We investigate bifurcations in the chain recurrent set for a particular class of one-parameter families of diffeomorphisms in the plane. We give necessary and sufficient conditions for a discontinuous change in the chain recurrent set to occur at a point of heteroclinic tangency. These are also necessary and sufficient conditions for an \Omega-explosion to occur at that point.

We construct an algebraic method for detecting bifurcations of discrete dynamical systems using the Conley index theory. We define the transition matrix pair for discrete semidynamical systems, which detects bifurcations of connecting orbits. As an application, we give a sufficient condition for the occurrence of homoclinic and heteroclinic tangencies by showing that the occurrence of a heterodimensional cycle in the projectivization of a dynamical system implies the tangency of the original system.

Connections are established between mixing or ergodic properties of maps on the one hand, and the convergence of the iterates of the map, or of the empirical measures of the iterates, to a constant measure-valued map, on the other. The uniqueness of an absolutely continuous ergodic measure can also be verified via the convergence. The technique helps to identify ergodic and mixing pairs and verify the uniqueness in specific examples.

For a fixed C^*-algebra A, we consider all non-commutative dynamical systems that can be generated by A. More precisely, an A-dynamical system is a triple (i,B,\alpha) where \alpha is a *-endomorphism of a C^*-algebra B, and i: A\subseteq B is the inclusion of A as a C^*-subalgebra with the property that B is generated by A\cup\alpha(A)\cup\alpha^2(A)\cup\dotsb. There is a natural hierarchy in the class of A-dynamical systems, and there is a universal one that dominates all others, denoted (i,\mathcal{P}A,\alpha). We establish certain properties of (i,\mathcal{P}A,\alpha) and give applications to some concrete issues of non-commutative dynamics.

For example, we show that every contractive completely positive linear map \varphi:A\to A gives rise to a unique A-dynamical system (i,B,\alpha) that is ‘minimal’ with respect to \varphi, and we show that its C^*-algebra B can be embedded in the multiplier algebra of A\otimes\mathcal{K}.

We answer in the negative a question of Viviane Baladi: in contrast to its deterministic or averaged versions, the random zeta function has no meromorphic extensions, hence no poles which would describe the Lyapunov exponents of the corresponding random transfer operator. In fact we show that the sequence of traces of the random iterates of this operator does not determine almost surely these exponents.

Dans cet article, nous complétons la classification analytique des feuilletages analytiques réduits de codimension 1 en dimension n\ge3, dans les cas résonnants. Nous montrons que, génériquement, l'espace de modules se réduit à un point.

We introduce transversality conditions for a piecewise smooth map in any dimension with a sufficiently regular discontinuity set. We prove that maps satisfying these conditions have certain nice properties and, among all maps with a common set of discontinuities, form an open and dense subset. This, in conjunction with other works of the author, implies the existence of stochastically stable invariant densities for an open and dense set of piecewise smooth expanding maps.

Hyperbolic sets are robust under perturbations: they persist on an open set of the parameter space. In this paper we investigate the boundary of this open set. Generalizing the theory of fixed points we define saddle-node bifurcations for hyperbolic sets K with one-dimensional unstable directions. In this bifurcation the geometrical splitting of the tangent space is preserved but the expansion in the unstable direction degenerates near a periodic orbit. The compact set K can be followed on a closed half-space bounded by a codimension-one manifold \mathcal{O}^0. On \mathcal{O}^0 the saddle-node bifurcation occurs. On one side of \mathcal{O}^0, K is hyperbolic and on the other side, it has disappeared.

We consider the collapse of a saddle of a horseshoe and a sink via a critical saddle-node bifurcation. In this way one obtains a saddle-node horseshoe. We prove that there is an open set of arcs of diffeomorphisms \{f_\mu\}_{\mu \in I} unfolding generically, say at \mu_0, a saddle-node horseshoe with Hausdorff dimension arbitrarily close to 1/2 so that there is an interval (\mu_0, \mu_0+\varepsilon) in the parameter line such that every diffeomorphism f_\mu, \mu\in (\mu_0, \mu_0+\delta), has a tangency.

We discuss results on engaging actions of a lattice \Gamma in a higher rank simple group G on a compact manifold M. An action is engaging if there is no loss of ergodicity in passing to lifts of the action on finite covers of M.

Suppose \mathbb{R}-rank (G)\geq 2 and \Gamma<G is a lattice. Let \Lambda be the group of lifts of the \Gamma action on M to the universal cover of M. Assume the \Gamma action on M is measure preserving and engaging. We show that the image of \Lambda under any linear representation \sigma is \mathfrak{s}-arithmetic. Also, associated to each \sigma we have a measurable quotient of the \Gamma action on M; this measurable quotient is a generalized affine action on a double coset space. Furthermore, the pushforward of the invariant measure on M is a Lebesgue measure on the quotient. The fundamental group of this quotient is closely related to the image of \pi_1(M) under \sigma.

Let M={\tilde M}/\Gamma be a closed negatively curved manifold with universal covering {\tilde M} and fundamental group \Gamma. Every Gibbs equilibrium state \nu of a Hölder continuous function on the unit tangent bundle T^1M of M projects to a \Gamma-invariant ergodic measure class mc(\nu_+) on the ideal boundary \partial{\tilde M} of {\tilde M}. We show that this measure class is also ergodic under the action of any normal subgroup \Gamma^\prime of \Gamma for which the factor group \Gamma/\Gamma^\prime is nilpotent.

Let \Omega be a metric space. There is a natural metric that can be put on the space of all measures on \Omega called the Wasserstein metric. It is obtained by taking the infimum over all couplings of the two measures of the average distance between two points. There is a simple coupling called the greedy coupling for the two measures, but operations researchers will assure you that it is terrible to use the value obtained by the greedy coupling as a substitute for the Wasserstein distance between two measures. The purpose of this paper, however, is to show that if all that you are interested in is the topology obtained by the metric, the value you get from the greedy coupling is good enough.

For an infinite word \alpha=\alpha_0\alpha_1\alpha_2\dots, over a finite alphabet A, we define the maximal pattern complexity by p_\alpha^*(k)=\sup_\tau\sharp\{\alpha_{n+\tau(0)} \alpha_{n+\tau(1)}\dots\alpha_{n+\tau(k-1)}; n=0,1,2,\dots\} where the ‘sup’ is taken over all subsequences 0=\tau(0)<\tau(1)<\dots<\tau(k-1) of integers of length k. We prove that \alpha is eventually periodic if and only if p_\alpha^*(k)\le 2k-1 for some k. Infinite words \alpha, with p_\alpha^*(k)=2k for any k, are called pattern Sturmian words and are studied.

For an infinite word \alpha=\alpha_0\alpha_1\alpha_2\dotsc over a finite alphabet, the authors introduced a new notion of complexity called maximal pattern complexity defined by p_\alpha^*(k):=\sup_\tau\sharp\{\alpha_{n+\tau(0)}\alpha_{n+\tau(1)}\dots\alpha_{n+\tau(k-1)};n=0,1,2,\dotsc\} where the supremum is taken over all sequences of integers 0=\tau(0)<\tau(1)<\dotsc<\tau(k-1) of length k. The authors proved that \alpha is aperiodic if and only if p_\alpha^*(k)\ge 2k for every k=1,2,\dotsc. A word \alpha with p_\alpha^*(k)=2k for every k\geq 1 is called pattern Sturmian. In this paper, we give a simple criterion to be pattern Sturmian and exhibit a new class of recurrent pattern Sturmian words which do not arise from rotations. We also investigate the maximal pattern complexity of various discrete dynamical systems including irrational rotations on the circle, and self-similar systems generated by substitutions. We show that, for each irrational rotation on the circle, there exists a twofold partition of the circle, with respect to which the system generated has full maximal pattern complexity with probability one. Using the arithmetic properties of the underlying numeration system associated to a substitution dynamical system, we prove that the maximal pattern complexity of the fixed point of the Rauzy substitution 1\mapsto 12, 2\mapsto 13, 3\mapsto 1 has exponential growth. It is well known that the system generated by the Rauzy substitution is isomorphic in measure to an irrational rotation on the 2-torus.

We prove a generalized version of the Lefschetz fixed point theorem, and use it to obtain a variety of periodic and aperiodic solutions for differential delay equations; in particular, of the type \dot x(t) = f(x(t-1)). Here f:\mathbb R \to \mathbb R is odd and two-periodic, and we obtain both strictly periodic solutions and solutions periodic modulo a multiple of two. The qualitative behavior of solutions can be coded by symbol sequences containing the sequence of levels about which these solutions oscillate.

We present some results of KAM type, comparable to the KAM theory for volume-preserving maps and flows, for generalized Hamiltonian systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd dimensional. Applications to the perturbation of three-dimensional steady Euler fluid particle path flows are considered with respect to the existence problem of barriers to fluid transport and mixing.

For any diffeomorphism f of the two torus T^2 isotopic to the identity, the Ruelle invariant {\mathcal R}_\mu(f) is shown to vanish for some f-invariant probability measure \mu.

Let \mathbb{M} be a monoid (e.g. \mathbb{N}, \mathbb{Z}, or \mathbb{M}^D), and \mathcal{A} an abelian group. \mathcal{A}^\mathbb{M} is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism \mathfrak{F}:\mathcal{A}^\mathbb{M}\longrightarrow\mathcal{A}^\mathbb{M} that commutes with all shift maps. Let \mu be a (possibly non-stationary) probability measure on \mathcal{A}^\mathbb{M}; we develop sufficient conditions on \mu and \mathfrak{F} so that the sequence \{\mathfrak{F}^N\mu\}_{N=1}^\infty weak* converges to the Haar measure on \mathcal{A}^\mathbb{M} in density (and thus, in Cesàro average as well). As an application, we show that, if \mathcal{A}=\mathbb{Z}_{/p} (p prime), \mathfrak{F} is any ‘non-trivial’ LCA on \mathcal{A}^{(\mathbb{Z}^D)}, and \mu belongs to a broad class of measures (including most Bernoulli measures (for D \geq 1) and ‘fully supported’ N-step Markov measures (when D=1)), then \mathfrak{F}^N\mu weak* converges to the Haar measure in density.

For a class of one-dimensional maps with two indifferent fixed points of the same order, a parallel to the arc-sine law for the number of positive partial sums of a real random walk is proved.

Let g be a polynomial automorphism of \mathbb{C}^2. We study the Hausdorff dimension and topological dimension of the Julia set of g. We show that when g is a hyperbolic mapping, then the Hausdorff dimension of the Julia set is strictly greater than its topological dimension. Moreover, the Julia set cannot be locally connected. We also provide estimates for the dimension of the Julia sets in the general (not necessarily hyperbolic) case.