A power-bounded operator T satisfying $\sup _n n\lVert T^n-T^{n+1}\rVert <\infty $ is a Ritt operator. For such operators, we study the generalized square function $Q_{\alpha ,s,r}^Tf=(\sum _n n^{\alpha } |T^n(I-T)^rf|^s)^{1/s}$. It is known that when T is a positive contraction and a Ritt operator on $L^p$, $1<p<\infty $, then for any integer $r\ge 1$, the square function $Q_{2r-1,2,r}^Tf$ defines a bounded operator [17] on $L^p$. In this work, we extend the theory to the endpoint case $p=1$. We show that if T is a Ritt operator on $L^1$, then the generalized square function $Q_{\alpha ,s,r}^Tf $ is bounded on $L^1$ whenever $\alpha +1<sr$. In the particular setting where T is a convolution operator of the form $T_{\mu }=\sum _k \mu (k) U^kf$, with $\mu $ a probability measure on $\mathbb Z$ and U the composition operator induced by an invertible measure-preserving transformation, we provide sufficient conditions on $\mu $ under which $Q_{2r-1,2,r}^{T_{\mu }}f$ is of weak type $(1,1)$, for $r>0$. We also establish bounds for variational and oscillation norms, $\lVert n^{\beta } T^n(1-T)^r\rVert _{v(s)}$ and $\lVert n^{\beta } T^n(1-T)^r\rVert _{o(s)}$, for Ritt operators, highlighting endpoint behavior.

Let $ ([0,1]^d,T,\mu ) $ be a measure-preserving dynamical system so that the correlations decay exponentially for Hölder continuous functions. Suppose that $ \mu $ is absolutely continuous with a density function $ h\in L^q(\mathcal L^d) $ for some $ q>1 $, where $ \mathcal L^d $ is the $ d $-dimensional Lebesgue measure. Under suitable conditions on the underlying dynamical system, we obtain a strong dynamical Borel–Cantelli lemma for recurrence: for any sequence $ \{R_n\} $ of hyperrectangles centered at the origin, with sides parallel to the axes and diameter going to $0$ as $n\to \infty $, where $ \mathbf {x}\in [0,1]^d $ and $ R_n+\mathbf {x} $ is the translation of $ R_n $. The result applies to the Gauss map, $\beta $-transformations, and expanding toral endomorphisms.

We prove that among the set of pairs ($C^2$-diffeomorphism, $C^1$-potential), there exists a $C^1$-open and dense subset such that either the Lagrange spectrum is finite and the dynamics is a Morse–Smale diffeomorphism or the Lagrange spectrum has positive Hausdorff dimension and the system has positive topological entropy. We also prove that such dichotomy does not hold for typical systems when replacing the Lagrange by the Markov spectrum.

Polynomial Julia sets with tree structure, typically Hubbard trees, play an important role in holomorphic dynamics. In this paper, we study the dynamics of ${f_\alpha (z)=z^2+\alpha \bar {z}}$ for $\alpha $ being real and the Julia sets being trees. We show that all such $\alpha $ form an interval $[1,4]$. This answers a question of G. Sienra. We further show that $f_\alpha $ exhibits non-trivial dynamics on the Fatou set which equals to the escaping set.

We classify generic unfoldings of germs of antiholomorphic diffeomorphisms with a parabolic point of codimension k (i.e. a fixed point of multiplicity $k+1$) under conjugacy. Such generic unfoldings depend real analytically on k real parameters. A preparation of the unfolding allows to identify real analytic canonical parameters, which are preserved by any conjugacy between two prepared generic unfoldings. A modulus of analytic classification is defined, which is an unfolding of the modulus assigned to the antiholomorphic parabolic point. Since the second iterate of such a germ is a real unfolding of a holomorphic parabolic point, the modulus is a special form of an unfolding of the Écalle–Voronin modulus of the second iterate of the antiholomorphic parabolic germ. We also solve the problem of the existence of an antiholomorphic square root to a germ of a generic analytic unfolding of a holomorphic parabolic germ.

Let G be a connected Lie group and $\mathrm {Sub}_G$ be the space of closed subgroups of G equipped with the Chabauty topology. In this article, we investigate the existence of invariant random subgroups of G supported on various orbits of the conjugation action of G on $\mathrm {Sub}_G$.

We prove that a formal curve $\Gamma $ that is invariant by a $C^{\infty }$ vector field $\xi $ of $\mathbb R^{m}$ has a geometrical realization, as soon as the Taylor expansion of $\xi $ is not identically zero along $\Gamma $. This means that there is a trajectory $\gamma \subset \mathbb R^{m}$ of $\xi $ which is asymptotic to $\Gamma $. This result solves a natural question proposed by Bonckaert [Smooth invariant curves of singularities of vector fields in R3. Ann. Inst. Henri Poincaré 3(2) (1986), 111–183] nearly forty years ago. We also construct an invariant $C^0$ manifold S in some open horn around $\Gamma $ which is composed entirely of trajectories asymptotic to $\Gamma $ and contains the germ of any such trajectory. If $\xi $ is analytic, we prove that there exists a trajectory $\gamma $ asymptotic to $\Gamma $ which is, moreover, non-oscillating with respect to subanalytic sets.

We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As two applications, we first obtain the upcrossing inequalities with exponential decay of ergodic averages and then provide an explicit bound on the convergence rate such that the ergodic averages with strongly continuous regular group actions are metastable (or locally stable) on a large interval. Before exploiting the transference techniques, we actually obtain a stronger result—the jump estimates on a metric space with a measure not necessarily doubling. The ideas or techniques involve martingale theory, non-doubling Calderón–Zygmund theory, almost orthogonality argument, and some delicate geometric argument involving the balls and the cubes on a group equipped with a not necessarily doubling measure.

We introduce a correlation number for two strictly positive locally Hölder continuous independent potentials with strong entropy gaps at infinity on a topologically mixing countable state Markov shift with big images and pre-images (BIP) property. We define in this way a correlation number for pairs of cusped Hitchin representations. Furthermore, we explore the connection between the correlation number and the Manhattan curve, along with several rigidity properties of this correlation number.

In this paper, we introduce a new technique to study the distribution in residue classes of sets of integers with digit and sum-of-digits restrictions. From our main theorem, we derive a necessary and sufficient condition for integers with missing digits to be uniformly distributed in arithmetic progressions, extending previous results going back to the work of Erdős, Mauduit and Sárközy. Our approach uses Markov chains and does not rely on Fourier analysis as many results of this nature do. Our results apply more generally to the class of multiplicatively invariant sets of integers. This class, defined by Glasscock, Moreira and Richter using symbolic dynamics, is an integer analogue to fractal sets and includes all missing digits sets. We address uniform distribution in this setting, partially answering an open question posed by the same authors.

We show that the canonical anticommutation relations (CAR) algebra admits a Cantor spectrum $\mathrm {C}^\ast $-diagonal that is not conjugate to the standard AF diagonal. We obtain this by classification theory of $\mathrm {C}^\ast $-algebras, and the diagonal arises by realizing the CAR algebra as the crossed product of a free minimal action on the Cantor space, where the acting group is the product of a locally finite group with the infinite dihedral group. The main ingredient in the construction is a binary subshift associated to the well-known regular paper-folding sequence. Moreover, we show that the CAR algebra in fact admits countably many, pairwise non-conjugate, Cantor spectrum diagonals which are distinguished by the different values of their diagonal dimension, as defined by Li, Liao and the second named author.

It is known that for a uniform morphic sequence $\boldsymbol u = \langle u_n\rangle _{n=0}^\infty $ and an algebraic number $\beta $ such that $|\beta |>1$, the number $[\![ \boldsymbol {u} ]\!] _\beta :=\sum _{n=0}^\infty ({u_n}/{\beta ^n})$ either lies in $\mathbb Q(\beta )$ or is transcendental. In this paper, we show a similar rational–transcendental dichotomy for sequences defined by irreducible Pisot morphisms on binary alphabets. Subject to the Pisot conjecture (an irreducible Pisot morphism has pure discrete spectrum), we generalise the latter result to arbitrary finite alphabets. In certain cases, we are able to show transcendence of $[\![ \boldsymbol {u}]\!] _{\beta }$ outright. In particular, for $k\geq 2$, if $\boldsymbol u$ is the k-Bonacci word, then $[\![ \boldsymbol {u}]\!] _{\beta }$ is transcendental.

An avoshift is a subshift where for each set C from a suitable family of subsets of the shift group, the set of all possible valid extensions of a globally valid pattern on C to the identity element is determined by a bounded subpattern. This property is shared (for various families of sets C) by, for example, cellwise quasigroup shifts, totally extremally permutive (TEP) subshifts, and subshifts of finite type (SFTs) with a safe symbol. In this paper, we concentrate on avoshifts on polycyclic groups, when the sets C are what we call ‘inductive intervals’. We show that then, avoshifts are a recursively enumerable subset of subshifts of finite type. Furthermore, we can effectively compute lower-dimensional projective subdynamics and certain factors (avofactors), and we can decide equality and inclusion for subshifts in this class. These results were previously known for group shifts, but our class also covers many non-algebraic examples as well as many SFTs without dense periodic points. The theory also yields new proofs of decidability of inclusion for SFTs on free groups, and SFTness of subshifts with the topological strong spatial mixing property.

Piecewise contractions (PCs) are piecewise smooth maps that decrease the distance between pairs of points in the same domain of continuity. The dynamics of a variety of systems is described by PCs. During the last decade, much effort has been devoted to proving that in parameterized families of one-dimensional PCs, the $\omega $-limit set of a typical PC consists of finitely many periodic orbits while there exist atypical PCs with Cantor $\omega $-limit sets. In this article, we extend these results to the multi-dimensional case. More precisely, we provide criteria to show that an arbitrary family $\{f_{\mu }\}_{\mu \in U}$ of locally bi-Lipschitz piecewise contractions $f_\mu :X\to X$ defined on a compact metric space X is asymptotically periodic for Lebesgue almost every parameter $\mu $ running over an open subset U of the M-dimensional Euclidean space $\mathbb {R}^M$. As a corollary of our results, we prove that piecewise affine contractions of $\mathbb {R}^d$ defined in generic polyhedral partitions are asymptotically periodic.

We construct various novel and elementary examples of dynamics with metric attractors that have intermingled basins. A main ingredient is the introduction of random walks along orbits of a given dynamical system. We develop the theory and use it in particular to provide examples of thick metric attractors with intermingled basins.

For full shifts on finite alphabets, Coelho and Quas [Criteria for $\overline {d}$-continuity. Trans. Amer. Math. Soc. 350(8) (1998), 3257–3268] showed that the map that sends a Hölder continuous potential $\phi $ to its equilibrium state $\mu _\phi $ is $\overline {d}$-continuous. We extend this result to the setting of full shifts on countable (infinite) alphabets. As part of the proof, we show that the map that sends a strongly positive recurrent potential to its normalization is continuous for potentials on mixing countable state Markov shifts.

We describe a new construction of equilibrium states for a class of partially hyperbolic systems. This generalizes our construction for Gibbs measures in the uniformly hyperbolic setting. This more general setting introduces new issues that we need to address carefully, in particular requiring additional assumptions on the transformation. We treat two cases: either the centre-stable manifold satisfies a bounded expansion condition; or the centre-unstable manifold satisfies a subexponential contraction condition which appears new in the context of equilibrium state constructions. The problem of constructing equilibrium states was previously raised by Pesin and Sinai and by Dolgopyat for the particular case of u-Gibbs measures, and by Climenhaga, Pesin and Zelerowicz for other equilibrium states.

In this work, we introduce the type and typeset invariants for equicontinuous group actions on Cantor sets; that is, for generalized odometers. These invariants are collections of equivalence classes of asymptotic Steinitz numbers associated to the action. We show the type is an invariant of the return equivalence class of the action. We introduce the notion of commensurable typesets and show that two actions which are return equivalent have commensurable typesets. Examples are given to illustrate the properties of the type and typeset invariants. The type and typeset invariants are used to define homeomorphism invariants for solenoidal manifolds.

We introduce Feldman–Katok convergence for invariant measures of a topological dynamical system. This can be seen as a counterpart to the convergence with respect to the $\bar {f}$-metric for finite-state stationary processes (shift-invariant measures on a symbolic space). Feldman–Katok convergence is based on a dynamically defined Feldman–Katok pseudometric. This convergence is stronger than weak$^*$ convergence. We prove that Feldman–Katok convergence preserves ergodicity and makes the Kolmogorov–Sinai entropy lower semicontinuous, thereby preserving zero entropy. We apply our findings to non-hyperbolic (having at least one vanishing Lyapunov exponent) ergodic measures constructed using the GIKN method as axiomatized by Bonatti, Díaz and Gorodetski [Nonlinearity, 23 (2010), 687–705]. The GIKN method, originally introduced by Gorodetski, Ilyashenko, Kleptsyn and Nalsky [Functional Analysis and its Applications, 39 (2005), 21–30], has been widely adapted to produce non-hyperbolic ergodic measures for diffeomorphisms of compact manifolds. We prove that an ergodic measure satisfying the conditions provided by the axiomatized GIKN method is the Feldman–Katok limit of a sequence of periodic measures, which implies that it is either a periodic measure or a loosely Kronecker measure (a measure Kakutani equivalent to an aperiodic ergodic rotation on a compact group) and has zero entropy. This classifies all these measures up to Kakutani equivalence and confirms that geometric constructions of non-hyperbolic measures via periodic approximations based on the axiomatized GIKN method presented in Bonatti et al. [op. cit.] systematically produce zero-entropy systems.

This article investigates the genericity of ergodic probability measures for the geodesic flow on Riemannian manifolds. We demonstrate that if the metric splits as a product metric within a tubular neighborhood of a geodesically complete submanifold containing a closed geodesic, then the closure of the set of ergodic measures does not encompass all invariant probability measures. Our findings notably provide an answer to the question of genericity of ergodic measures concerning a specific example of 3-manifold introduced by Gromov.