In this work, we study the entropies of subsystems of shifts of finite type (SFTs) and sofic shifts on countable amenable groups. We prove that for any countable amenable group G, if X is a G-SFT with positive topological entropy $h(X)> 0$, then the entropies of the SFT subsystems of X are dense in the interval $[0, h(X)]$. In fact, we prove a ‘relative’ version of the same result: if X is a G-SFT and $Y \subset X$ is a subshift such that $h(Y) < h(X)$, then the entropies of the SFTs Z for which $Y \subset Z \subset X$ are dense in $[h(Y), h(X)]$. We also establish analogous results for sofic G-shifts.

The isometric extensions of a multidimensional Bernouli shift are classified completely, up to C-isomorphism, and up to isomorphism. If such an extension is weakly mixing then it must be Bernoulli; otherwise, it has a rotation factor, which has a Bernoulli complementary algebra. This result is extended to multidimensional Bernoulli flows and Bernoulli shifts of infinite entropy.

For a mapping F:ℝ2→ℝ2 being a lift of an isotopic to the identity homeomorphism of the two-dimensional torus the rotation set ρ(F) consists of limit points of all sequences where xi, ∈ ℝ2 and ni→∞. It is known that if ρ(F) has nonempty interior then h(f) (the topological entropy of ƒ) is positive. We provide an estimate from below of h(f) in terms of ρ(F).

The pointwise spectral radii of irreducible matrices whose entries are polynomials with positive, integral coefficients are studied in this paper. Most results are derived in the case that the resulting algebraic function, the beta function of S. Tuncel, is in fact a polynomial. We show that the set of beta functions forms a semiring, and the spectral radius of a matrix of beta functions is again a beta function. We also show that the coefficients of a polynomial beta function p must be real algebraic integers, and p satisfies (after a change of variables if necessary) the inequality for non-zero (and not all positive) complex numbers z1,…,zd. If and the ordered sequence of exponents appearing in p is of the form (m,m+1,…,M−,1,M) for some integers m and M, the same inequality is necessary and sufficient for p to be a beta function.

Let $K$ be the attractor of a linear iterated function system (IFS) $S_{j}(x)={\it\rho}_{j}x+b_{j},j=1,\ldots ,m$, on the real line $\mathbb{R}$ satisfying the generalized finite type condition (whose invariant open set ${\mathcal{O}}$ is an interval) with an irreducible weighted incidence matrix. This condition was recently introduced by Lau and Ngai [A generalized finite type condition for iterated function systems. Adv. Math.208 (2007), 647–671]as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of $K$ coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let ${\it\alpha}$ be the dimension of $K$. In this paper, we state that

$$\begin{eqnarray}{\mathcal{H}}^{{\it\alpha}}(K\cap J)\leq |J|^{{\it\alpha}}\end{eqnarray}$$

$J\subset \overline{{\mathcal{O}}}$

$$\begin{eqnarray}{\mathcal{P}}^{{\it\alpha}}(K\cap J)\geq |J|^{{\it\alpha}}\end{eqnarray}$$

$J\subset \overline{{\mathcal{O}}}$

$K$

${\mathcal{H}}^{{\it\alpha}}$

${\it\alpha}$

${\mathcal{P}}^{{\it\alpha}}$

${\it\alpha}$

$K$

${\mathcal{H}}^{{\it\alpha}}(K)$

${\mathcal{P}}^{{\it\alpha}}(K)$

$\mathbb{R}$

It is well known that in a small neighbourhood of a parabolic fixed point a real-analytic diffeomorphism of (ℝ2,0) embeds in a smooth autonomous flow. In this paper we show that the complex-analytic situation is completely different and a generic diffeomorphism cannot be embedded in an analytic flow in a neighbourhood of its parabolic fixed point. We study two analytic invariants with respect to local analytic changes of coordinates. One of the invariants was introduced earlier by one of the authors. These invariants vanish for time-one maps of analytic flows. We show that one of the invariants does not vanish on an open dense subset. A complete analytic classification of the maps with a parabolic fixed point in ℂ2 is not available at the present time.

We consider divergent orbits of the group of diagonal matrices in the space of lattices in Euclidean space. We define two natural numerical invariants of such orbits: the discriminant—an integer—and the type—an integer vector. We then study the question of the limit distributional behavior of these orbits as the discriminant goes to infinity. Using entropy methods we prove that, for divergent orbits of a specific type, virtually any sequence of orbits equidistributes as the discriminant goes to infinity. Using measure rigidity for higher-rank diagonal actions, we complement this result and show that, in dimension three or higher, only very few of these divergent orbits can spend all of their life-span in a given compact set before they diverge.

A sequence (sn) of integers is good for the mean ergodic theorem if for each invertible measure-preserving system (X,ℬ,μ,T) and any bounded measurable function f, the averages (1/N)∑ Nn=1f(Tsnx) converge in the L2(μ) norm. We construct a sequence (sn) which is good for the mean ergodic theorem but such that the sequence (s2n) is not. Furthermore, we show that for any set of bad exponents B, there is a sequence (sn) where (skn) is good for the mean ergodic theorem exactly when k is not in B. We then extend this result to multiple ergodic averages of the form (1/N)∑ Nn=1f1(Tsnx)f2(T2snx)⋯fℓ(Tℓsnx). We also prove a similar result for pointwise convergence of single ergodic averages.

We consider interval exchange transformations of $n$ intervals with $k$ flips, or $(n,k)$-IETs for short, for positive integers $k,n$ with $k\leq n$. Our main result establishes the existence of minimal uniquely ergodic $(n,k)$-IETs when $n\geq 4$; moreover, these IETs are self-induced for $2\leq k\leq n-1$. This result extends the work on transitivity in Gutierrez et al [Transitive circle exchange transformations with flips. Discrete Contin. Dyn. Syst. 26(1) (2010), 251–263]. In order to achieve our objective we make a direct construction; in particular, we use the Rauzy induction to build a periodic Rauzy graph whose associated matrix has a positive power. Then we use a result in the Perron–Frobenius theory [Pullman, A geometric approach to the theory of non-negative matrices. Linear Algebra Appl. 4 (1971) 297–312] which allows us to ensure the existence of these minimal self-induced and uniquely ergodic $(n,k)$-IETs, $2\leq k\leq n-1$. We then find other permutations in the same Rauzy class generating minimal uniquely ergodic $(n,1)$- and $(n,n)$-IETs.

Let $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\ldots ,m_1-1\} \times \{0,\ldots ,m_2-1\})^{\mathbb {N}^*}$ that are invariant under the action of the semigroup of multiplicative integers. These sets are defined following Kenyon, Peres, and Solomyak and using a fixed integer $q \geq 2$ . We compute the Hausdorff and Minkowski dimensions of the projection of these sets onto an affine grid of the unit square. The proof of our Hausdorff dimension formula proceeds via a variational principle over some class of Borel probability measures on the studied sets. This extends well-known results on self-affine Sierpiński carpets. However, the combinatoric arguments we use in our proofs are more elaborate than in the self-similar case and involve a new parameter, namely $j = \lfloor \log _q ( {\log (m_1)}/{\log (m_2)} ) \rfloor $ . We then generalize our results to the same subsets defined in dimension $d \geq 2$ . There, the situation is even more delicate and our formulas involve a collection of $2d-3$ parameters.

In a previous paper [P. Mardešić and M. Resman. Analytic moduli for parabolic Dulac germs. Russian Math. Surveys, to appear, 2021, arXiv:1910.06129v2.] we determined analytic invariants, that is, moduli of analytic classification, for parabolic generalized Dulac germs. This class contains parabolic Dulac (almost regular) germs, which appear as first-return maps of hyperbolic polycycles. Here we solve the problem of realization of these moduli.

We will show that the sequence appearing in the double recurrence theorem is a good universal weight for the Furstenberg averages. That is, given a system $(X,{\mathcal{F}},\unicode[STIX]{x1D707},T)$ and bounded functions $f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$, there exists a set of full-measure $X_{f_{1},f_{2}}$ in $X$ that is independent of integers $a$ and $b$ and a positive integer $k$ such that, for all $x\in X_{f_{1},f_{2}}$, for every other measure-preserving system $(Y,{\mathcal{G}},\unicode[STIX]{x1D708},S)$ and for each bounded and measurable function $g_{1},\ldots ,g_{k}\in L^{\infty }(\unicode[STIX]{x1D708})$, the averages

$$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)g_{1}\circ S^{n}g_{2}\circ S^{2n}\cdots g_{k}\circ S^{kn}\end{eqnarray}$$

$L^{2}(\unicode[STIX]{x1D708})$

Let M be a closed n-dimensional manifold, and let U be a n-dimensional isolating block such that U is smoothly embedded in M. Let φ be a smooth semi-flow on U and let Λ contained in U, be isolated and invariant under φ Then there exists a semi-flow φ′ on M which extends φ such that φ′ is Morse-Smale outside of U, and no new recurrence is introduced in U. The theorem is true for any finite number of pairwise-disjoint Ui. Furthermore, if Λ is hyperbolic, topologically transitive and is the closure of periodic orbits, then φ′ is an Axiom A flow and is Ω-stable. In dimensions two and three, we have the stronger result that φ′ is structurally stable. Also, as a corollary, we give sufficient conditions for the flow φ′ to be nonsingular. One application of the corollary permits the formation of allowable knots and links in three-manifolds such that there exists a structurally stable nonsingular Morse-Smale flow φ′ which contains the specified knots and links in Ω(φ′) Moreover, the knots and links can be specified to be any combination of attractors, repellers or saddles.

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.

We study the independence density for finite families of finite tuples of sets for continuous actions of discrete groups on compact metrizable spaces. We use it to show that actions with positive naive entropy are Li–Yorke chaotic and untame. In particular, distal actions have zero naive entropy. This answers a question of Lewis Bowen.

We prove that every smooth diffeomorphism group valued cocycle over certain $\mathbb{Z}^{k}$ Anosov actions on tori (and more generally on infranilmanifolds) is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialization apply, via the existing global rigidity results, to maximal Cartan $\mathbb{Z}^{k}$ ($k\geq 3$) actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over $\mathbb{Z}^{k}$ actions with values in diffeomorphism groups which does not require any restrictions on the smallness of the cocycle or on the diffeomorphism group.

We present the general notion of Borel fields of metric spaces and show some properties of such fields. Then we make the study specific to the Borel fields of proper CAT(0) spaces and we show that the standard tools we need behave in a Borel way. We also introduce the notion of the action of an equivalence relation on Borel fields of metric spaces and we obtain a rigidity result for the action of an amenable equivalence relation on a Borel field of proper finite dimensional CAT(0) spaces. This main theorem is inspired by the result obtained by Adams and Ballmann regarding the action of an amenable group on a proper CAT(0) space.

We prove pointwise ergodic theorems for general families of radial averages on semi-simple Lie groups, as well as strong Lp-maximal inequalities for the corresponding maximal functions, for p > 1. The methods of proof make essential use of exponential volume growth on the group, and of spectral considerations utilizing the existence of spectral gaps in measure-preserving actions of simple groups of real rank at least two. We also consider the case of the action by convolutions on the symmetric space, where we establish maximal inequalities of weak-type, going beyond Lp, p > 1. The proof here is based on some new convolution inequalities which are established by direct geometric comparison arguments. These inequalities can be viewed as a quantitative version of the wave front lemma. The maximal inequalities and pointwise ergodic theorems we establish are in sharp contrast to what one would be lead to expect by comparison with the analogous averages on the Abelian groups $\mathbb{R}^n$. We explain how the contrast arises from the difference between the polynomial volume growth on $\mathbb{R}^n$ and the exponential volume growth on semi-simple groups.

We prove that, up to topological conjugacy, every Smale space admits an Ahlfors regular Bowen measure. Bowen’s construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry’s measure, which is Ahlfors regular. By extending Bowen’s construction, we create a tool for transferring the Ahlfors regularity of the Parry measure down to the Bowen measure of the Smale space. An essential part of our method uses a refined notion of approximation graphs over compact metric spaces. Moreover, we obtain new estimates for the Hausdorff, box-counting and Assouad dimensions of a large class of Smale spaces.

Nous étendons des théorèmes sur le spectre de Markoff et de Lagrange connus dans le cas rationnel au cas analogues imaginaires quadratiques. Pour ce faire, nous nous ramenons à un problème de géométrie hyperbolique.