We consider a $C^\infty$ Lagrangian function $L: T^*M\rightarrow \mathbb{R}$ which is superlinear and convex in the fibers and has one antisymplectic symmetry. We prove that:

1. in every energy level strictly above the critical one, there exists a Mather set which is the union of some periodic orbits;

2. for $L$ generic, these orbits are hyperbolic;

3. on the torus, these orbits have one homoclinic orbit.

Fathi and Franks showed that a pseudo-Anosov diffeomorphism $f$ with orientable foliations and dilation coefficient $\lambda$ with no conjugates (over ${\mathbb Q}$) in the unit circle factors onto a (homologically non-trivial) invariant subset of a hyperbolic toral automorphism. After recounting this result, we show that the factor map is either almost everywhere one-to-one or almost everywhere $m$-to-one for some $m>1$ and the pseudo-Anosov map $f$ is an $m$-to-one ramified covering of another pseudo-Anosov (or Anosov) map on a surface of smaller genus. As a corollary, any pseudo-Anosov diffeomorphism with orientable foliations and hyperbolic action on the first homologies almost everywhere embeds into a hyperbolic toral automorphism.

We obtain explicit formulas for the average density, also called order-two density, of an arbitrary conformal repeller of a $C^{1+\epsilon}$ transformation. We note that unlike with the pointwise densities, the average densities exist almost everywhere on each repeller, and are in fact constant almost everywhere. Thus, they are natural parameters to describe the geometric structure of the corresponding invariant sets, and as such it is of interest to obtain explicit formulas for their almost constant value. We obtain simultaneously formulas without using symbolic dynamics, and formulas based on the symbolic dynamics generated by a given Markov partition. The proofs strongly depend on the use of appropriate Markov partitions, and are also based on methods of Falconer and of Zähle.

Let $\Omega$ be an abelian group. A set $R\subset \Omega$ is a set of recurrence if, for any probability measure-preserving system $(X,{\mathcal B},\mu,\{T_g\}_{g\in \Omega})$ and any $A\in {\mathcal A}$ with $\mu(A)>0,\ \mu(A\cap T_g A)>0$ for some $g\in R$. If $(x_i)_{i=1}^\infty$ is a sequence in $\Omega$, the set of its finite sums $\{ x_{i_1}+x_{i_2}+\dotsb + x_{i_k}: i_1<i_2<\dotsb <i_k\}$ is called an IP-set. In Bergelson et al (Erg. Theor. Dyn. Sys.16 (1996), 963–974) it is shown that if $p:{\mathbb Z}^d{\rightarrow} {\mathbb Z}^k$ is a polynomial vanishing at zero and F is an IP-set in ${\mathbb Z}^d$ then $\{ p(n):n\in F\}$ is a set of recurrence in ${\mathbb Z}^k$. Here we extend this result to an analogous family of generalized polynomials, that is functions formed from regular polynomials by iterated use of the greatest integer function, as a consequence of a theorem establishing a much wider class of recurrence sets occurring in any (possibly non-finitely generated) abelian group. While these sets do in a sense have a distinctively ‘polynomial’ nature, this far-ranging class includes, even in ${\mathbb Z}$, such examples as $\big\{\!\sum_{i, j\in \alpha, i<j} 2^i3^j : \alpha\subset \mathbb{N}, 0<|\alpha|<\infty\big\}$, where the connection to conventional polynomials is somewhat distant.

In this paper, which is a sequel to our previous work (On a general Conley index continuation principle for singular perturbation problems. Ergod. Th. & Dynam. Sys.22 (2002), 729–755) we establish a general singular nested index filtration principle which, in particular, implies a continuation principle for homology index braids and connection matrices in singular perturbation problems.

In 1993, E. Lesigne proved a polynomial extension of the Wiener–Wintner theorem and asked two questions: does this result have a uniform counterpart and can an assumption of total ergodicity be replaced by ergodicity? The purpose of this article is to answer these questions, the first one positively and the second one negatively.

The notion of the maximal pattern complexity of words was introduced by Kamae and Zamboni. In this paper, we obtain an almost exact formula for the maximal pattern complexity $p^*_\alpha(k)$ of Toeplitz words $\alpha$ on an alphabet ${\mathbb A}$ defined by a sequence of coding words $(\eta^{(n)})^\infty\in({\mathbb A}\cup\{?\})^{\mathbb N}(n=1,2,\dotsc)$ including just one ‘?’ in their cycles $\eta^{(n)}$. Using this formula, we characterize pattern Sturmian words (i.e. $p^*_\alpha(k)=2k$ (for all $k$)) in this class. Moreover, we give a characterization of simple Toeplitz words in the sense of Kamae and Zamboni in terms of pattern complexity. In the case where $\eta^{(1)}=\eta^{(2)}=\dotsb$, we obtain the value $\lim_{k\to\infty}p_\alpha^*(k)/k$. We construct a Toeplitz word $\alpha\in{\mathbb A}^{\mathbb N}$ with $\#\A=2$ such that $p^*_\alpha(k)=2^k~(k=1,2,\dotsc)$, while Toeplitz words in our sense always have discrete spectra.

We construct a homeomorphism from the filled Julia set of a given polynomial into the connectedness locus of the family of all polynomials of higher degree by using intertwining surgery. A polynomial in the image of such a homeomorphism has a renormalization of ‘capture’ type and the inverse map is given by the ‘marking’ of the captured critical orbit.

It is well known that in the dynamics of a piecewise strictly monotone (that is, piecewise embedding) map $f$ on an interval, the topological entropy can be expressed in terms of the growth of the number (that is, the lap number) of strictly monotone intervals for $f^n$. Recently, there has been an increase in the importance of fractal sets in the sciences, and many geometric and dynamical properties of fractal sets have been studied. In the present paper, we shall study topological entropy of some maps on regular curves, which are contained in the class of fractal sets. We generalize the theorem of Misiurewicz–Szlenk and Young to the cases of regular curves and dendrites.

We consider a topological dynamical system $(\Sigma , \sigma ) on the real line which is topologically conjugate to a topologically mixing one-sided subshift <formula form="inline" disc="math" id="ffm002"><formtex notation="AMSTeX">$(\Sigma_{A}^{+},\sigma_A )$ of finite type with structure matrix $A$. Moreover, we assume that it satisfies some additional conditions so that the map $\sigma$ can have an expanding extension with nice properties in a neighborhood of $\Sigma$. Let $u$ be an eventually positive Lipschitz continuous function on $\Sigma$ and $\zeta_u (s)$ the corresponding dynamical zeta function. We show that $\zeta_u (s)$ has a meromorphic extension in the half-plane $\text{Re }s > -\beta_u$ for some $\beta_u >0$ and its special value at the origin is given by $\zeta_u (0) =1/\text{det}(I-A)$. As an application, we can see that the zeta function $\zeta_Q (s)$ for the two-dimensional dispersing billiard table $Q$ without eclipse has a meromorphic extension in the half-plane $\text{Re }s > -\beta$ for some $\beta>0$ and $\zeta_Q (0)=-1/(J-2)2^{J-1}$ holds, where $J$ is the number of scatterers.

In this paper, we consider tilings of the hyperbolic 2-space ${\mathbb H}^2$, built with a finite number of polygonal tiles, up to affine transformation. To such a tiling $T$, we associate a space of tilings: the continuous hull $\Omega(T)$ on which the affine group acts. This space $\Omega(T)$ inherits a solenoid structure whose leaves correspond to the orbits of the affine group. First, we prove that the finite harmonic measures of this laminated space correspond to finite invariant measures for the affine group action. Then we give a complete combinatorial description of these finite invariant measures. Finally, we give examples with an arbitrary number of ergodic invariant probability measures.

Let ${\mathbb{M}}={\mathbb{Z}}^D$ be a $D$-dimensional lattice, and let $({\mathcal{A}},+)$ be an abelian group. ${\mathcal{A}}^{\mathbb{M}}$ is then a compact abelian group under componentwise addition. A continuous function $\Phi:{\mathcal{A}}^{\mathbb{M}}\longrightarrow{\mathcal{A}}^{\mathbb{M}}$ is called a linear cellular automaton if there is a finite subset ${\mathbb{F}}\subset{\mathbb{M}}$ and non-zero coefficients $\varphi_{\textsf{f}}\in{\mathcal{Z}}$ so that, for any ${\bf{a}}\in{\mathcal{A}}^{\mathbb{M}}\, \Phi({\bf{a}}) = \sum_{{\textsf{f}}\in{\mathbb{F}}}\varphi_{\textsf{f}}\cdot\sigma^{{\textsf{f}}}({\bf{a}})$. Suppose that $\mu$ is a probability measure on ${\mathcal{A}}^{\mathbb{M}}$ whose support is a subshift of finite type or sofic shift. We provide sufficient conditions (on $\Phi$ and $\mu$) under which $\Phi$asymptotically randomizes$\mu$, meaning that $\mathrm{wk}^*-\lim_{{\mathbb{J}}\ni j\rightarrow\infty} \Phi^j\mu = \eta$, where $\eta$ is the Haar measure on ${\mathcal{A}}^{\mathbb{M}}$, and ${\mathbb{J}}\subset{\mathbb{N}}$ has Cesàro density one. In the case when $\Phi=1+\sigma$ and ${\mathcal{A}}=({{\mathbb{Z}}_{/p}})^s$ ($p$ prime), we provide a condition on $\mu$ that is both necessary and sufficient. We then use this to construct zero-entropy measures which are randomized by $1+\sigma$.

Let ${\mathbb{M}}={\mathbb{N}}^D$ be the positive orthant of a $D$-dimensional lattice and let $({\mathcal{G}},+)$ be a finite abelian group. Let ${\mathfrak{G}} \subseteq{\mathcal{G}}^{\mathbb{M}}$ be a subgroup shift, and let $\mu$ be a Markov random field whose support is ${\mathfrak{G}}$. Let $\Phi: {\mathfrak{G}} \longrightarrow {\mathfrak{G}}$ be a linear cellular automaton. Under broad conditions on ${\mathcal{G}}$, we show that the Cesaro average $N^{-1} \sum_{n=0}^{N-1} \Phi^n (\mu)$ converges to a measure of maximal entropy for the shift action on ${\mathfrak{G}}$.

Let $f$ be a transcendental meromorphic function and $U$ a Baker domain of $f$. We obtain new estimates for the behaviour of the iterates of $f$ in $U$ and we use these estimates to improve an earlier result relating the geometric properties of $U$ to the proximity of $f$ to the identity function in $U$. We also apply these estimates to Baker domains of transcendental meromorphic functions $f$ of the form

\begin{gather*} f(z) = az + bz^ke^{-z}(1+o(1)) \quad \text{as } \Re (z) \rightarrow \infty, \end{gather*}

where $k \in {\mathbb N},\ a > 1$ and $b > 0$, and show that these Baker domains contain an unbounded set of critical points and an unbounded set of critical values.

This is a study of the invariant measures of a synchronized subshift (in particular, a sofic shift). Among other results we give necessary and sufficient conditions for the existence of a maximal measure of full support, and a method to determine if the shift is intrinsically ergodic.