We continue studying branched covering maps of the plane with expanding properties, which we call expanding polymodials. They are analogous to piecewise expanding interval maps and have properties similar to those of complex polynomials (in particular, the Julia set and a lot of other notions from complex dynamics can be defined for expanding polymodials). In this paper we include the case when the Julia set is disconnected, study its topological properties and its Hausdorff dimension.

We show that the renewal theory developed by Sarig and Gouëzel in the context of non-uniformly expanding dynamical systems applies also to the study of compact group extensions of such systems. As a consequence, we obtain results on subexponential decay of correlations for equivariant Hölder observations.

We prove measurable Livšic theorems for dynamical systems modelled by Markov towers. Our regularity results apply to solutions of cohomological equations posed on Hénon-like mappings and a wide variety of non-uniformly hyperbolic systems. We consider both Hölder cocycles and cocycles with singularities of prescribed order.

The basic goal of this paper is to describe the dynamics of an m + 1-parameter family of piecewise conformal automorphisms of the Riemann sphere. In the ‘symmetric’ figure shown later (Figure 1), we observe the regions of attraction of periodic attractors. In fact, we prove that the only $\omega$-limit sets are periodic cycles. This shows that the conformality of the functions is stronger than the discontinuity.

We show that the set of cocycles with integral separateness is open and dense in the space of all bounded $Gl(d,{\mathbb R})$-cocycles equipped with uniform topology. As a consequence, a generic bounded linear cocycle has simple Lyapunov spectrum and dominated Oseledets splitting, and a generic bounded $Sl(2,\mathbb R)$-cocycle is uniformly hyperbolic.

The geometry of Cantor systems associated with an asymptotically non-hyperbolic family $(f_\epsilon)_{0\le\epsilon\le \epsilon_0}$ was studied by Jiang (Geometry of Cantor systems. Trans. Amer. Math. Soc.351 (1999), 1975–1987). By applying the geometry studied there, we prove that the Hausdorff dimension of the maximal invariant set of $f_\epsilon$ behaves like $1- K\epsilon^{1/\gamma}$ asymptotically, as was conjectured by Jiang (Generalized Ulam–von Neumann transformations. PhD Thesis, CUNY Graduate Center, May 1999).

Measure equivalence (ME) is the measure theoretic counterpart of quasi-isometry. This field has grown considerably over the past few years, developing tools to distinguish between different ME classes of countable groups. On the other hand, constructions of ME groups are very rare. We present a new method, based on a notion of measurable free factor, and we apply it to exhibit a new family of groups that are measure equivalent to the free group. We also present a quite extensive survey on results about ME for countable groups.

In this paper we study topological cocycles for minimal homeomorphisms on a compact metric space. We introduce a notion of an essential range for topological cocycles with values in a locally compact group, and we show that this notion coincides with the well-known topological essential range if the group is abelian. We then define a regularity condition for cocycles and prove several results on the essential ranges and the orbit closures of the skew product of regular cocycles. Furthermore, we show that recurrent cocycles for a minimal rotation on a locally connected compact group are always regular, assuming that their ranges are in a nilpotent group, and then their essential ranges are almost connected.

Nous montrons que l'entropie topologique des applications méromorphes n'est pas un invariant biméromorphe. Cela fournit un contre-exemple à une conjecture de Friedland. Nous proposons une version raffinée de cette conjecture.

Nous démontrons l'égalité, pour les convexes divisibles, entre la régularité Hölder du bord du convexe et la régularité Hölder du bord du convexe dual.

We study the multifractal analysis of the pointwise dimension for equilibrium measures on countable Markov shifts. The main difficulty is that the space is not compact. In order to overcome this, we use an approximation argument based on the theory of convergence of Fenchel pairs developed by Wijsman. The results of Pesin and Weiss on multifractal analysis for compact spaces are used as well. We also prove a Bowen formula for countable Markov shifts. It turns out that, in this setting, this formula provides the Hausdorff dimension of the set of recurrent points.

In this paper we extend the results of an earlier paper, which deal with a description of the smooth untwisted cohomology for ${\mathbb Z}^k$-actions by hyperbolic automorphisms of a torus, to the partially hyperbolic case. Along the way we correct an error found in one of the steps in the proof for the hyperbolic case.

We investigate topological mixing for $\mathbb Z$ and $\mathbb R$ actions associated with primitive substitutions on two letters. The characterization is complete if the second eigenvalue $\theta_2$ of the substitution matrix satisfies $|\theta_2|\ne 1$. If $|\theta_2|<1$, then (as is well known) the substitution system is not topologically weak mixing, so it is not topologically mixing. We prove that if $|\theta_2|> 1$, then topological mixing is equivalent to topological weak mixing, which has an explicit arithmetic characterization. The case $|\theta_2|=1$ is more delicate, and we only obtain some partial results.

A $\lambda$-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. The present author has previously constructed a C*-algebra $\mathcal{ O}_{\mathfrak L}$ associated with a $\lambda$-graph system $\mathfrak L$ that is a generalization of the Cuntz–Krieger algebras. In this paper, we introduce an entropic quantity for the $\lambda$-graph system $\mathfrak L$, called the volume entropy for $\mathfrak L$ and written as $h_{\rm vol}(\mathfrak L)$. The volume entropy $h_{\rm vol}(\mathfrak L)$ is invariant under shift equivalence of $\lambda$-graph systems, and yields a new topological conjugacy invariant of subshifts. We prove that Voiculescu's non-commutative topological entropy for the canonical completely positive map of the C*-algebra $\mathcal{O}_{\mathfrak L}$ is the volume entropy $h_{\rm vol}(\mathfrak L)$.

Let M be the four-dimensional compact manifold $M=T^2\times S^2$ and let $k\ge2$. We construct a $C^\infty$ diffeomorphism $F:M\to M$ with precisely k intermingled minimal attractors $A_1,\dotsc, A_k$. Moreover the union of the basins is a set of full Lebesgue measure. This means that Lebesgue almost every point in M lies in the basin of attraction of Aj for some j, but every non-empty open set in M has a positive measure intersection with each basin. We also construct $F:M\to M$ with a countable infinity of intermingled minimal attractors.

This paper deals with families of conformal iterated function systems (CIFS). The space of all CIFS, with common seed space X and alphabet I, is successively endowed with the topology of pointwise convergence and a new, weaker topology called $\lambda$-topology. It is proved that the pressure and the Hausdorff dimension of the limit set are continuous with respect to the topology of pointwise convergence when I is finite, and are lower semi-continuous, though generally not continuous, when I is infinite. It is then shown that these two functions are, in any case, continuous in the $\lambda$-topology. The concepts of analytic, regularly analytic and plane-analytic families of CIFS are also introduced. It is established that if a family of CIFS is regularly analytic, then the Hausdorff dimension function is real-analytic; if a family is plane-analytic, then the Hausdorff dimension function is continuous and subharmonic, though not necessarily real-analytic. These results are then applied to finite parabolic CIFS. Counter-examples highlighting breakdowns of real-analyticity in the Hausdorff dimension among analytic, but not regularly analytic, families are further provided. Such families often exhibit a phenomenon known as phase transition. Sufficient conditions preventing the occurrence of such transitions are supplemented.

We describe a pair of invariants for actions of finite groups on shifts of finite type, the left-reduced and right-reduced shifts. The left-reduced shift was first constructed by Fiebig, who showed that its zeta function is an invariant, and in fact equal to the zeta function of the quotient dynamical system. We also give conditions for expansivity of the quotient, and applications to combinatorial group theory, knot theory and topological quantum field theory.