Every subgroup of the symmetric group defines a natural factor of the Cartesian power of a transformation. We calculate the set of values of the spectral multiplicity function of such factors (under certain conditions on the transformation) in terms of the number of orbits of diagonal actions of these subgroups. An analogous statement also holds on the unitary level for operators that preserve 1. In particular, we prove that for every positive integer n, there exists a transformation which is mixing of all orders and has a staircase multiplicity function of length n; that is, the essential values of the spectral multiplicity function are {1,2,…,n}.

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 present a construction of hedgehogs for holomorphic maps with an indifferent fixed point. We construct, for a family of commuting nonlinearizable maps, a common hedgehog of Hausdorff dimension one, the minimum possible.

In this paper the authors find examples of translation surfaces that have infinitely generated Veech groups, satisfy the topological dichotomy property that for every direction the flow in that direction is either completely periodic or minimal, and yet have minimal but non-uniquely ergodic directions.

A polynomial skew product of ℂ2 is a map of the form f(z,w)=(p(z),q(z,w)), where p and q are polynomials, such that f extends holomorphically to an endomorphism of ℙ2 of degree at least two. For polynomial maps of ℂ, hyperbolicity is equivalent to the condition that the closure of the postcritical set is disjoint from the Julia set; further, critical points either iterate to an attracting cycle or infinity. For polynomial skew products, Jonsson [Dynamics of polynomial skew products on C2. Math. Ann.314(3) (1999), 403–447] established that f is Axiom A if and only if the closure of the postcritical set is disjoint from the right analog of the Julia set. Here we present an analogous conclusion: critical orbits either escape to infinity or accumulate on an attracting set. In addition, we construct new examples of Axiom A maps demonstrating various postcritical behaviors.

We prove that every one-parameter family of area-preserving maps unfolding a homoclinic tangency has a sequence of parameter intervals, approaching the bifurcation parameter, where the dynamics exhibits wild hyperbolic sets accumulated by elliptic isles. This is a parametric conservative analogue of a famous theorem of Newhouse on the abundance of wild hyperbolic sets.

It is well known that in a small neighbourhood of a parabolic fixed point a real-analytic diffeomorphism of (ℝ2,0) embeds in a smooth autonomous flow. In this paper we show that the complex-analytic situation is completely different and a generic diffeomorphism cannot be embedded in an analytic flow in a neighbourhood of its parabolic fixed point. We study two analytic invariants with respect to local analytic changes of coordinates. One of the invariants was introduced earlier by one of the authors. These invariants vanish for time-one maps of analytic flows. We show that one of the invariants does not vanish on an open dense subset. A complete analytic classification of the maps with a parabolic fixed point in ℂ2 is not available at the present time.

For a generic C1 expanding map of the circle, the Lyapunov maximizing measure is unique and fully supported, and has zero entropy.

Let Γ be a countable group and let Γ0 be an infinite abelian subgroup of Γ. We prove that if the pair (Γ,Γ0) satisfies some combinatorial condition called (SS), then the abelian subalgebra A=L(Γ0) is a singular MASA in M=L(Γ) which satisfies a weakly mixing condition. If, moreover, it satisfies a stronger condition called (ST), then it provides a singular MASA with a strictly stronger mixing property. We describe families of examples of both types coming from free products, Higman–Neumann–Neumann extensions and semidirect products, and in particular we exhibit examples of singular MASAs that satisfy the weak mixing condition but not the strong mixing one.

We consider analytic Hamiltonian systems with two degrees of freedom, and prove that every Hamiltonian on the strong local stable manifold of the renormalization group fixed point obtained in Koch [A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete Contin. Dyn. Syst. A 11 (2004), 881–909] has a non-differentiable golden invariant torus (conjugacy to a linear flow).

We study the relationship between free curves and periodic points for torus homeomorphisms in the homotopy class of the identity. By free curve we mean a homotopically non-trivial simple closed curve that is disjoint from its image. We prove that every rational point in the rotation set is realized by a periodic point provided that there is no free curve and the rotation set has empty interior. This gives a topological version of a theorem of Franks. Using this result, and inspired by a theorem of Guillou, we prove a version of the Poincaré–Birkhoff theorem for torus homeomorphisms: in the absence of free curves, either there is a fixed point or the rotation set has non-empty interior.

We investigate the relationship between Poincaré recurrence and topological entropy of a dynamical system (X,f). For , let D(α,β) be the set of x with lower and upper recurrence rates α and β, respectively. Under the assumptions that the system is not minimal and that the map f is positively expansive and satisfies the specification condition, we show that for any open subset , has the full topological entropy of X. This extends a result of Feng and Wu [The Hausdorff dimension of recurrence sets in symbolic spaces. Nonlinearity14 (2001), 81–85] for symbolic spaces.

Suppose ℛ is a finite commutative ring of prime characteristic, 𝒜 is a finite ℛ-module, 𝕄:=ℤD×ℕE, and Φ is an ℛ-linear cellular automaton on 𝒜𝕄. If μ is a Φ-invariant measure which is multiply σ-mixing in a certain way, then we show that μ must be the Haar measure on a coset of some submodule shift of 𝒜𝕄. Under certain conditions, this means that μ must be the uniform Bernoulli measure on 𝒜𝕄.

Veech showed that if a translation surface has a stabilizer which is a lattice in SL(2,ℝ), then any direction for the corresponding constant slope flow is either completely periodic or uniquely ergodic. We show that the converse does not hold: there are translation surfaces that satisfy Veech’s dichotomy but for which the corresponding stabilizer subgroup is not a lattice. The construction relies on work of Hubert and Schmidt.

Let M be a positive integer and let f be a holomorphic mapping from a ball Δn={x∈ℂn:∣x∣<δ} into ℂn such that the origin 0 is an isolated fixed point of both f and fM, the Mth iteration of f. Then one can define the number 𝒪M(f,0), interpreted as the number of periodic orbits of f with period M that are hidden at the fixed point 0. For an n×n matrix A whose eigenvalues are all the same primitive Mth root of unity, we give a sufficient and necessary condition on A such that for any holomorphic mapping f:Δn→ℂn with f(0)=0 and Df(0)=A, if 0 is an isolated fixed point of the Mth iteration fM , then 𝒪M (f,0)≥2.