It is a known fact that an iterated function system (IFS) of entire functions is not necessarily ergodic. In this paper we show that if an IFS of analytic functions is defined in a domain whose boundary contains more than two points (in the extended complex plane) then the system possesses an ergodic property.

We prove that the set of linear cocycles with simple Lyapunov spectrum is dense in the space of all linear cocycles equipped with the uniform topology.

We consider in this paper the space of all orientation preserving and fixed point free homeomorphisms of the plane, endowed with the compact-open topology. Our main theorem asserts that this space is locally contractible. Then we give a new proof of the path connectedness. Finally, it is shown that the inclusion of this space into the space of all orientation preserving planar homeomorphisms induces the trivial map on the fundamental groups.

We prove the existence of the average linking number, also called the Hopf invariant, for any invariant measure under a differentiable flow on $S^3$ without singularities, which has no periodic orbit of positive measure.

We show that the Patterson–Sullivan measure on the limit set of a geometrically finite Kleinian group with cusps can be recovered as a weak limit of sums of Dirac masses placed on an appropriate orbit of each parabolic fixed point. A corollary is a sharp asymptotic estimate for a natural counting function associated to a cuspidal subgroup. We also discuss the connection between the above counting and the Riemann hypothesis in some examples of arithmetical lattices.

We study the natural symbolic dynamics associated with piecewise continuous, non-invertible, dynamical systems. Our study is centered primarily on the relationship between the point-set topological properties of the partition of the system and the symbolic coding. We prove that for a class of maps locally preserving distances with regular partition, the associated symbolic dynamics cannot embed subshifts of finite type of positive entropy. Hence, in particular, almost sofic subshifts obtained from the symbolic dynamics have zero entropy. However, there are examples in Euclidean spaces of systems with non-regular partitions for which the coding maps can be surjective, particularly embedding all subshifts. For all such examples, the associated group of isometries is a subgroup of $O(\mathbb{R}, N)$.

Given a free action of a group $G$ on a directed graph $E$ we show that the crossed product of $C^* (E)$, the universal $C^*$-algebra of $E$, by the induced action is strongly Morita equivalent to $C^* (E/G)$. Since every connected graph $E$ may be expressed as the quotient of a tree $T$ by an action of a free group $G$ we may use our results to show that $C^* (E)$ is strongly Morita equivalent to the crossed product $C_0 ( \partial T ) \times G$, where $\partial T$ is a certain zero-dimensional space canonically associated to the tree.

For every irrational number $\alpha$ satisfying the property $\lim_{n\to\infty}|\!\sin \pi\alpha n|^{-1/n}=1$ and for every number $\beta>1$, it is shown that the difference equation $$ \xi_{n+1}+\xi_{n-1} +2\beta\cos(2\pi\alpha n+\th)\xi_n=0, \quad n\in\mathbb{Z} $$ has a non-trivial solution $\{\xi_n\}$ satisfying $\mathop{\overline{\lim}}_{|n|\to\infty}|\xi_n|^{1/|n|}\le|\beta|^{-1}$ if and only if $\theta=2\pi\alpha n+2\pi k\pm{\pi/2}$ for some $n,k\in\mathbb{Z}$.

The problem under consideration is: when is a Markov endomorphism (one-sided shift) $T=T_P$ with transition matrix $P$, isomorphic to a Bernoulli endomorphism $\tilde{T}_\rho$ with an appropriate stationary vector $\rho=\{\rho_i\}_{i\in I}$? An obvious necessary condition is that there exists an independent complement $\delta$ of the measurable partition $T^{-1}\varepsilon$ with $\distr \delta = \rho$. In this case the cofiltration (decreasing sequence of measurable partitions) $\xi(T)=\{T^{-n}\varepsilon\}^{\infty}_{n=1}$ generated by $T$ is finitely isomorphic to the standard Bernoulli cofiltration $\xi(\tilde{T}_{\rho})=\{\tilde{T}^{-n}_{\rho}\varepsilon\}^{\infty}_{n=1}$ and $T$ is called finitely $\rho$-Bernoulli.

We show that every ergodic Markov endomorphism $T_P$, which is finitely Bernoulli, can be represented as a skew product over $\tilde{T}_{\rho}$ with $d$-point fibres $(d\in \mathbb{N})$. We compute the minimal $d=d(T_P)$ in these skew-product representations by means of the transition matrix, and obtain necessary and sufficient conditions under which $d(T_P)=1$, i.e. $T_P$ is isomorphic to $\tilde{T}_{\rho}$. The cofiltration $\xi(T)$ of any finitely Bernoulli ergodic Markov endomorphism $T=T_P$ is represented as a $d$-point extension of the standard cofiltration $\xi(\tilde{T}_{\rho})$, and we show that the minimal $d=d_{\xi}(T)$ in these extensions is equal to $d(T)$. In particular, $d(T)=1\Longleftrightarrow d_{\xi}(T)=1$, that is, a Markov endomorphism $T$ is isomorphic to $\tilde{T}_{\rho}$ iff $\xi(T)$ is isomorphic to the Bernoulli cofiltration $\xi(\tilde{T}_{\rho})$.

We establish a generalized thermodynamic formalism for topological Markov shifts with a countable number of states. We offer a definition of topological pressure and show that it satisfies a variational principle for the metric entropies. The pressure of $\phi =0$ is the Gurevic entropy. This pressure may be finite even if the topological entropy is infinite. Let $L_\phi$ denote the Ruelle operator for $\phi$. We offer a definition of positive recurrence for $\phi$ and show that it is a necessary and sufficient condition for a Ruelle–Perron–Frobenius theorem to hold: there exist a $\sigma$-finite measure $\nu $, a continuous function $h>0$ and $\lambda >0$ such that $L_\phi ^{*}\nu =\lambda \nu$, $L_\phi h=\lambda h $ and $\lambda ^{-n}L_\phi ^nf\rightarrow h\int f\,d\nu$ for suitable functions $f$. We show that under certain conditions this convergence is uniform and exponential. We prove a decomposition theorem for positive recurrent functions and construct conformal measures and equilibrium measures. We give complete characterization of the situation when the equilibrium measure is a Gibbs measure. We end by giving examples where positive recurrence can be verified. These include functions of the form $$ \phi =\log f\left( \cfrac{1}{x_0+ \cfrac{1}{x_1+\dotsb }}\right), $$ where $f$ is a suitable function on a suitable shift $X$.

We study the behavior of multifractal spectra on the boundary of their domains of definition. In particular, we show that the dimension of the set of points having minimal (or maximal) pointwise dimension is not necessarily zero. However, this situation may be neglected in the sense of Baire category for Gibbs measures on expanding conformal repellers.

The wreath-product construction is used to give a complete combinatorial description of the dynamics of period-doubling quadratic maps leading to the Feigenbaum map. An explicit description of the action on periodic points uses the Thue–Morse sequence. In particular, a wreath-product construction of this sequence is given. The combinatorial renormalization operator on the period-doubling family of maps is invertible.

Sur tout espace riemannien compact, localement symétrique de type non compact et de rang au moins égal à deux, nous construisons explicitement une métrique de Finsler donnant à l'espace le même volume que la métrique localement symétrique mais dont l'entropie volumique est strictement inférieure à l'entropie volumique de cette dernière. De plus, cette métrique de Finsler est l'unique minimum de l'entropie volumique parmi les métriques de Finsler $G$-invariantes normalisées par le volume de la variété.

D'autre part, concernant le rang un, nous prouvons que les métriques hyperboliques réelles sont des points critiques de l'entropie topologique parmi les métriques de Finsler sur une variété compacte, et ce, en normalisant aussi bien par le volume de la variété en dimension quelconque que par le volume de Liouville des fibrés unitaires en dimension deux.

We present Sharkovskii's theorem for multidimensional perturbations of one-dimensional maps. We show that if an unperturbed one-dimensional map has a point of period $n$, then sufficiently close multidimensional perturbations of this map have periodic points of all periods which are allowed by Sharkovskii's theorem.

Lemma 2.2 on p. 704 was quoted from the Ph.D. thesis of B. Praggastis [2, 1.4] incorrectly. The local isomorphism property ($=$ repetitivity) of a $\phi$-subdividing tiling does imply that the subdivision matrix $M$ is primitive (that is, $M^k>0$ for some $k\ge1$), however, the converse is false, in general. I am grateful to A. E. Robinson Jr. who showed me a counter-example.