On $ n $-tuplewise IP-sensitivity and thick sensitivity

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M2">\begin{document}$ (X,T) $\end{document}</tex-math></inline-formula> be a topological dynamical system and <inline-formula><tex-math id="M3">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>. We say that <inline-formula><tex-math id="M4">\begin{document}$ (X,T) $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive (resp. <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive) if there exists a constant <inline-formula><tex-math id="M7">\begin{document}$ \delta&gt;0 $\end{document}</tex-math></inline-formula> with the property that for each non-empty open subset <inline-formula><tex-math id="M8">\begin{document}$ U $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M9">\begin{document}$ X $\end{document}</tex-math></inline-formula>, there exist <inline-formula><tex-math id="M10">\begin{document}$ x_1,x_2,\dotsc,x_n\in U $\end{document}</tex-math></inline-formula> such that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Bigl\{k\in \mathbb{N}\colon \min\limits_{1\le i&lt;j\le n}d(T^k x_i,T^k x_j)&gt;\delta\Bigr\} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is an IP-set (resp. a thick set).</p><p style='text-indent:20px;'>We obtain several sufficient and necessary conditions of a dynamical system to be <inline-formula><tex-math id="M11">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive or <inline-formula><tex-math id="M12">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive and show that any non-trivial weakly mixing system is <inline-formula><tex-math id="M13">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive for all <inline-formula><tex-math id="M14">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>, while it is <inline-formula><tex-math id="M15">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive if and only if it has at least <inline-formula><tex-math id="M16">\begin{document}$ n $\end{document}</tex-math></inline-formula> minimal points. We characterize two kinds of sensitivity by considering some kind of factor maps. We introduce the opposite side of pairwise IP-sensitivity and pairwise thick sensitivity, named (almost) pairwise IP<inline-formula><tex-math id="M17">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuity and (almost) pairwise syndetic equicontinuity, and obtain dichotomies results for them. In particular, we show that a minimal system is distal if and only if it is pairwise IP<inline-formula><tex-math id="M18">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuous. We show that every minimal system admits a maximal almost pairwise IP<inline-formula><tex-math id="M19">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuous factor and admits a maximal pairwise syndetic equicontinuous factor, and characterize them by the factor maps to their maximal distal factors.</p>

Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centres

In this paper we consider the unfolding of saddle-node \[ X= \frac{1}{xU_a(x,y)}\Big(x(x^{\mu}-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \] parametrized by $(\varepsilon,\,a)$ with $\varepsilon \approx 0$ and $a$ in an open subset $A$ of $ {\mathbb {R}}^{\alpha },$ and we study the Dulac time $\mathcal {T}(s;\varepsilon,\,a)$ of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative $\partial _s\mathcal {T}(s;\varepsilon,\,a)$ tends to $-\infty$ as $(s,\,\varepsilon )\to (0^{+},\,0)$ uniformly on compact subsets of $A.$ This result is addressed to study the bifurcation of critical periods in the Loud's family of quadratic centres. In this regard we show (theorem 1.2) that no bifurcation occurs from certain semi-hyperbolic polycycles.

Regulous Functions over Real Closed Fields

Let $X$ be a quasi-projective algebraic variety over a real closed field $R$, and let $f \colon U \to R$ be a function defined on an open subset $U$ of the set $X(R)$ of $R$-rational points of $X$. Assume that either the function $f$ is locally semialgebraic or the field $R$ is uncountable. If for every irreducible algebraic curve $C \subset X$ the restriction $f|_{U \cap C}$ is continuous and admits a rational representation, then $f$ is continuous and admits a rational representation. There are also suitable versions of this theorem with algebraic curves replaced by algebraic arcs. Heretofore, results of such a type have been known only for $R={\mathbb{R}}$. The transition from ${\mathbb{R}}$ to $R$ is not automatic at all and requires new methods.

Floer Theory of Higher Rank Quiver 3-folds

We study threefolds Y fibred by $$A_m$$ A m -surfaces over a curve S of positive genus. An ideal triangulation of S defines, for each rank m, a quiver $$Q(\Delta _m)$$ Q ( Δ m ) , hence a $$CY_3$$ C Y 3 -category $$\mathcal {C}(W)$$ C ( W ) for any potential W on $$Q(\Delta _m)$$ Q ( Δ m ) . We show that for $$\omega $$ ω in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of $$(Y,\omega )$$ ( Y , ω ) is quasi-isomorphic to $$(\mathcal {C},W_{[\omega ]})$$ ( C , W [ ω ] ) for a certain generic potential $$W_{[\omega ]}$$ W [ ω ] . This partially establishes a conjecture of Goncharov (in: Algebra, geometry, and physics in the 21st century, Birkhäuser/Springer, Cham, 2017) concerning ‘categorifications’ of cluster varieties of framed $${\mathbb {P}}GL_{m+1}$$ P G L m + 1 -local systems on S, and gives a symplectic geometric viewpoint on results of Gaiotto et al. (Ann Henri Poincaré 15(1):61–141, 2014) on ‘theories of class $${\mathcal {S}}$$ S ’.

On the uniqueness of inverse problems for the reduced wave equation with unknown embedded obstacles

In this paper, we consider inverse problems associated with the reduced wave equation on a bounded domain Ω belongs to R^N (N >= 2) for the case where unknown obstacles are embedded in the domain Ω. We show that, if both the leading and 0-order coefficients in the equation are a priori known to be piecewise constant functions, then both the coefficients and embedded obstacles can be simultaneously recovered in terms of the local Dirichlet-to-Neumann map defined on an arbitrary small open subset of the boundary \partial Ω. The method depends on a well-defined coupled PDE-system constructed for the reduced wave equations in a sufficiently small domain and the singularity analysis of solutions near the interface for the model.

Hyperbolicity for Log Smooth Families with Maximal Variation

We prove that the base space of a log smooth family of log canonical pairs of log general type is of log general type as well as algebraically degenerate, when the family admits a relative good minimal model over a Zariski open subset of the base and the relative log canonical model is of maximal variation.

Tropical Bisectors and Voronoi Diagrams

In this paper we initiate the study of tropical Voronoi diagrams. We start out with investigating bisectors of finitely many points with respect to arbitrary polyhedral norms. For this more general scenario we show that bisectors of three points are homeomorphic to a non-empty open subset of Euclidean space, provided that certain degenerate cases are excluded. Specializing our results to tropical bisectors then yields structural results and algorithms for tropical Voronoi diagrams.

Caloric Measure Null Sets

We give a systematic treatment of caloric measure null sets on the essential boundary $\partial_eE$ of an arbitrary open set $E$ in ${\bf R}$. We discuss two characterisations of such sets and present some basic properties. We investigate the dependence of caloric measure null sets on the open set $E$. Thus, if $D$ is an open subset of $E$ and $Z\subseteq\partial_eE\cap\partial_eD$, we show that $Z$ is caloric measure null for $D$ if it is caloric measure null for $E$. We also give conditions on $E$ and $Z$ which imply that the reverse implication is true. We know from \cite{watson2011} that any polar subset of $\partial_eD$ is caloric measure null for $D$, but the reverse implication is not generally true. In our final result we show that, for subsets of a certain component of $\partial_eD$, caloric measure null sets are necessarily polar.

A note on a family of surfaces with $$p_g=q=2$$ and $$K^2=7$$

We study a family of surfaces of general type with $$p_g=q=2$$ p g = q = 2 and $$K^2=7$$ K 2 = 7 , originally constructed by C. Rito in [35]. We provide an alternative construction of these surfaces, that allows us to describe their Albanese map and the corresponding locus $$\mathcal {M}$$ M in the moduli space of surfaces of general type. In particular we prove that $$\mathcal {M}$$ M is an open subset, and it has three connected components, all of which are 2-dimensional, irreducible and generically smooth

Study on Strong Sensitivity of Systems Satisfying the Large Deviations Theorem

Let [Formula: see text] be a nontrivial compact metric space with metric [Formula: see text] and [Formula: see text] be a continuous self-map, [Formula: see text] be the sigma-algebra of Borel subsets of [Formula: see text], and [Formula: see text] be a Borel probability measure on [Formula: see text] with [Formula: see text] for any open subset [Formula: see text] of [Formula: see text]. This paper proves the following results : (1) If the pair [Formula: see text] has the property that for any [Formula: see text], there is [Formula: see text] such that [Formula: see text] for any open subset [Formula: see text] of [Formula: see text] and all [Formula: see text] sufficiently large (where [Formula: see text] is the characteristic function of the set [Formula: see text]), then the following hold : (a) The map [Formula: see text] is topologically ergodic. (b) The upper density [Formula: see text] of [Formula: see text] is positive for any open subset [Formula: see text] of [Formula: see text], where [Formula: see text]. (c) There is a [Formula: see text]-invariant Borel probability measure [Formula: see text] having full support (i.e. [Formula: see text]). (d) Sensitivity of the map [Formula: see text] implies positive lower density sensitivity, hence ergodical sensitivity. (2) If [Formula: see text] for any two nonempty open subsets [Formula: see text], then there exists [Formula: see text] satisfying [Formula: see text] for any nonempty open subset [Formula: see text], where [Formula: see text] there exist [Formula: see text] with [Formula: see text].