In this paper, we establish uniform oscillation estimates on L^p(X)$L^p(X)$ with p\in (1,\infty )$p\in (1,\infty )$ for the polynomial ergodic averages. This result contributes to a certain problem about uniform oscillation bounds for ergodic averages formulated by Rosenblatt and Wierdl in the early 1990s [Pointwise ergodic theorems via harmonic analysis. Proceedings of Conference on Ergodic Theory (Alexandria, Egypt, 1993) (London Mathematical Society Lecture Notes, 205). Eds. K. Petersen and I. Salama. Cambridge University Press, Cambridge, 1995, pp. 3–151]. We also give a slightly different proof of the uniform oscillation inequality of Jones, Kaufman, Rosenblatt, and Wierdl for bounded martingales [Oscillation in ergodic theory. Ergod. Th. & Dynam. Sys. 18(4) (1998), 889–935]. Finally, we show that oscillations, in contrast to jump inequalities, cannot be seen as an endpoint for r-variation inequalities.

We study the dynamics of the map f:\mathbb {A}^N\to \mathbb {A}^N$f:\mathbb {A}^N\to \mathbb {A}^N$ defined by

\begin{align*} f(\mathbf{X})=A\mathbf{X}^d+\mathbf{b}, \end{align*} $$\begin{align*} f(\mathbf{X})=A\mathbf{X}^d+\mathbf{b}, \end{align*}$$

for A\in \operatorname {SL}_N$A\in \operatorname {SL}_N$ , \mathbf {b}\in \mathbb {A}^N$\mathbf {b}\in \mathbb {A}^N$ , and d\geq 2$d\geq 2$ , a class which specializes to the unicritical polynomials when N=1$N=1$ . In the case k=\mathbb {C}$k=\mathbb {C}$ we obtain lower bounds on the sum of Lyapunov exponents of f, and a statement which generalizes the compactness of the Mandelbrot set. Over \overline {\mathbb {Q}}$\overline {\mathbb {Q}}$ we obtain estimates on the critical height of f, and over algebraically closed fields we obtain some rigidity results for post-critically finite morphisms of this form.

In this paper, we investigate pigeonhole statistics for the fractional parts of the sequence \sqrt {n}$\sqrt {n}$. Namely, we partition the unit circle \mathbb {T} = \mathbb {R}/\mathbb {Z}$\mathbb {T} = \mathbb {R}/\mathbb {Z}$ into N intervals and show that the proportion of intervals containing exactly j points of the sequence (\sqrt {n} + \mathbb {Z})_{n=1}^N$(\sqrt {n} + \mathbb {Z})_{n=1}^N$ converges in the limit as N \to \infty $N \to \infty$. More generally, we investigate how the limiting distribution of the first sN$sN$ points of the sequence varies with the parameter s \geq 0$s \geq 0$. A natural way to examine this is via point processes—random measures on [0,\infty )$[0,\infty )$ which represent the arrival times of the points of our sequence to a random interval from our partition. We show that the sequence of point processes we obtain converges in distribution and give an explicit description of the limiting process in terms of random affine unimodular lattices. Our work uses ergodic theory in the space of affine unimodular lattices, building upon work of Elkies and McMullen [Gaps in \sqrt {n}$\sqrt {n}$ mod 1 and ergodic theory. Duke Math. J. 123 (2004), 95–139]. We prove a generalisation of equidistribution of rational points on expanding horocycles in the modular surface, working instead on nonlinear horocycle sections.

We consider the unique measure of maximal entropy for proper 3-colorings of \mathbb {Z}^{2}$\mathbb {Z}^{2}$ , or equivalently, the so-called zero-slope Gibbs measure. Our main result is that this measure is Bernoulli, or equivalently, that it can be expressed as the image of a translation-equivariant function of independent and identically distributed random variables placed on \mathbb {Z}^{2}$\mathbb {Z}^{2}$ . Along the way, we obtain various estimates on the mixing properties of this measure.

We obtain quenched almost sure invariance principles (with convergence rates) for random Young towers if the average measure of the tail of return times to the base of random towers decays sufficiently fast. We apply our results to some independent and identically distributed perturbations of some non-uniformly expanding maps. These imply that the random systems under study tend to a Brownian motion under various scalings.

Dirichlet’s theorem, including the uniform setting and asymptotic setting, is one of the most fundamental results in Diophantine approximation. The improvement of the asymptotic setting leads to the well-approximable set (in words of continued fractions)

\begin{align*} \mathcal{K}(\Phi):=\{x:a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}; \end{align*} $$\begin{align*} \mathcal{K}(\Phi):=\{x:a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}; \end{align*}$$

\begin{align*} \mathcal{G}(\Phi):=\{x:a_{n}(x)a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}. \end{align*} $$\begin{align*} \mathcal{G}(\Phi):=\{x:a_{n}(x)a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}. \end{align*}$$

\Phi _{1},\Phi _{2}:[1,+\infty )\rightarrow \mathbb {R}^{+}$\Phi _{1},\Phi _{2}:[1,+\infty )\rightarrow \mathbb {R}^{+}$

\mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$\mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$

\mathcal {G}(\Phi )$\mathcal {G}(\Phi )$

\mathcal {K}(\Phi )$\mathcal {K}(\Phi )$

\Phi $\Phi$

\mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$\mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$

\Phi _1$\Phi _1$

In this paper we apply Conley index theory in a covering space of an invariant set S, possibly not isolated, in order to describe the dynamics in S. More specifically, we consider the action of the covering translation group in order to define a topological separation of S which distinguishes the connections between the Morse sets within a Morse decomposition of S. The theory developed herein generalizes the classical connection matrix theory, since one obtains enriched information on the connection maps for non-isolated invariant sets, as well as for isolated invariant sets. Moreover, in the case of an infinite cyclic covering induced by a circle-valued Morse function, one proves that the Novikov differential of f is a particular case of the p-connection matrix defined herein.

For any self-similar measure \mu $\mu$ in \mathbb {R}$\mathbb {R}$ , we show that the distribution of \mu $\mu$ is controlled by products of non-negative matrices governed by a finite or countable graph depending only on the iterated function system of similarities (IFS). This generalizes the net interval construction of Feng from the equicontractive finite-type case. When the measure satisfies the weak separation condition, we prove that this directed graph has a unique attractor. This allows us to verify the multifractal formalism for restrictions of \mu $\mu$ to certain compact subsets of \mathbb {R}$\mathbb {R}$ , determined by the directed graph. When the measure satisfies the generalized finite-type condition with respect to an open interval, the directed graph is finite and we prove that if the multifractal formalism fails at some q\in \mathbb {R}$q\in \mathbb {R}$ , there must be a cycle with no vertices in the attractor. As a direct application, we verify the complete multifractal formalism for an uncountable family of IFSs with exact overlaps and without logarithmically commensurable contraction ratios.

We consider the family \mathrm {MC}_d$\mathrm {MC}_d$ of monic centered polynomials of one complex variable with degree d \geq 2$d \geq 2$, and study the map \widehat {\Phi }_d:\mathrm {MC}_d\to \widetilde {\Lambda }_d \subset \mathbb {C}^d / \mathfrak {S}_d$\widehat {\Phi }_d:\mathrm {MC}_d\to \widetilde {\Lambda }_d \subset \mathbb {C}^d / \mathfrak {S}_d$ which maps each f \in \mathrm {MC}_d$f \in \mathrm {MC}_d$ to its unordered collection of fixed-point multipliers. We give an explicit formula for counting the number of elements of each fiber \widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$\widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$ for every \bar {\unicode{x3bb} } \in \widetilde {\Lambda }_d$\bar {\unicode{x3bb} } \in \widetilde {\Lambda }_d$ except when the fiber \widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$\widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$ contains polynomials having multiple fixed points. This formula is not a recursive one, and is a drastic improvement of our previous result [T. Sugiyama. The moduli space of polynomial maps and their fixed-point multipliers. Adv. Math. 322 (2017), 132–185] which gave a rather long algorithm with some induction processes.

Baker and Rumely, Favre, Rivera, and Letelier, and Chambert-Loir proved an important arithmetic equidistribution theorem for points of small height associated to an adelic measure. To broaden the scope in which arithmetic equidistribution may be employed, we generalize the notion of an adelic measure to that of a quasi-adelic measure and show that arithmetic equidistribution holds for quasi-adelic measures as well. We exhibit examples of non-adelic, quasi-adelic measures arising from the dynamics of quadratic rational maps. In fact, we show that the measures that arise in applications of arithmetic equidistribution theorems are typically not adelic. Finally, we motivate our definition of a quasi-adelic measure by relating it to a seemingly different problem in arithmetic dynamics arising from results of Call, Tate, and Silverman in the study of abelian varieties.

A D_{\infty }$D_{\infty }$-topological Markov chain is a topological Markov chain provided with an action of the infinite dihedral group D_{\infty }$D_{\infty }$. It is defined by two zero-one square matrices A and J satisfying AJ=JA^{\textsf {T}}$AJ=JA^{\textsf {T}}$ and J^2=I$J^2=I$. A flip signature is obtained from symmetric bilinear forms with respect to J on the eventual kernel of A. We modify Williams’ decomposition theorem to prove the flip signature is a D_{\infty }$D_{\infty }$-conjugacy invariant. We introduce natural D_{\infty }$D_{\infty }$-actions on Ashley’s eight-by-eight and the full two-shift. The flip signatures show that Ashley’s eight-by-eight and the full two-shift equipped with the natural D_{\infty }$D_{\infty }$-actions are not D_{\infty }$D_{\infty }$-conjugate. We also discuss the notion of D_{\infty }$D_{\infty }$-shift equivalence and the Lind zeta function.

Let M be a compact smooth manifold without boundary. Based on results by Good and Meddaugh [Invent. Math. 220 (2020), 715–736], we prove that a strong distributional chaos is C^0$C^0$ -generic in the space of continuous self-maps (respectively, homeomorphisms) of M. The results contain answers to questions by Li, Li and Tu [Chaos 26 (2016), 093103] and Moothathu [Topology Appl. 158 (2011), 2232–2239] in the zero-dimensional case. A related counter-example on the chain components under shadowing is also given.

Furstenberg–Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure-preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogs of those algebraic and geometric descriptions are sought. Although the unconditional dichotomy and the characterizations are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying groups and spaces. The aim of this article is to remove these restrictions in the relative situation and establish a Furstenberg–Zimmer structure theory in full generality. As an independent byproduct, we establish a connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi.

Motivated by fractal geometry of self-affine carpets and sponges, Feng and Huang [J. Math. Pures Appl. 106(9) (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems, and proved variational principles for them. We introduce a new approach to this theory. Our new definitions of weighted topological entropy and pressure are very different from the original definitions of Feng and Huang. The equivalence of the two definitions seems highly non-trivial. Their equivalence can be seen as a generalization of the dimension formula for the Bedford–McMullen carpet in purely topological terms.

In this paper it is shown that for a minimal system (X,T)$(X,T)$ and d,k\in \mathbb {N}$d,k\in \mathbb {N}$ , if (x,x_{i})$(x,x_{i})$ is regionally proximal of order d for 1\leq i\leq k$1\leq i\leq k$ , then (x,x_{1},\ldots ,x_{k})$(x,x_{1},\ldots ,x_{k})$ is (k+1)$(k+1)$ -regionally proximal of order d. Meanwhile, we introduce the notion of \mathrm {IN}^{[d]}$\mathrm {IN}^{[d]}$ -pair: for a dynamical system (X,T)$(X,T)$ and d\in \mathbb {N}$d\in \mathbb {N}$ , a pair (x_{0},x_{1})\in X\times X$(x_{0},x_{1})\in X\times X$ is called an \mathrm {IN}^{[d]}$\mathrm {IN}^{[d]}$ -pair if for any k\in \mathbb {N}$k\in \mathbb {N}$ and any neighborhoods U_{0} ,U_{1} $U_{0} ,U_{1}$ of x_{0}$x_{0}$ and x_{1}$x_{1}$ respectively, there exist different (p_{1}^{(i)},\ldots ,p_{d}^{(i)})\in \mathbb {N}^{d} , 1\leq i\leq k$(p_{1}^{(i)},\ldots ,p_{d}^{(i)})\in \mathbb {N}^{d} , 1\leq i\leq k$ , such that

\begin{align*} \bigcup_{i=1}^{k}\{ p_{1}^{(i)}\epsilon(1)+\cdots+p_{d}^{(i)} \epsilon(d):\epsilon(j)\in \{0,1\},1\leq j\leq d\}\backslash \{0\}\in \mathrm{Ind}(U_{0},U_{1}), \end{align*} $$\begin{align*} \bigcup_{i=1}^{k}\{ p_{1}^{(i)}\epsilon(1)+\cdots+p_{d}^{(i)} \epsilon(d):\epsilon(j)\in \{0,1\},1\leq j\leq d\}\backslash \{0\}\in \mathrm{Ind}(U_{0},U_{1}), \end{align*}$$

\mathrm {Ind}(U_{0},U_{1})$\mathrm {Ind}(U_{0},U_{1})$

(U_{0},U_{1})$(U_{0},U_{1})$

\mathrm {IN}^{[d]}$\mathrm {IN}^{[d]}$

We prove the convergence and ergodicity of a wide class of real and higher-dimensional continued fraction algorithms, including folded and \alpha $\alpha$ -type variants of complex, quaternionic, octonionic, and Heisenberg continued fractions, which we combine under the framework of Iwasawa continued fractions. The proof is based on the interplay of continued fractions and hyperbolic geometry, the ergodicity of geodesic flow in associated modular manifolds, and a variation on the notion of geodesic coding that we refer to as geodesic marking. As a corollary of our study of markable geodesics, we obtain a generalization of Serret’s tail-equivalence theorem for almost all points. The results are new even in the case of some real and complex continued fractions.

We show that the image of a subshift X under various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, respectively a sofic subshift, if and only if so is X. Similarly, let G be a countable monoid and let A, B be Artinian modules over a ring. We prove that for every closed subshift submodule \Sigma \subset A^G$\Sigma \subset A^G$ and every injective G-equivariant uniformly continuous module homomorphism \tau \colon \! \Sigma \to B^G$\tau \colon \! \Sigma \to B^G$, a subshift \Delta \subset \Sigma $\Delta \subset \Sigma$ is of finite type, respectively sofic, if and only if so is the image \tau (\Delta )$\tau (\Delta )$. Generalizations for admissible group cellular automata over admissible Artinian group structure alphabets are also obtained.