We compactify and regularise the space of initial values of a planar map with a quartic invariant and use this construction to prove its integrability in the sense of algebraic entropy. The system has certain unusual properties, including a sequence of points of indeterminacy in $\mathbb {P}^{1}\!\times \mathbb {P}^{1}$ . These indeterminacy points lie on a singular fibre of the mapping to a corresponding QRT system and provide the existence of a one-parameter family of special solutions.

Let f be a germ of a holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in ${\mathbb C}$ (i.e. $f(0) = 0, f'(0) = e^{2\pi i \alpha }, \alpha \in {\mathbb R} - {\mathbb Q}$ ). Pérez-Marco [Fixed points and circle maps. Acta Math.179(2) (1997), 243–294] showed the existence of a unique continuous monotone one-parameter family of non-trivial invariant full continua containing the fixed point called Siegel compacta, and gave a correspondence between germs and families $(g_t)$ of circle maps obtained by conformally mapping the complement of these compacts to the complement of the unit disk. The family of circle maps $(g_t)$ is the orbit of a locally defined semigroup $(\Phi _t)$ on the space of analytic circle maps, which we show has a well-defined infinitesimal generator X. The explicit form of X is obtained by using the Loewner equation associated to the family of hulls $(K_t)$ . We show that the Loewner measures $(\mu _t)$ driving the equation are 2-conformal measures on the circle for the circle maps $(g_t)$ .

We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order $\ell $ of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to $0$ under the dynamics of the tower for corresponding $\ell $ . That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when $\ell $ tends to $\infty $ . We also prove the convergence of the drifts to a finite limit, which can be expressed purely in terms of the limiting tower, which corresponds to a Feigenbaum map with a flat critical point.

We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis [Logarithm laws for flows on homogeneous spaces. Invent. Math.138(3) (1999), 451–494] resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. Selecta Math.10 (2004), 479–523], hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson–Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW’s sufficient conditions for extremality. In the first of this series of papers [Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. Selecta Math.24(3) (2018), 2165–2206], we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, as well as proving the ‘inherited exponent of irrationality’ version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson–Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying.

We show that condenser capacity varies continuously under holomorphic motions, and the corresponding family of the equilibrium measures of the condensers is continuous with respect to the weak-star convergence. We also study the behavior of uniformly perfect sets under holomorphic motions.

We focus on various dynamical invariants associated to monomial correspondences on toric varieties, using algebraic and arithmetic geometry. We find a formula for their dynamical degrees, relate the exponential growth of the degree sequences to a strict log-concavity condition on the dynamical degrees and compute the asymptotic rate of the growth of heights of points of such correspondences.

We consider the dynamics of complex rational maps on $\widehat{\mathbb{C}}$. We prove that, after reducing their orbits to a fixed number of positive values representing the Fubini–Study distances between finitely many initial elements of the orbit and the origin, ergodic properties of the rational map are preserved.

Our main result states that, under an exponential map whose Julia set is the whole complex plane, on each piecewise smooth Jordan curve there is a point whose orbit is dense. This has consequences for the boundaries of nice sets, used in induction methods to study ergodic and geometric properties of the dynamics.

We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show a simple and natural example of a homogeneous polynomial with an orbit that is at the same time $\unicode[STIX]{x1D6FF}$-dense (the orbit meets every ball of radius $\unicode[STIX]{x1D6FF}$), weakly dense and such that $\unicode[STIX]{x1D6E4}\cdot \text{Orb}_{P}(x)$ is dense for every $\unicode[STIX]{x1D6E4}\subset \mathbb{C}$ that either is unbounded or has 0 as an accumulation point. Moreover, we generalize the construction to arbitrary infinite-dimensional separable Banach spaces. To prove this, we study Julia sets of homogeneous polynomials on Banach spaces.

We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $(\mathbb{C}^{2},0)$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$, and that at least one of these parabolic manifolds is or contains a parabolic curve.

We consider a certain two-parameter family of automorphisms of the affine plane over a complete, locally compact non-Archimedean field. Each of these automorphisms admits a chaotic attractor on which it is topologically conjugate to a full two-sided shift map, and the attractor supports a unit Borel measure which describes the distribution of the forward orbit of Haar-almost all points in the basin of attraction. We also compute the Hausdorff dimension of the attractor, which is non-integral.

Let $X$ be a nonempty set and ${\mathcal{P}}(X)$ the power set of $X$. The aim of this paper is to identify the unital subrings of ${\mathcal{P}}(X)$ and to compute its cardinality when it is finite. It is proved that any topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$, where $\unicode[STIX]{x1D70F}^{c}=\{U^{c}\mid U\in \unicode[STIX]{x1D70F}\}$, is a unital subring of ${\mathcal{P}}(X)$. It is also shown that $X$ is finite if and only if any unital subring of ${\mathcal{P}}(X)$ is a topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$ if and only if the set of unital subrings of ${\mathcal{P}}(X)$ is finite. As a consequence, if $X$ is finite with cardinality $n\geq 2$, then the number of unital subrings of ${\mathcal{P}}(X)$ is equal to the $n$th Bell number and the supremum of the lengths of chains of unital subalgebras of ${\mathcal{P}}(X)$ is equal to $n-1$.

We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.

We study topological properties of the escaping endpoints and fast escaping endpoints of the Julia set of complex exponential $\exp (z)+a$ when $a\in (-\infty ,-1)$. We show neither space is homeomorphic to the whole set of endpoints. This follows from a general result stating that for every transcendental entire function $f$, the escaping Julia set $I(f)\cap J(f)$ is first category.

We discuss the relation between the existence of fixed points of the Ruelle operator acting on different Banach spaces, and Sullivan’s conjecture in holomorphic dynamics.

We study the iteration of transcendental self-maps of $\mathcal{C}^*:\=\mathcal{C}\{0}$, that is, holomorphic functions $\fnof:\mathcal{C}^*:\rarr\mathcal{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0},\infin$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\mathcal{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

We develop some foundational results in a higher-dimensional foliated Mori theory, and show how these results can be used to prove a structure theorem for the Kleiman–Mori cone of curves in terms of the numerical properties of $K_{{\mathcal{F}}}$ for rank 2 foliations on threefolds. We also make progress toward realizing a minimal model program (MMP) for rank 2 foliations on threefolds.

We prove that all transcendental entire functions have infinite topological entropy.

Let $f$ be an $n$-dimensional holomorphic map defined in a neighborhood of the origin such that the origin is an isolated fixed point of all of its iterates, and let ${\mathcal{N}}_{M}(f)$ denote the number of periodic orbits of $f$ of period $M$ hidden at the origin. Gorbovickis gives an efficient way of computing ${\mathcal{N}}_{M}(f)$ for a large class of holomorphic maps. Inspired by Gorbovickis’ work, we establish a similar method for computing ${\mathcal{N}}_{M}(f)$ for a much larger class of holomorphic germs, in particular, having arbitrary Jordan matrices as their linear parts. Moreover, we also give another proof of the result of Gorbovickis [On multi-dimensional Fatou bifurcation. Bull. Sci. Math.138(3)(2014) 356–375] using our method.

Let $B$ be a rational function of degree at least two that is neither a Lattès map nor conjugate to $z^{\pm n}$ or $\pm T_{n}$. We provide a method for describing the set $C_{B}$ consisting of all rational functions commuting with $B$. Specifically, we define an equivalence relation $\underset{B}{{\sim}}$ on $C_{B}$ such that the quotient $C_{B}/\underset{B}{{\sim}}$ possesses the structure of a finite group $G_{B}$, and describe generators of $G_{B}$ in terms of the fundamental group of a special graph associated with $B$.