The Mahler measure M(α) of an algebraic integer α is the product of the absolute value of the conjugates of α which lie outside the unit circle. The quantity log M(α) occurs in ergodic theory as the entropy of an endomorphism of the torus. Adler and Marcus showed that if β = M(α) then β is a Perron number which is a unit if α is a unit. They asked whether the Perron number β whose minimal polynomial is tm −t −1 is the measure of any algebraic integer. We show here that the answer is negative for all m > 3.

We show that entire functions which are critically finite and which meet certain growth conditions admit ‘Cantor bouquets’ in their Julia sets. These are invariant subsets of the Julia set which are homeomorphic to the product of a Cantor set and the line [0, ∞). All of the curves in the bouquet tend to ∞ in the same direction, and the map behaves like the shift automorphism on the Cantor set. Hence the dynamics near ∞ for these types of maps may be analyzed completely. Among the entire maps to which our methods apply are exp (z), sin (z), and cos (z).

We show that for every ergodic ℤd-action T, there is a mixing ℤd-action S which is orbit equivalent to T via an orbit equivalence that is a weak a-equivalence for all a ≥ 1 and a strong b-equivalence for all b ∈ (0, 1). If T has positive entropy, then S can be taken to have completely positive entropy. If the dimension d is greater than one, the orbit equivalence may be taken to be bounded and a strong b-equivalence for all b > 0.

Parmi les systèmes dynamiques définis par des substitutions, les mieux connus sont ceux qui proviennent de substitutions à longueur constante. F. M. Dekking [1] a déterminé les valeurs propres de ces systèmes, et, plus récemment, M. Queffelec a étudié leur type spectral maximal. On se propose ici de déterminer les valeurs propres de ces systèmes dans le cas général des substitutions à longueur variable; dans cette direction, un résultat partiel a été obtenu par J. C. Martin [3] dans le cas des substitutions de 2 lettres, pour les valeurs propres correspondant à des fonctions propres continues.

We study the various notions of spectrum for an action α of a locally compact abelian group G on a type IC*-algebra A, and discuss how these are related to the structure of the crossed product A ⋊αG. In the case where A has continuous trace and the action of G on  is minimal, we completely describe the ideal structure of the crossed product. A key role is played by the restriction of α to a certain ‘symmetrizer subgroup’ S of the common stabilizer in G of the points of Â. We show by example that, contrary to a conjecture of Bratteli, it is possble for A⋊G to be primitive but not simple, provided that S is not discrete. In such cases, the Connes spectrum Γ(α) differs from the strong Connes spectrum of Kishimoto. The counterexamples come from subtle phenomena in topological dynamics.

For each d ≥ 2 there exists a polynomial p with real coefficients such that the associated Newton function z–[p(z)/p′(z)] has 2d–2 distinct attracting periodic orbits in the complex plane. According to a theorem of G. Julia, this is the maximal number of attracting orbits that any rational function of degree d can possess.

For every Perron number λ we construct an infinite collection of topological Markov shifts with entropy log λ whose spectra are disjoint except for the necessary conjugates of λ. This is used to show that Marcus' theorem about every Markov shift of entropy log n factoring onto the full n-shift does not extend to certain entropy values.

We present here an investigation of the degree to which mixing properties can be shown to lift to cocycle extensions of an ergodic map by ℤn and ℝn weakly mixing actions. A number of other results on such extensions are also included.

A natural geometrical representation of the geodesic flow on a surface M of constant negative curvature is given in which the base transformation is the shift on a (finite type) space of shortest words relative to a fixed generating set for π1(M) and the height function is the hyperbolic distance across a fundamental region for π1(M). This representation is obtained by comparing cutting sequences on M with generalised continued fraction expansions of endpoints on ℝ

Lyapunov exponents for two-dimensional matrix cocycles are shown to be stable under certain stochastic perturbations.

We study actions of arithmetic groups on compact manifolds which preserve a connection.