Real Analytic Families Research Articles

Semi-continuity in real analytic families

For a real analytic family of almost complex structures (M,Jt), t∈T, on an even dimensional compact real analytic manifold M, we prove upper semi-continuity with respect to t of the dimension of the space of Δ∂‾t-harmonic (p,q)-forms in Zariski sense. More general statements in terms of the dimension of the kernel of a family of self-adjoint strongly elliptic operators on a vector bundle over a manifold in real analytic settings are proved.

EULER CHARACTERISTIC AND SIGNATURE OF REAL SEMI-STABLE DEGENERATIONS

AbstractWe give a motivic proof of the fact that for nonsingular real tropical complete intersections, the Euler characteristic of the real part is equal to the signature of the complex part. This was originally proved by Itenberg in the case of surfaces in $\mathbb {C}P^{3}$ , and has been successively generalized by Bertrand and by Bihan and Bertrand. Our proof, different from previous approaches, is an application of the motivic nearby fiber of semistable degenerations. In particular, it extends the original result by Itenberg, Bertrand, and Bihan to real analytic families admitting a $\mathbb {Q}$ -nonsingular tropical limit.

Real entropy rigidity under quasi-conformal deformations

We set up a real entropy function h R h_\Bbb {R} on the space M d ′ \mathcal {M}’_d of Möbius conjugacy classes of real rational maps of degree d d by assigning to each class the real entropy of a representative f ∈ R ( z ) f\in \Bbb {R}(z) ; namely, the topological entropy of its restriction f ↾ R ^ f\restriction _{\hat {\Bbb {R}}} to the real circle. We prove a rigidity result stating that h R h_\Bbb {R} is locally constant on the subspace determined by real maps quasi-conformally conjugate to f f . As examples of this result, we analyze real analytic stable families of hyperbolic and flexible Lattès maps with real coefficients along with numerous families of degree d d real maps of real entropy log ⁡ ( d ) \log (d) . The latter discussion moreover entails a complete classification of maps of maximal real entropy.

Solenoidal attractors with bounded combinatorics are shy

We show that in a generic finite-dimensional real-analytic family of real-analytic multimodal maps, the subset of parameters on which the corresponding map has a solenoidal attractor with bounded combinatorics is a set with zero Lebesgue measure.

Re-parameterizing and reducing families of normal operators

We present a new proof of results of Kurdyka & Paunescu, and of Rainer, about real-analytic multi-parameters generalizations of classical results by Rellich and Kato about the reduction in families of univariate deformations of normal operators over real or complex vector spaces of finite dimensions. Given a real analytic family of normal operators over a finite dimensional real or complex vector space, there exists a locally finite composition of blowings-up with smooth centers re-parameterizing the given family such that at each point of the source space of the re-parameterizing mapping, there exists a neighbourhood of any given point over which exists a real analytic orthonormal frame in which the pull back of the operator is in reduced form at every point of the neighbourhood. A free by-product of our proof is the local real analyticity of the eigen-values, which in all prior works was a prerequisite step to get local regular reducing bases.

Controllability of ensembles of linear dynamical systems

We investigate the task of controlling ensembles of initial and terminal state vectors of parameter-dependent linear systems by applying parameter-independent open loop controls. Necessary, as well as sufficient, conditions for ensemble controllability are established, using tools from complex approximation theory. For real analytic families of linear systems it is shown that ensemble controllability holds only for systems with at most two independent parameters. We apply the results to networks of linear systems and address the question of open-loop robust synchronization.

Real analytic families of harmonic functions in a planar domain with a small hole

We consider a Dirichlet problem in a planar domain with a hole of diameter proportional to a real parameter ϵ and we denote by uϵ the corresponding solution. The behavior of uϵ for ϵ small and positive can be described in terms of real analytic functions of two variables evaluated at (ϵ,1/log⁡ϵ). We show that under suitable assumptions on the geometry and on the boundary data one can get rid of the logarithmic behavior displayed by uϵ for ϵ small and describe uϵ by real analytic functions of ϵ. Then it is natural to ask what happens when ϵ is negative. The case of boundary data depending on ϵ is also considered. The aim is to study real analytic families of harmonic functions which are not necessarily solutions of a particular boundary value problem.

Local and global liftings of analytic families of idempotents in Banach algebras

Generalizing results of our earlier paper, we investigate the following question. Let $\pi(\lambda) : A \to B$ be an analytic family of surjective homomorphisms between two Banach algebras, and $q(\lambda)$ an analytic family of idempotents in $B$. We want to find an analytic family $p(\lambda)$ of idempotents in $A$, lifting $q(\lambda)$, i.e., such that $\pi(\lambda)p(\lambda) = q(\lambda)$, under hypotheses of the type that the elements of $\text{Ker}\, \pi(\lambda)$ have small spectra. For spectra which do not disconnect $\Bbb C$ we obtain a local lifting theorem. For real analytic families of surjective $^*$-homomorphisms (for continuous involutions) and self-adjoint idempotents we obtain a local lifting theorem, for totally disconnected spectra. We obtain a global lifting theorem if the spectra of the elements in $\text{Ker}\, \pi(\lambda)$ are $\{0\}$, both in the analytic case, and, for $^*$-algebras (with continuous involutions) and self-adjoint idempotents, in the real analytic case. Here even an at most countably infinite set of mutually orthogonal analytic families of idempotents can be lifted to mutually orthogonal analytic families of idempotents. In the proofs, spectral theory is combined with complex analysis and general topology, and even a connection with potential theory is mentioned.

Regular or stochastic dynamics in families of higher-degree unimodal maps

AbstractWe construct a lamination of the space of unimodal maps with critical points of fixed degree $d\geq 2$ by the hybrid classes. The structure of the lamination yields a partition of the parameter space for one-parameter real analytic families of unimodal maps and allows us to transfer a priori bounds in the phase space to the parameter space. This implies that almost every map in such a family is either regular or stochastic.

Characterization of the generic unfolding of a weak focus

In this paper we prove that the orbital class of a generic real analytic family unfolding a weak focus is determined by the conjugacy class of its Poincaré monodromy and vice versa. We solve the embedding problem by means of quasiconformal surgery on the formal normal form. The surgery yields an integrable abstract almost complex 2-manifold equipped with an elliptic foliation. The monodromy of the latter coincides with the second iterate of a germ of prescribed family of real analytic diffeomorphisms undergoing a flip bifurcation.

Real analytic families of harmonic functions in a domain with a small hole

Let n⩾3. Let Ωi and Ωo be open bounded connected subsets of Rn containing the origin. Let ϵ0>0 be such that Ωo contains the closure of ϵΩi for all ϵ∈]−ϵ0,ϵ0[. Then, for a fixed ϵ∈]−ϵ0,ϵ0[∖{0} we consider a Dirichlet problem for the Laplace operator in the perforated domain Ωo∖ϵΩi. We denote by uϵ the corresponding solution. If p∈Ωo and p≠0, then we know that under suitable regularity assumptions there exist ϵp>0 and a real analytic operator Up from ]−ϵp,ϵp[ to R such that uϵ(p)=Up[ϵ] for all ϵ∈]0,ϵp[. Thus it is natural to ask what happens to the equality uϵ(p)=Up[ϵ] for ϵ negative. We show a general result on continuation properties of some particular real analytic families of harmonic functions in domains with a small hole and we prove that the validity of the equality uϵ(p)=Up[ϵ] for ϵ negative depends on the parity of the dimension n.

Propagation of boundary CR foliations and Morera type theorems for manifolds with attached analytic discs

We prove that homologically nontrivial generic smooth ( 2 n − 1 ) -parameter families of analytic discs in C n , n ⩽ 2 , attached by their boundaries to a CR-manifold Ω, test CR-functions in the following sense: if a smooth function on Ω analytically extends into any analytic discs from the family, then the function satisfies tangential CR-equations on Ω. In particular, we give an answer (Theorem 1) to the following long standing open question, so called strip-problem, earlier solved only for special families (mainly for circles): given a smooth one-parameter family of Jordan curves in the plane and a function f admitting holomorphic extension inside each curve, must f be holomorphic on the union of the curves? We prove, for real-analytic functions and arbitrary generic real-analytic families of curves, that the answer is “yes,” if no point is surrounded by all curves from the family. The latter condition is essential. We generalize this result to characterization of complex curves in C 2 as real 2-manifolds admitting nontrivial families of attached analytic discs (Theorem 4). The main result implies fairly general Morera type characterization of CR-functions on hypersurfaces in C 2 in terms of holomorphic extensions into three-parameter families of attached analytic discs (Theorem 2). One of the applications is confirming, in real-analytic category, the Globevnik–Stout conjecture (Theorem 3) on boundary values of holomorphic functions. It is proved that a smooth function on the boundary of a smooth strictly convex domain in C n extends holomorphically inside the domain if it extends holomorphically into complex lines tangent to a given strictly convex subdomain. The proofs are based on a universal approach, namely, on the reduction to a problem of propagation, from the boundary to the interior, of degeneracy of CR-foliations of solid torus type manifolds (Theorem 2.2).

Isometric Embeddings of Families of Special Lagrangian Submanifolds

We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi–Yau manifolds. For example, we prove that given any real-analytic one parameter family of Riemannian metrics gt on a three-dimensional manifold Y with volume form independent of t and with a real-analytic family of nowhere vanishing harmonic one forms θt, then (Y,gt) can be realized as a family of special Lagrangian submanifolds of a Calabi–Yau manifold X. We also prove that certain principal torus bundles can be equivariantly and isometrically embedded inside Calabi-Yau manifolds with torus action. We use this to construct examples of n-parameter families of special Lagrangian tori inside n + k-dimensional Calabi–Yau manifolds with torus symmetry. We also compute McLean's metric of 3-dimensional special Lagrangian fibrations with T2-symmetry.

Bifurcation of unimodal maps

Here we review some recent results that give a rather complete description of the dynamics of almost all mappings in real analytic families of unimodal maps

Regular or stochastic dynamics in real analytic families of unimodal maps

In this paper we prove that in any non-trivial real analytic family of quasiquadratic maps, almost any map is either regular (i.e., it has an attracting cycle) or stochastic (i.e., it has an absolutely continuous invariant measure). To this end we show that the space of analytic maps is foliated by codimension-one analytic submanifolds, “hybrid classes”. This allows us to transfer the regular or stochastic property of the quadratic family to any non-trivial real analytic family.

Tracking eigenvalues to the frontier of moduli space II: Limits for eigenvalue branches

We prove the existence of limits of real-analytic Laplace eigenvalue branches for real-analytic families of metrics that degenerate along a compact hypersurface.

Modified analytic trivialization for weighted homogeneous function-germs

We show that a real analytic family of real analytic functions admits a modified analytic trivialization (MAT) via an appropriate toric modification, if their weighted initial forms define isolated singularities near zero.

Misiurewicz Maps are Rare

In any real-analytic family of S-unimodal maps with non-constant combinatorics, the set of parameters corresponding to maps with Misiurewicz (non-recurrent) dynamics has Lebesgue measure zero. In particular, this is the case for the logistic family, answering an old question.

Natural extensions of holomorphic motions

We consider an arbitrary real analytic family Xz,\(z \in \bar D\), over the closed unit disc\(\bar D\), of real analytic plane Jordan curves Xz. Ifjeiθ,eiθ ∋ ∂D, is an arbitrary real-analytic family of orientation-reversing homeomorphisms of\(\bar {\mathbb{C}}\) fixingXeiθ pointwise, we show that there is a unique holomorphic motion of\(\bar {\mathbb{C}}\) extending the given motion of Jordan curves and consistent with the given family of involutions. If these generalized reflections are defined using the barycentric extension construction of Douady-Earle-Nag, then the resulting extension method for holomorphic motions of X is natural, that is Moebius-invariant and continuous with respect to variation of the given motion of X0.

Abundance of generic homoclinic tangencies in real-analytic families of diffeomorphisms

We consider one-parameter families of two-dimensional diffeomorphisms with homoclinic tangencies. Various authors considered the dynamical complexities due to such tangencies satisfying certain nondegeneracy conditions. In this paper we provide methods to actually verify, for real analytic families, that there are homoclinic tangencies which satisfy these (generic) nondegeneracy generic conditions.