We prove density modulo $1$ of the sets of the form$$\{{\mu }^m{\lambda }^n\xi +r_m:n,m\in \mathbb{N}\},$$where $\lambda ,\mu \in ℝ$ is a pair of rationally independent algebraic integers of degree $d\ge 2,$ satisfying some additional assumptions, $\xi \ne 0,$ and $r_m$ is any sequence of real numbers.

We study the enclosing problem for discrete and continuous dynamical systems in the context of computer assisted proofs. We review and compare the existing methods and emphasize the importance of developing a suitable set arithmetic for efficient algorithms solving the enclosing problem.

For every $k\in \mathbb{N}$, we produce a set of integers which is $k$-recurrent but not $(k+1)$-recurrent. This extends a result of Furstenberg who produced a $1$-recurrent set which is not $2$-recurrent. We discuss a similar result for convergence of multiple ergodic averages. We also point out a combinatorial consequence related to Szemerédi’s theorem.

We study differentiability of topological conjugacies between expanding piecewise $C^{1+ϵ}$ interval maps. If these conjugacies are not C¹, then their derivative vanishes Lebesgue almost everywhere. We show that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Moreover, by employing the thermodynamic formalism, we show that this Hausdorff dimension can be determined explicitly in terms of the Lyapunov spectrum....

We study natural measures on sets of $\beta$-expansions and on slices through self similar sets. In the setting of $\beta$-expansions, these allow us to better understand the measure of maximal entropy for the random $\beta$-transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading...

A Borel subset of the unit square whose vertical and horizontal sections are two-point sets admits a natural group action. We exploit this to discuss some questions about Borel subsets of the unit square on which every function is a sum of functions of the coordinates. Connection with probability measures with prescribed marginals and some function algebra questions is discussed.

A unified geometric approach to nonholonomic constrained mechanical systems is applied to several concrete problems from the classical mechanics of particles and rigid bodies. In every of these examples the given constraint conditions are analysed, a corresponding constraint submanifold in the phase space is considered, the corresponding constrained mechanical system is modelled on the constraint submanifold, the reduced equations of motion of this system (i.e. equations of motion defined on the...

The topology and combinatorial structure of the Mandelbrot set ${ℳ}^d$ (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in ${ℳ}^d$. Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, ${\Lambda }^d$. In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence is realized....

We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain expanding maps have shadowing, and generalize some known results in this area. We also investigate the impact of our theory on maps of the interval.

The main result of this paper is that a map f: X → X which has shadowing and for which the space of ω-limits sets is closed in the Hausdorff topology has the property that a set A ⊆ X is an ω-limit set if and only if it is closed and internally chain transitive. Moreover, a map which has the property that every closed internally chain transitive set is an ω-limit set must also have the property that the space of ω-limit sets is closed. As consequences of this result, we show that interval maps with...

We study shadowing properties of continuous actions of the groups ${\mathbb{Z}}^p$ and ${\mathbb{Z}}^p\times {ℝ}^p$. Necessary and sufficient conditions are given under which a linear action of ${\mathbb{Z}}^p$ on ${ℂ}^m$ has a Lipschitz shadowing property.

We show that the class of expansive ${\mathbb{Z}}^d$ actions with P.O.T.P. is wider than the class of actions topologically hyperbolic in some direction $\nu \in {\mathbb{Z}}^d$. Our main tool is an extension of a result by Walters to the multi-dimensional symbolic dynamics case.

We present a scheme for constructing various Conley indices for locally defined maps. In particular, we extend the shape index of Robbin and Salamon to the case of a locally defined map in a locally compact Hausdorff space. We compare the shape index with the cohomological Conley index for maps. We also prove the commutativity property of the Conley index, which is analogous to the commutativity property of the fixed point index.

We extend the shape index, introduced by Robbin and Salamon and Mrozek, to locally defined maps in metric spaces. We show that this index is additive. Thus our construction answers in the affirmative two questions posed by Mrozek in [12]. We also prove that the shape index cannot be arbitrarily complicated: the shapes of q-adic solenoids appear as shape indices in natural modifications of Smale's horseshoes but there is not any compact isolated invariant set for any locally defined map in a locally...

We show that the Sharkovskiĭ ordering of periods of a continuous real function is also valid for every function with connected $G_{\delta }$ graph. In particular, it is valid for every DB₁ function and therefore for every derivative. As a tool we apply an Itinerary Lemma for functions with connected $G_{\delta }$ graph.

Let $ℱ:L\to TS$ be a foliation on a complex, smooth and irreducible projective surface $S$, assume $ℱ$ admits a holomorphic first integral $f:S\to {\mathbb{P}}^1$. If $h^0(S,{𝒪}_S(-n{𝒦}_S\left)\right)\>0$ for some $n\ge 1$ we prove the inequality: $(2n-1)(g-1)\le h^1(S,{ℒ}^{\text{'}-1}(-(n-1)K_S\left)\right)+h^0(S,{ℒ}^{\text{'}})+1$. If $S$ is rational we prove that the direct image sheaves of the co-normal sheaf of $ℱ$ under $f$ are locally free; and give some information on the nature of their decomposition as direct sum of invertible sheaves.