ is not infinitely renormalizable and all its periodic orbits are hyperbolic repelling, then some iterate of $f$ is expanding. In this case, $f$ admits an absolutely continuous invariant probability measure.

We study certain metrics on subshifts of finite type for which we define the discrete analogs of Lyapunov exponents. We prove Young's formula for $\mu$-Hausdorff dimension. We give sufficient conditions on the above metrics for which the Hausdorff dimension is given by thermodynamic formalism. We apply these results to the Hausdorff dimension of the limit sets of geometrically finite, purely loxodromic, Kleinian groups.

We develop a pluripotential theory for random iteration on ${\mathbf P}^k$. We show the existence of a positive closed $(1,1)$ current and a measure on ${\mathbf P}^k$ which are invariant, in a certain sense, and which attract all positive closed $(1,1)$ currents and all measures, respectively, under normalized pull-back and averaging by the maps. Thus the concept of an exceptional set disappears as soon as we allow a slight amount of randomness in our system. We also consider the problem of push-forward of measures, and describe certain limiting measures in this case also, supported near the attractors for the perturbed map.

Consider a finite Abelian group $(G,+)$, with $|G|=p^r$, $p$ a prime number, and $\varphi: G^\mathbb{N} \to G^\mathbb{N}$ the cellular automaton given by $(\varphi x)_n=\mu x_n+\nu x_{n+1}$ for any $n\in \mathbb{N}$, where $\mu$ and $\nu$ are integers coprime to $p$. We prove that if $\mathbb{P}$ is a translation invariant probability measure on $G^\mathbb{Z}$ determining a chain with complete connections and summable decay of correlations, then for any ${\underline w}= (w_i:i<0)$the Cesàro mean distribution $${\cal M}_{\mathbb{P}_{\underline w}} =\lim_{M\to\infty} \frac{1}{M} \sum^{M-1}_{m=0}\mathbb{P}_{\underline w}\circ\varphi^{-m},$$ where $\mathbb{P}_{\underline w}$ is the measure induced by $\mathbb{P}$ on $G^\mathbb{N}$ conditioned by $\underline w$, exists and satisfies ${\cal M}_{\mathbb{P}_{\underline w}}=\lambda^\mathbb{N}$, the uniform product measure on $G^\mathbb{N}$. The proof uses a regeneration representation of $\mathbb{P}$.

In this paper we will show that any number normal with respect to the regular continued fraction (RCF) is also normal with respect to the continued fraction from below (BCF) and the nearest integer continued fraction (NICF). Furthermore, we will show that in case of the NICF a number is normal with respect to the RCF-expansion if and only if it is NICF-normal. This can be generalized to a larger class of semi-regular continued fraction expansions.

On semiflows on Banach manifolds, Quandt (1988 and 1989) stated that some special semiflows were inverse limit stable. However, the concepts and proofs used to show inverse limit stability of semiflows are rough and ambiguous. We consider semiflows on finite-dimensional compact manifolds or on finite-dimensional compact non-singular branched manifolds. In this paper we show that more special semiflows are inverse limit stable. That is, Anosov semiflows are inverse limit stable.

In this paper we introduce and explore conformal parabolic iterated function systems. We define and study topological pressure, Perron–Frobenius-type operators, semiconformal and conformal measures and the Hausdorff dimension of the limit set. With every parabolic system we associate an infinite hyperbolic conformal iterated function system and we employ it to study geometric and dynamical features (properly defined invariant measures for example) of the limit set.

We compute dynamical entropy in Connes, Narnhofer and Thirring sense for a Bogoliubov action of a torsion-free abelian group $G$ on the CAR-algebra. A formula analogous to that found by Størmer and Voiculescu in the case $G={\Bbb Z}$ is obtained. The singular part of a unitary representation of $G$ is shown to give zero contribution to the entropy. A proof of these results requires new arguments since a torsion-free group may have no finite index proper subgroups. Our approach allows us to overcome these difficulties, it differs from that of Størmer–Voiculescu.

A unimodal map $f:[0,1] \to [0,1]$ is renormalizable if there is a sub-interval $I \subset [0,1]$ and an $n > 1$ such that $f^n|_I$ is unimodal. The renormalization of $f$ is $f^n|_I$ rescaled to the unit interval.

We extend the well-known classification of limits of renormalization of unimodal maps with bounded combinatorics to a classification of the limits of renormalization of unimodal maps with essentially bounded combinatorics. Together with results of Lyubich on the limits of renormalization with essentially unbounded combinatorics, this completes the combinatorial description of limits of renormalization. The techniques are based on the towers of McMullen and on the local analysis around perturbed parabolic points. We define a parabolic tower to be a sequence of unimodal maps related by renormalization or parabolic renormalization. We state and prove the combinatorial rigidity of bi-infinite parabolic towers with complex bounds and essentially bounded combinatorics, which implies the main theorem.

As an example we construct a natural unbounded analogue of the period-doubling fixed point of renormalization, called the essentially period-tripling fixed point.

We study degree $n$ polynomial perturbations of quadratic reversible Hamiltonian vector fields with one center and one saddle point. It was recently proved that if the first Poincaré–Pontryagin integral is not identically zero, then the exact upper bound for the number of limit cycles on the finite plane is $n-1$. In the present paper we prove that if the first Poincaré–Pontryagin function is identically zero, but the second is not, then the exact upper bound for the number of limit cycles on the finite plane is $2(n-1)$. In the case when the perturbation is quadratic ($n=2$) we obtain a complete result—there is a neighborhood of the initial Hamiltonian vector field in the space of all quadratic vector fields, in which any vector field has at most two limit cycles.

We study asymptotic physical measures in certain dynamical systems formed by a skew product. The skew product is formed by two ‘nice’ systems via a skewing function which is asymptotically Brownian. In this setting, no analogue of the Sinai–Ruelle–Bowen measure exists.

We characterize the Hamiltonian flows in the plane which are structurally stable (in a global sense) among Hamiltonian flows. This notion is closely related to, but distinct from, the topological stability of the generating function as a map from the plane to the line.

Let $M$ be a two-sided surface of genus $g>1$. In 1936 A. Weil singled out the problem of generalization of the Poincarè rotation numbers to the minimal flows $\phi$ on $M$. In this paper we suggest a solution to this problem by constructing the rotation numbers which we call Artin's. These numbers are invariants of the $K_0$-group of a crossed product $C^*$-algebra $C(X)\rtimes_{\phi}\Z$. It is shown that the dynamics of $\phi$ is completely subordinate to diophantine properties of the Artin numbers.

We show that the $n$-point correlation function for the fractional parts of a random linear form in $m$ variables has a limit distribution with power-like tail. The existence of the limit distribution follows from the mixing property of flows on ${\rm SL}(m+1,{\Bbb R})/{\rm SL}(m+1,{\Bbb Z})$. Moreover, we prove similar limit theorems (i) for the probability to find the fractional part of a random linear form close to zero and (ii) also for related trigonometric sums. For large $m$, all of the above limit distributions approach the classical distributions for independent uniformly distributed random variables.

We construct Bratteli–Vershik models for Toeplitz flows and characterize a class of properly ordered Bratteli diagrams corresponding to these flows. We use this result to extend by a novel approach—using basic theory of dimension groups—an interesting and non-trivial result about Toeplitz flows, first shown by Downarowicz. (Williams had previously obtained preliminary results in this direction.) The result states that to any Choquet simplex $K$, there exists a $0$–$1$ Toeplitz flow $(Y,\psi)$, so that the set of invariant probability measures of $(Y,\psi)$ is affinely homeomorphic to $K$. Not only do we give a conceptually new proof of this result, we also show that we may choose $(Y,\psi)$ to have zero entropy and to have full rational spectrum.

Furthermore, our Bratteli–Vershik model for a given Toeplitz flow explicitly exhibits the factor map onto the maximal equicontinuous (odometer) factor. We utilize this to give a simple proof of the existence of a uniquely ergodic 0–1 Toeplitz flow of zero entropy having a given odometer as its maximal equicontinuous factor and being strongly orbit equivalent to this factor. By the same token, we show the existence of 0–1 Toeplitz flows having the 2-odometer as their maximal equicontinuous factor, being strong orbit equivalent to the same, and assuming any entropy value in $[0,\ln 2)$.

Finally, we show by an explicit example, using Bratteli diagrams, that Toeplitz flows are not preserved under Kakutani equivalence (in fact, under inducing)—contrasting what is the case for substitution minimal systems. In fact, the example we exhibit is an induced system of a 0–1 Toeplitz flow which is conjugate to the Chacon substitution system, thus it is prime, i.e. it has no non-trivial factors.

The thrust of our paper is to demonstrate the relevance and usefulness of Bratteli–Vershik models and dimension group theory for the study of minimal symbolic systems. This is also exemplified in recent papers by Forrest and by Durand, Host and Skau, treating substitution minimal systems, and by papers by Boyle, Handelman and by Ormes.

Let $G$ and $H$ be locally compact groups. Assume that $G$ acts on a standard probability space $(S,\mu)$, $\mu$ being $G$-invariant. We prove that if there exists a Borel cocycle $\alpha:S\times G\longrightarrow H$ which is proper in an appropriate sense, then $G$ inherits some approximation properties of $H$, for instance amenability or the so-called Haagerup Approximation Property. On the other hand, if $G_{0}$ is a closed subgroup of $G$, if the pair $(G,G_{0})$ has the relative property (T) of Margulis [19] and if either $H$ has Haagerup Approximation Property, or if it is the unitary group of a finite von Neumann algebra with a similar property, then we give rigidity results analogous to that in [23] and [1].

We extend our previous work by proving that for translation invariant Gibbs states on ${\mathbb Z}^d$ with a translation invariant interaction potential $\Psi=(\Psi_A)$ satisfying $\sum_{A \ni 0}|A|^{-1}[\diam(A)]^d\|\Psi_A\|<\infty$ the following hold: (1) the Kolmogorov-property implies a trivial full tail and (2) the Bernoulli-property implies Følner independence. The existence of bilaterally deterministic Bernoulli Shifts tells us that neither (1) nor (2) is, in general, true for random fields without some further assumption (even when $d=1$).

In this paper we prove that permutations of even length with maximal entropy are not unique. There are in fact two distinct permutations, $\theta_n$ and $\widetilde{\theta_n}$, with maximal entropy, both of which are self-dual and $\widetilde{\theta_n}$ is the reverse of $\theta_n$.