For $\beta>1$, consider the $\beta$-transformation $T_\beta$. When $\beta$ is an integer, the natural extension of $T_\beta$ can be represented explicitly as a map on the unit square with an invariant measure: the corresponding two-dimensional Lebesgue measure. We show that, under certain conditions on $\beta$, the natural extension is defined on a simply connected region and an invariant measure is a constant multiple of the Lebesgue measure.

We characterize those $\beta$ in terms of the $\beta$-expansion of one, and study the structure and size of the set of all such $\beta$.

To study the geometry of a Fibonacci map $f$ of even degree $\ell\geq 4$, Lyubich (Dynamics of quadratic polynomials, I–II. Acta Mathematica178 (1997), 185–297) defined a notion of generalized renormalization, so that $f$ is renormalizable infinitely many times. van Strien and Nowicki (Polynomial maps with a Julia set of positive Lebesgue measure: Fibonacci maps. Preprint, Institute for Mathematical Sciences, SUNY at Stony Brook, 1994) proved that the generalized renormalizations ${\cal R}^{\circ n}(f)$ converge to a cycle $\{f_1,f_2\}$ of order two depending only on $\ell$. We will explicitly relate $f_1$ and $f_2$ and show the convergence in shape of Fibonacci puzzle pieces to the Julia set of an appropriate polynomial-like map.

Any polynomial automorphism of $\mathbb{C}^2$ with nontrivial dynamics is conjugate to a diffeomorphism of the 4-ball such that this diffeomorphism extends to a diffeomorphism of the closed 4-ball. Moreover, the conjugating map is a smooth bijection of $\mathbb{C}^2$ to itself. On the sphere at infinity, the extension has an attracting and a repelling solenoid, and the dynamics near these invariant solenoids are described by conjugation to a model solenoidal map.

In this paper, it is proved that the distribution of values of $N^{-1/2}\sum_{n=1}^N f_1(\theta^{p_1(n)}x)\dots f_K(\theta^{p_K(n)}x)$ converges to normal distribution. Here $p_k(n)$ are polynomials.

This paper treats the Pinsker algebra of a dynamical system in a way which avoids the use of an ordering on the acting group. This enables us to prove some of the classical results about entropy and the Pinsker algebra in the general setup of measure-preserving dynamical systems, where the acting group is a discrete countable amenable group. We prove a basic disjointness theorem which asserts the relative disjointness in the sense of Furstenberg, of $0$-entropy extensions from completely positive entropy (c.p.e.) extensions. This theorem is used to prove several classical results in the general setup. For example, we show that the Pinsker factor of a product system is equal to the product of the Pinsker factors of the component systems. Another application is to obtain a generalization (as well as a simpler proof) of the quasifactor theorem for $0$-entropy systems of Glasner and Weiss.

Equilibrium states of rational maps for Hölder continuous potentials are not $\phi$-mixing, mainly due to the presence of critical points. Here we prove that for disks the normalized return times of arbitrary orders are, in the limit, Poisson distributed as the radius of the disks go to zero. The return times are normalized by the measure of the disks. We also show that rational maps are weakly Bernoulli with respect to the partition given by Denker and Urbanski.

It is shown that a polynomial with a Cremer periodic orbit has a non-accessible critical point in its Julia set provided that the Cremer periodic orbit is approximated by small cycles. Also, this paper contains a new proof of the Douady–Shishikura inequality for the number of non-repelling cycles of a complex polynomial.

In this paper we will show that any number normal with respect to the regular continued fraction (RCF) is also normal with respect to the continued fraction from below (BCF) and the nearest integer continued fraction (NICF). Furthermore, we will show that in case of the NICF a number is normal with respect to the RCF-expansion if and only if it is NICF-normal. This can be generalized to a larger class of semi-regular continued fraction expansions.

In this paper we introduce and explore conformal parabolic iterated function systems. We define and study topological pressure, Perron–Frobenius-type operators, semiconformal and conformal measures and the Hausdorff dimension of the limit set. With every parabolic system we associate an infinite hyperbolic conformal iterated function system and we employ it to study geometric and dynamical features (properly defined invariant measures for example) of the limit set.

Let $M$ be a two-sided surface of genus $g>1$. In 1936 A. Weil singled out the problem of generalization of the Poincarè rotation numbers to the minimal flows $\phi$ on $M$. In this paper we suggest a solution to this problem by constructing the rotation numbers which we call Artin's. These numbers are invariants of the $K_0$-group of a crossed product $C^*$-algebra $C(X)\rtimes_{\phi}\Z$. It is shown that the dynamics of $\phi$ is completely subordinate to diophantine properties of the Artin numbers.

In the present paper we give an improvement of a previous result of the paper [M. M. Skriganov. Ergodic theory on $SL(n)$, diophantine approximations and anomalies in the lattice point problem. Inv. Math.132(1), (1998), 1–72, Theorem 2.2] on logarithmically small errors in the lattice point problem for polyhedra. This improvement is based on an analysis of hidden symmetries of the problem generated by the Weyl group for $SL(n,\mathbb{B})$. Let $UP$ denote a rotation of a given compact polyhedron $P\subset\mathbb{B}^n$ by an orthogonal matrix $U\in SO(n)$, $tUP$ a dilation of $UP$ by a parameter $t>0$ and $N(tUP)$ the number of integer points $\gamma\in\mathbb{Z}^n$ which fall into the polyhedron $tUP$. We show that for almost all rotations $U$ (in the sense of the Haar measure on $SO(n)$) the following asymptotic formula \[ N(t\UP)=t^n{\rm vol} P+ O((\log t)^{n-1+\varepsilon}),\quad t\to\infty, \] holds with arbitrarily small $\varepsilon>0$.

To each function we associate a period, a polytope, a group $\Gamma$, a subgroup $\Delta \subset\Gamma$ and a coset of $\Gamma/\Delta$, and show these objects to be universal for the class of Markov chains with that beta function. One consequence is that polynomial matrices employed in coding problems of Markov chains can be chosen canonically.

We study a weak Gibbs property of equilibrium states for potentials of weak bounded variation and for maps admitting indifferent periodic points. We further establish statistical properties of the weak Gibbs measures and bounds of their pointwise dimension. We apply our results to higher-dimensional maps (which are not necessarily conformal) with indifferent periodic points and show that their absolutely continuous finite invariant measures are weak Gibbs measures.

We consider piecewise twice differentiable maps $T$ on $[0,1]$ with indifferent fixed points giving rise to infinite invariant measures, and we study their behaviour on ergodic components. As we do not assume the existence of a Markov partition but only require the first image of the fundamental partition to be finite, we use canonical Markov extensions to first prove pointwise dual-ergodicity, which, together with an identification of wandering rates, leads to distributional limit theorems. We show that $T$ satisfies Rohlin's formula and prove a variant of the Shannon–McMillan–Breiman theorem. Moreover, we give a stronger limit theorem for the transfer operator providing us with a large collection of uniform and Darling–Kac sets. This enables us to apply recent results from fluctuation theory.

The function $d$ defined in (0.3) may not be a metric unless \begin{equation} \phi_p(a_1,\dots, a_p)\le \phi_{p+1} (a_1,\dots, a_{p+1}),\quad \text{for all } (a_i)^{p+1}_1 \in \Gamma^{p+1},\ p=1,\dots.\tag{A} \end{equation} Let $$ \tilde\phi_p(a_1,\dots, a_p)=\max_{1\le i\le p}\phi_1 (a_1,\dots, a_i),\quad (a_i)^p_1\in\Gamma^p,\ p=1,\dots. $$ Then $\tilde\phi$ satisfies (A) and (0.1). Note that \begin{equation} \phi_p(a_1,\dots, a_p) \le\tilde\phi_p (a_1,\dots, a_p)\le p \max_{1\le i\le n} \phi_1(a_i).\tag{B} \end{equation} In (0.3) replace the equality $d(a,b)=e^{-\phi_p(a_1,\dots,a_p)}$ by $d(a,b)=e^{-\tilde\phi_p(a_1,\dots,a_p)}$. Then $d(a,b)\le \max(d(a,c), d(c,b))$. Let \begin{gather*} \tilde\psi_p(x):=\max_{1\le i\le p} \psi_i(x),\ x\in\Gamma^\infty,\quad \tilde\alpha_p(\mu) :=\int \tilde\psi_p\,d\mu,\ p=1, \dots\\ \tilde\alpha(\mu) :=\lim_{p\rightarrow\infty} \frac{\tilde\alpha_p(\mu)}{p}. \end{gather*} Then (B) yields $$ \lim_{p\rightarrow\infty} \frac{\psi_p(x)}{p} = s\Rightarrow \lim_{p\rightarrow\infty} \frac{\tilde\psi_p(x)}{p}=s. $$