Given, in the Lagrangian torus fibration $R^4\to R^2$, a Lagrangian submanifold $L$, endowed with a trivial flat connection, the corresponding mirror object is constructed on the dual fibration by means of a family of Morse homologies associated to the generating function of $L$, and it is provided with a holomorphic structure. Morse homology, however, is not defined along the caustic $C$ of $L$ or along the bifurcation locus $B$, where the family does not satisfy the Morse-Smale condition. The holomorphic structure is extended to the subset $C\cup B$, except cusps, yielding the so called quantum corrections to the mirror object.
We show the existence of open sets of bifurcations near Latt{è}s maps of sufficiently high degree. In particular, every Latt{è}s map has an iterate which is in the closure of the interior of the bifurcation locus. To show this, we design a method to intersect the limit set of some particular type of IFS with a well-oriented curve. Then, we show that a Latt{è}s map of sufficiently high degree can be perturbed to exhibit this geometry.
Consider a quadratic rational self-map of the Riemann sphere such that one critical point is periodic of period 2, and the other critical point lies on the boundary of its immediate basin of attraction. We will give explicit topological models for all such maps. We also discuss the corresponding parameter picture.
We unfold the codimension-two simultaneous occurrence of a border-collision bifurcation and a period-doubling bifurcation for a general piecewise-smooth, continuous map. We find that, with sufficient non-degeneracy conditions, a locus of period-doubling bifurcations emanates non-tangentially from a locus of border-collision bifurcations. The corresponding period-doubled solution undergoes a border-collision bifurcation along a curve emanating from the codimension-two point and tangent to the period-doubling locus here. In the case that the map is one-dimensional local dynamics are completely c…
We define a family B(t) of compact subsets of the unit interval which generalizes the sets of numbers whose continued fraction expansion has bounded digits. We study how the set B(t) changes as one moves the parameter t, and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behavior as the usual family of quadratic polynomials. The set E of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension 1. We also show that the Hausdorff dimension of B(t) varies continuously with the parameter, and the d…
We study the bifurcation loci of quadratic (and unicritical) polynomials and exponential maps. We outline a proof that the exponential bifurcation locus is connected; this is an analog to Douady and Hubbard's celebrated theorem that (the boundary of) the Mandelbrot set is connected. For these parameter spaces, a fundamental conjecture is that hyperbolic dynamics is dense. For quadratic polynomials, this would follow from the famous stronger conjecture that the bifurcation locus (or equivalently the Mandelbrot set) is locally connected. It turns out that a formally slightly weaker statement is …
In this paper we continue the study, began in \cite{CJK2}, of the bifurcation locus of a family of meromorphic functions with two asymptotic values, no critical values and an attracting fixed point. If we fix the multiplier of the fixed point, either of the two asymptotic values determines a one-dimensional parameter slice for this family. We proved that the bifurcation locus divides this parameter slice into three regions, two of them analogous to the Mandelbrot set and one, the shift locus, analogous to the complement of the Mandelbrot set. In \cite{FK, CK} it was proved that the points in t…
We give an example of a family of endomorphisms of $\mathbb P^2 (\mathbb C)$ whose Julia set depends continuously on the parameter and whose bifurcation locus has non empty interior.
We give an estimate for the Hausdorff dimension of the bifurcation locus of a family of endomorphisms of $\mathbb{P}^k (\mathbb{C})$. This dimension is maximal near isolated Lattès examples.
We study holomorphic families of polynomial-like maps depending on a parameter s. We prove that the partial sums of largest Lyapunov exponents are plurisubharmonic functions of s. We also study their continuity and introduce the bifurcation locus as the support of bifurcation currents.
Motivated by mirror symmetry, we consider a Lagrangian fibration $X\to B$ and Lagrangian maps $f:L\hookrightarrow X\to B$, when $L$ has dimension 2, exhibiting an unstable singularity, and study how their caustic changes, in a neighbourhood of the unstable singularity, when slightly perturbed. The integral curves of $\nabla f_x$, for $x\in B$, where $f_x(y)=f(y)-x\cdot y$, called ``gradient lines'', are then introduced, and a study of them, in order to analyse their bifurcation locus, is carried out.
We study conformal quantities at generic parameters with respect to the harmonic measure on the boundary of the connectedness loci ${\cal M}_d$ for unicritical polynomials $f_c(z)=z^d+c$. It is known that these parameters are structurally unstable and have stochastic dynamics. We prove $C^{1+\fracα{d}-ε}$-conformality, $α= 2-\mbox{HD}\,({\cal J}_{c_0})$, of the parameter-phase space similarity maps $Υ_{c_0}(z):\mathbb{C}\mapsto \mathbb{C}$ at typical $c_0\in \partial {\cal M}_d$ and establish that globally quasiconformal similarity maps $Υ_{c_0}(z)$, $c_0\in \partial {\cal M}_d$, are $C^1$-con…
Motivated by mirror symmetry, we consider the Lagrangian fibration $\R^4\to\R^2$ and Lagrangian maps $f:L\hookrightarrow \R^4\to \R^2$, exhibiting an unstable singularity, and study how the bifurcation locus of gradient lines, the integral curves of $\nabla f_x$, for $x\in B$, where $f_x(y)=f(y)-x\cdot y$, changes when $f$ is slightly perturbed. We consider the cases when $f$ is the germ of a fold, of a cusp and, particularly, of an elliptic umbilic.
We show the existence of computable complex numbers $λ$ for which the bifurcation locus of the one parameter complex family $f_{b}(z) = λz + b z^{2} + z^{3}$ is not Turing computable.
Let (ρ_λ)_{λ\in Λ} be a holomorphic family of representations of a finitely generated group G into PSL(2,C), parameterized by a complex manifold Λ. We define a notion of bifurcation current in this context, that is, a positive closed current on Λdescribing the bifurcations of this family of representations in a quantitative sense. It is the analogue of the bifurcation current introduced by DeMarco for holomorphic families of rational mappings on the Riemann sphere. Our definition relies on the theory of random products of matrices, so it depends on the choice of a probability measure μon G. We…
Toward the understanding of bifurcation phenomena of dynamics on the Berkovich projective line $\mathbb{P}^{1,an}$ over non-archimedean fields, we study the stability (or passivity) of critical points of families of polynomials parametrized by analytic curves. We construct the activity measure of a critical point of a family of rational functions, and study its properties. For a family of polynomials, we study more about the activity locus such as its relation to boundedness locus, i.e., the Mandelbrot set, and to the normality of the sequence of the forward orbit.
We initiate a parametric study of holomorphic families of polynomial skew products, i.e., polynomial endomorphisms of $\mathbb{C}^2$ of the form $F(z,w)= (p(z), q(z,w))$ that extend to holomorphic endomorphisms of $\mathbb{P}^2(\mathbb{C})$. We prove that dynamical stability in the sense of arXiv:1403.7603 preserves hyperbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family $z^2 +c$. We also precisely describe the geometry of the bifurcation locus and current …
We study the dynamics of a family K_alpha of discontinuous interval maps whose (infinitely many) branches are Moebius transformations in SL(2, Z), and which arise as the critical-line case of the family of (a, b)-continued fractions. We provide an explicit construction of the bifurcation locus E_KU for this family, showing it is parametrized by Farey words and it has Hausdorff dimension zero. As a consequence, we prove that the metric entropy of K_alpha is analytic outside the bifurcation set but not differentiable at points of E_KU, and that the entropy is monotone as a function of the parame…
