In this paper, we show the existence of Markov covers for $C^2$ surface diffeomorphisms with a dominated splitting under some assumptions. Using a Markov cover, we can reduce the dynamics to a one-dimensional dynamical system having a good metric property. As an application, we show finiteness of periodic attractors for the above diffeomorphisms.

We prove that interval maps for which $\omega$-limit sets of all critical points are minimal are dense in the space of all interval maps of class $C^2$.

We present a sufficient condition for the existence of ergodic continuous invariant measures (ECIMs) for continuous folding transformations. We apply this result to prove the existence of ECIMs for general set expanding Markov transformations and $n$-dimensional tent maps.

We consider deterministic perturbations $\ddot q^\varepsilon(t)+F'(q^\varepsilon(t))=\varepsilon b(\dot q^\varepsilon(t),q^\varepsilon(t))$ of an oscillator $\ddot q+F'(q)=0$, $q\in{\mathbb R}^1$. Assume that $\lim_{|q|\to\infty}F(q)=\infty$ and that $F'(q)$ has a finite number of nondegenerate zeros. For a generic $F$, if $\partial b/\partial\dot q<0$ (as in the case of friction), then typical orbits are attracted to points where $F$ has a local minimum. For $0<\varepsilon\ll1$, the equilibrium to which the trajectory is attracted is ‘random’. To study this randomness, which is caused by the sensitive behavior of trajectories near the saddle points, we consider the graph $\Gamma$ homeomorphic to the space of connected components of the level sets of the Hamiltonian $H(p,q)=p^2/2+F(q)$. We show that, as $\varepsilon\to0$, the slow component of $(p^\epsilon(t/\epsilon),q^\epsilon(t/\epsilon))$ tends to a certain stochastic process on $\Gamma$ which is deterministic inside the edges and branches at the interior vertices into adjacent edges with probabilities which can be calculated through the Hamiltonian $H$ and the perturbation $b$.

We consider a $C^{1+1}$ diffeomorphism $f$ of a compact manifold $M$ which preserves an ergodic probability measure $\mu$. We conclude that $\mu$-a.e. $x \in M$ is contained in a disk $D_x \subset W^u(x)$, with $D_x$ open in the $W^u(x)$ topology, which exhibits an exponential volume growth rate greater than or equal to the measure-theoretic entropy of $f$ with respect to $\mu$. Drawing on results of Newhouse and Yomdin, we then find that when $f$ is $C^\infty$ and $\mu$ is a measure of maximal entropy, this exponential volume growth rate equals the topological entropy of $f$ for $\mu$-a.e. $x$.

We study the most generic nilpotent singularity of a vector field in ${\mathbb R}^3$ which is equivariant under reflection with respect to a line, say the $z$-axis. We prove the existence of eight equivalence classes for $C^0$-equivalence, all determined by the 2-jet. We also show that in certain cases, the ${\mathbb Z}_2$-equivariant unfoldings generically contain codimension one heteroclinic cycles which are comparable to the Shil'nikov-type homoclinic cycle in non-equivariant unfoldings. The heteroclinic cycles are accompanied by infinitely many horseshoes and also have a reasonable possibility of generating suspensions of Hénon-like attractors, and even Lorenz-like attractors.

We consider analytically coupled circle maps (uniformly expanding and analytic) on the ${\mathbb Z}^d$-lattice with exponentially decaying interaction. We introduce Banach spaces for the infinite-dimensional system that include measures whose finite-dimensional marginals have analytic, exponentially bounded densities. Using residue calculus and ‘cluster expansion’-like techniques we define transfer operators on these Banach spaces. We get a unique (in the considered Banach spaces) probability measure that exhibits exponential decay of correlations.

We study certain metrics on subshifts of finite type for which we define the discrete analogs of Lyapunov exponents. We prove Young's formula for $\mu$-Hausdorff dimension. We give sufficient conditions on the above metrics for which the Hausdorff dimension is given by thermodynamic formalism. We apply these results to the Hausdorff dimension of the limit sets of geometrically finite, purely loxodromic, Kleinian groups.

A unimodal map $f:[0,1] \to [0,1]$ is renormalizable if there is a sub-interval $I \subset [0,1]$ and an $n > 1$ such that $f^n|_I$ is unimodal. The renormalization of $f$ is $f^n|_I$ rescaled to the unit interval.

We extend the well-known classification of limits of renormalization of unimodal maps with bounded combinatorics to a classification of the limits of renormalization of unimodal maps with essentially bounded combinatorics. Together with results of Lyubich on the limits of renormalization with essentially unbounded combinatorics, this completes the combinatorial description of limits of renormalization. The techniques are based on the towers of McMullen and on the local analysis around perturbed parabolic points. We define a parabolic tower to be a sequence of unimodal maps related by renormalization or parabolic renormalization. We state and prove the combinatorial rigidity of bi-infinite parabolic towers with complex bounds and essentially bounded combinatorics, which implies the main theorem.

As an example we construct a natural unbounded analogue of the period-doubling fixed point of renormalization, called the essentially period-tripling fixed point.

We extend our previous work by proving that for translation invariant Gibbs states on ${\mathbb Z}^d$ with a translation invariant interaction potential $\Psi=(\Psi_A)$ satisfying $\sum_{A \ni 0}|A|^{-1}[\diam(A)]^d\|\Psi_A\|<\infty$ the following hold: (1) the Kolmogorov-property implies a trivial full tail and (2) the Bernoulli-property implies Følner independence. The existence of bilaterally deterministic Bernoulli Shifts tells us that neither (1) nor (2) is, in general, true for random fields without some further assumption (even when $d=1$).

We study one-dimensional dynamical systems with indifferent fixed points $p$ and $q$. The dynamical systems we study in this paper have ergodic, infinite, invariant measures. We consider the limit of the ratio of the sojourn time of the trajectory in a small neighborhood of $p$ to that in a small neighborhood of $q$. We show that the limit does not exist under some conditions, which has been announced in a previous paper. In fact we prove that the limit supreme of the ratio is $\infty$ and that the limit infimum is 0.

We develop a new technique for calculating the first cohomology of certain classes of actions of higher-rank abelian groups (${\mathbb Z}^k$ and ${\mathbb R}^k$, $k\ge 2$) with values in a linear Lie group. In this paper we consider the discrete-time case. Our results apply to cocycles of different regularity, from Hölder to smooth and real-analytic. The main conclusion is that the corresponding cohomology trivializes, i.e. that any cocycle from a given class is cohomologous to a constant cocycle. The principal novel feature of our method is its geometric character; no global information about the action based on harmonic analysis is used. The method can be developed to apply to cocycles with values in certain infinite dimensional groups and to rigidity problems.

Let $T$ be a tree, ${\mathop{\rm End}\nolimits}(T)$ be the number of ends of $T$ and let $L(T)$ be the infimum of topological entropies of transitive maps of $T$. We give an elementary approach to the estimate that $L(T)\ge (1/{\mathop{\rm End}\nolimits} (T))\log 2$. We also divide the set of all trees (up to homeomorphisms) into pairwise disjoint subsets ${\cal P}(i)$, $i\in \{0\}\cup\N$ and prove that $L(T)=(1/({\mathop{\rm End}\nolimits} (T)-i))\log 2$ if $T\in {\cal P}(i)$ with $i=0,1$, and $L(T)\le (\text{respectively} =) (1/({\mathop{\rm End}\nolimits} (T)-i))\log 2$ if $T\in {\cal P}(i)$ (respectively $T\in {\cal P}'(i))$ with $i\ge 2$, where ${\cal P}'(i)$ is an infinite subset of ${\cal P}(i)$. Furthermore, we show that there is a tree $T$ such that the topological entropy of each transitive map of $T$ is larger than $L(T)$, and hence disprove a conjecture of Alseda et al (1997).