We define a free holomorphic function to be a function that is locally, with respect to the free topology, a bounded nc-function. We prove that free holomorphic functions are the functions that are locally uniformly approximable by free polynomials. We prove a realization formula and an Oka-Weil theorem for free analytic functions.

In this paper we prove a commutative algebraic extension of a generalized Skolem–Mahler– Lech theorem. Let $A$ be a finitely generated commutative $K$–algebra over a field of characteristic 0, and let $\sigma$ be a $K$–algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that ${{\sigma }^{m}}(I)\,\supseteq \,J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $\sigma \,\in \,\text{Au}{{\text{t}}_{k}}(X)$, and $Y$ and $Z$ any two closed subschemes of $X$, the set of integers $m$ with ${{\sigma }^{m}}(Z)\,\subseteq \,Y$ is as above. We present examples showing that this result may fail to hold if the affine scheme $X$ is not of finite type, or if $X$ is of finite type but the field $K$ has positive characteristic.

In this article we study the geometry of the eigenvarieties of unitary groups at points corresponding to tempered non-stable representations with an anti-ordinary (a.k.a evil) refinement. We prove that, except in the case where the Galois representation attached to the automorphic form is a sum of characters, the eigenvariety is non-smooth at such a point, and that (under some additional hypotheses) its tangent space is big enough to account for all the relevant Selmer group. We also study the local reducibility locus at those points, proving that in general, in contrast with the case of the eigencurve, it is a proper subscheme of the fiber of the eigenvariety over the weight space.

We study the hyperspace dynamics induced fromgeneric continuous maps and fromgeneric homeomorphisms of the Cantor space, with emphasis on the notions of Li– Yorke chaos, distributional chaos, topological entropy, chain continuity, shadowing, and recurrence.

We study a multimarginal optimal transportation problem in one dimension. For a symmetric, repulsive cost function, we show that, given a minimizing transport plan, its symmetrization is induced by a cyclical map, and that the symmetric optimal plan is unique. The class of costs that we consider includes, in particular, the Coulomb cost, whose optimal transport problem is strictly related to the strong interaction limit of Density Functional Theory. In this last setting, our result justifies some qualitative properties of the potentials observed in numerical experiments.

It is proved that if $G\,=\,{{G}_{1}}\,{{*}_{{{G}_{3}}}}\,{{G}_{2}}$ is free product of probability measure preserving $s$–regular ergodic discrete groupoids amalgamated over an amenable subgroupoid ${{G}_{3}}$, then the sofic dimension $s(G)$ satisfies the equality

$$s(G)=\mathfrak{h}(G_{1}^{0})s({{G}_{1}})+\mathfrak{h}(G_{2}^{0})s\left( {{G}_{2}} \right)-\,\mathfrak{h}(G_{3}^{0})s({{G}_{3}}),$$

where $\mathfrak{h}$ is the normalized Haar measure on $G$.

We show that if $u$ and $v$ are any two unitaries in a unital ${{C}^{*}}$–algebra such that $\left\| uv\,-\,vu \right\|\,<\,2$ and $uv{{u}^{*}}{{v}^{*}}$ commutes with $u$ and $v$, then the ${{C}^{*}}$–subalgebra ${{A}_{u,v}}$ generated by $u$ and $v$ is isomorphic to a quotient of some rotation algebra ${{A}_{\theta }}$, provided that ${{A}_{u,v}}$ has a unique tracial state. We also show that the Exel trace formula holds in any unital ${{C}^{*}}$–algebra. Let $\theta \,\in \,\left( -1/2,\,1/2 \right)$ be a real number. For any $\in \,>\,0$, we prove that there exists $\delta \,>\,0$ satisfying the following: if $u$ and $v$ are two unitaries in any unital simple ${{C}^{*}}$–algebra $A$ with tracial rank zero such that

$$\left\| uv\,-\,{{e}^{2\pi i\theta }}vu \right\|\,<\,\delta \,\,\,\text{and}\,\,\frac{1}{2\pi i}\tau \left( \log \left( uv{{u}^{*}}{{v}^{*}} \right) \right)\,=\,\theta ,$$

for all tracial states $\tau$ of $A$, then there exists a pair of unitaries $\widetilde{u}$ and $\widetilde{v}$ in $A$ such that

$$\widetilde{u}\widetilde{v}\,=\,{{e}^{2\pi i\theta }}\widetilde{v}\widetilde{u},\,\,\,\,\,\,\,\left\| u\,-\,\widetilde{u} \right\|\,<\,\in \,\,\,\text{and}\,\,\left\| v\,-\,\widetilde{v} \right\|\,<\,\in.$$

We study the Mahler measures of certain families of Laurent polynomials in two and three variables. Each of the known Mahler measure formulas for these families involves $L$–values of at most one newform and/or at most one quadratic character. In this paper we show, either rigorously or numerically, that the Mahler measures of some polynomials are related to $L$–values of multiple newforms and quadratic characters simultaneously. The results suggest that the number of modular $L$–values appearing in the formulas significantly depends on the shape of the algebraic value of the parameter chosen for each polynomial. As a consequence, we also obtain new formulas relating special values of hypergeometric series evaluated at algebraic numbers to special values of $L$–functions.

We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.