We study genericity of dynamical properties in the space of homeomorphisms of the Cantor set and in the space of subshifts of a suitably large shift space. These rather different settings are related by a Glasner–King type correspondence: genericity in one is equivalent to genericity in the other. By applying symbolic techniques in the shift-space model we derive new results about genericity of dynamical properties for transitive and totally transitive homeomorphisms of the Cantor set. We show that the isomorphism class of the universal odometer is generic in the space of transitive systems. On the other hand, the space of totally transitive systems displays much more varied dynamics. In particular, we show that in this space the isomorphism class of every Cantor system without periodic points is dense and the following properties are generic: minimality, zero entropy, disjointness from a fixed totally transitive system, weak mixing, strong mixing and minimal self joinings. The latter two stand in striking contrast to the situation in the measure-preserving category. We also prove a correspondence between genericity of dynamical properties in the measure-preserving category and genericity of systems supporting an invariant measure with the same property.

For certain analytic data, we show that the eigenvalue sequence of the associated transfer operator ℒ is insensitive to the holomorphic function space on which ℒ acts. Explicit bounds on this eigenvalue sequence are established.

We consider a subclass of tilings: the tilings obtained by cut-and-projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent α in terms of the ranks of certain groups which appear in the construction. We give bounds for α. These computations apply to some well-known tilings, such as the octagonal tilings, or tilings associated with billiard sequences. A link is made between the exponent of the complexity, and the fact that the cohomology of the associated tiling space is finitely generated over ℚ. We show that such a link cannot be established for more general tilings, and we present a counterexample in dimension one.

Symmetric S-unimodal functions with positive Liapunov exponent of the critical value have an invariant measure absolutely continuous with respect to Lebesgue measure.

We consider small random perturbations of matrix cocycles over Lipschitz homeomorphisms of compact metric spaces. Lyapunov exponents are shown to be stable provided that our perturbations satisfy certain regularity conditions. These results are applicable to dynamical systems, particularly to volume-preserving diffeomorphisms.

We study which algebraic integers λ ≥ l arise as the growth rate of a mapping class of a surface and give conditions that are necessary and perhaps sufficient. Flow equivalence and twisted Lefschetz zeta functions are used to generate families of λ's. Examples and open problems are included

Alves, Bonatti and Viana have recently shown that physical (or Sinai–Ruelle–Bowen) invariant measures exist for partially hyperbolic diffeomorphisms with mostly expanding center-unstable direction. In this paper we prove that such systems are statistically stable, that is, nearby diffeomorphisms have nearby physical measures.

We prove that on any surface there is a C^\infty diffeomorphism exhibiting a wandering domain D with the following ergodic property: for any orbit starting in D the corresponding Birkhoff mean of Dirac measures converges to the invariant measure supported on a hyperbolic horseshoe \Lambda which is equivalent to the unique non-trivial Hausdorff measure in \Lambda. The construction is obtained by perturbation of a diffeomorphism such that the unstable and stable foliations of this horseshoe \Lambda are relatively thick and in tangential position. We describe, in addition, the set of accumulation points of orbits starting in D.

Let N be the class of meromorphic functions f with the following properties: f has finitely many poles;f′ has finitely many multiple zeros; the superattracting fixed points of f are zeros of f′ and vice versa, with finitely many exceptions; f has finite order. It is proved that if f ∈ N, then f does not have wandering domains. Moreover, if f ∈ N and if ∞ is among the limit functions of fn in a cycle of periodic domains, then this cycle contains a singularity of f−1. (Here fn denotes the nth iterate of f) These results are applied to study Newton's method for entire functions g of the form where p and q are polynomials and where c is a constant. In this case, the Newton iteration function f(z) = z − g(z)/g′(z) is in N. It follows that fn(z) converges to zeros of g for all z in the Fatou set of f, if this is the case for all zeros z of g″. Some of the results can be extended to the relaxed Newton method.

We consider the Markov chain on a compact manifold M generated by a sequence of random diffeomorphisms, i.e. a sequence of independent Diff2(M)-valued random variables with common distribution. Random diffeomorphisms appear for instance when diffusion processes are considered as solutions of stochastic differential equations. We discuss the global dynamics of Markov chains with continuous transition densities and construct non-random stable foliations for random diffeomorphisms.

Periodic points of piecewise monotonic transformations are investigated using an oriented graph, whose closed paths represent the periodic orbits of the transformation. In the first part it is shown that the inverse of the ζ-function is a kind of characteristic polynomial of this graph, which is a power series if the graph is infinite. In the second part, the sets are determined, which can occur as{ with and where T is a monotonic mod one transformation.

We extend the theory of transience to general dynamical systems with no Markov structure assumed. This is linked to the theory of phase transitions. We also provide new examples to illustrate different kinds of transient behaviour.

Set-valued functions from an interval into the closed subsets of an interval arise in various areas of science and mathematical modeling. Research has shown that the dynamics of a single-valued function on a compact space are closely linked to the dynamics of the shift map on the inverse limit with the function as the sole bonding map. For example, it has been shown that with Devaney’s definition of chaos the bonding function is chaotic if and only if the shift map is chaotic. One reason for caring about this connection is that the shift map is a homeomorphism on the inverse limit, and therefore the topological structure of the inverse-limit space must reflect in its richness the dynamics of the shift map. In the set-valued case there may not be a natural definition for chaos since there is not a single well-defined orbit for each point. However, the shift map is a continuous single-valued function so it together with the inverse-limit space form a dynamical system which can be chaotic in any of the usual senses. For the set-valued case we demonstrate with theorems and examples rich topological structure in the inverse limit when the shift map is chaotic (on certain invariant sets). We then connect that chaos to a property of the set-valued function that is a natural generalization of an important chaos producing property of continuous functions.

A basic nilsequence is a sequence of the form ψ(n)=f(Tnx), where x is a point of a compact nilmanifold X, T is a translation on X, and f∈C(X); a nilsequence is a uniform limit of basic nilsequences. Let X=G/Γ be a compact nilmanifold, Y be a subnilmanifold of X, g(n) be a polynomial sequence in G, and f∈C(X); we show that the sequence ∫ g(n)Yf, n∈ℤ, is the sum of a basic nilsequence and a sequence that converges to zero in uniform density. This implies that, given an ergodic invertible measure-preserving system (W,ℬ,μ,T), with μ(W)<∞, polynomials p1,…,pk∈ℤ[n], and sets A1,…,Ak∈ℬ, the sequence μ(Tp1(n)A1∩⋯∩Tpk(n)Ak) is the sum of a nilsequence and a sequence that converges to zero in uniform density. We also obtain a version of this result for the case where pi are polynomials in several variables.

For a large class of non-uniformly expanding maps of ℝm, with indifferent fixed points and unbounded distortion and that are non-necessarily Markovian, we construct an absolutely continuous invariant measure. We extend previously used techniques for expanding maps on quasi-Hölder spaces to our case. We give general conditions and provide examples to which our results apply.

In this note we answer a question raised by Friedland and Milnor in [FM] concerning the topological entropy of polynomial diffeomorphisms of C2.

Let X be a compact Hausdorff space, f ∈ C0(X, X) and A ⊂ X a minimal set of f. We first introduce a new topological invariant, the D-function of a minimal set, by the investigation of the decomposition of the minimal set A under the action of fn, n ∈ N. Then important properties about the invariant and the existence of minimal set with a given D-function in some subshift of finite type are discussed. Finally Sharkovskii's theorem is generalized to minimal sets of continuous mappings from the interval into itself.

We prove that if F is an orientation-preserving homeomorphism of the plane that leaves invariant a continuum Λ which irreducibly separates the plane into exactly two domains, then the convex hull of the rotation set of F restricted to Λ is a closed interval and each reduced rational in this interval is the rotation number of a periodic orbit in Λ. We also show that the interior and exterior rotation numbers of F associated with Λ are contained in the convex hull of the rotation set of F restricted to Λ and that if this set is nondegenerate then Λ is an indecomposable continuum.

The notions of ergodicity, strong mixing and weak mixing are defined and studied for arbitrary sequences of measure-preserving transformations of a probability space. Several results, notably ones connected with mean ergodic theorems, are generalized from the case of the sequence of all powers of a single transformation to this case. The conditions for ergodicity, strong mixing and weak mixing of sequences of affine transformations of compact groups are investigated.

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 $O(\mathbb{R}, N)$.