In this paper, we analyze a certain family of holomorphic correspondences on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove double-struck upper C With caret times ModifyingAbove double-struck upper C With caret"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> <mml:mo>×</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\hat {\mathbb {C}}\times \hat {\mathbb {C}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and prove their equidistribution properties. In particular, for any correspondence in this family we prove that the naturally associated multivalued map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is such that for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a element-of double-struck upper C"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">a\in \mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we have that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper F Superscript n Baseline right-parenthesis Subscript asterisk Baseline left-parenthesis delta Subscript a Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>F</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∗</mml:mo> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>δ</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(F^n)_*(\delta _a)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> converges to a probability measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript upper F"> <mml:semantics> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>F</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mu _F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript asterisk Baseline left-parenthesis mu Subscript upper F Baseline right-parenthesis equals mu Subscript upper F Baseline d"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mo>∗</mml:mo> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>F</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>F</mml:mi> </mml:msub> <mml:mi>d</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">F_*(\mu _F)=\mu _F d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the degree of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This result is used to show that the minimal Hutchinson invariant set, introduced by P. Alexandersson, P. Brändén, and B. Shapiro [<italic>An inverse problem in Pólya–Schur theory. I. Non-degenerate and degenerate operators</italic>, preprint, 2024], of a large class of operators and for sufficiently large <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> exists and is the support of the aforementioned measure. We prove that under a minor additional assumption, the minimal Hutchinson-invariant set is a Cantor set.

Nils Hemmingsson

Conformal Geometry and Dynamics of the American Mathematical Society

Equidistribution of iterations of holomorphic correspondences and Hutchinson invariant sets

In this work we show that every quotient of a torus endomorphism has a parabolic orbifold, answering a question of Mario Bonk and Daniel Meyer posed in [<italic>Expanding Thurston maps</italic>, American Mathematical Society, Providence, RI, 2017].

Sofía Llavayol

Juliana Xavier

Quotients of torus endomorphisms have parabolic orbifolds

The induced action on the Picard group of a rational surface automorphism with positive entropy can be identified with an element of the Coxeter group associated to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript n Baseline comma n greater-than-or-equal-to 10"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>10</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">E_n, n\ge 10</mml:annotation> </mml:semantics> </mml:math> </inline-formula> diagram. It follows that the set of dynamical degrees of rational surface automorphisms is a subset of the spectral radii of elements in the Coxeter group. This article concerns the realizability of an element of the Coxeter group as an automorphism on a rational surface with an irreducible reduced anticanonical curve. For any unrealizable element, we explicitly construct a realizable element with the same spectral radius. Hence, we show that the set of dynamical degrees and the set of spectral radii of the Coxeter group are, in fact, identical. This has been shown by Uehara [Ann. Inst. Fourier (Grenoble) 66 (2016), pp. 377–432] by explicitly constructing a rational surface automorphism. This construction depends on a decomposition of an element of the Coxeter group. Our proof is conceptual and provides a simple description of elements of the Coxeter group, which are realized by automorphisms on anticanonical rational surfaces.

Kyounghee Kim

The dynamical degrees of rational surface automorphisms

The Teichmüller space of closed Riemann surfaces of genus two is considered as a subset of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript 7"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>7</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^7</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We present some mapping classes acting on the model of the Teichüller space and investigate its action especially on the extremal surfaces with respect to a certain metric. As a result we estimate the distance between two orbits of distinct extremal surfaces from above.

Gou Nakamura

Actions of some mapping classes on the closed Riemann surfaces of genus two

This paper is part of a general program in complex dynamics to understand parameter spaces of transcendental maps with finitely many singular values. The simplest families of such functions have two asymptotic values and no critical values. These families, up to affine conjugation, depend on two complex parameters. Understanding their parameter spaces is key to understanding families with more asymptotic values, just as understanding quadratic polynomials was for rational maps more generally. The first such families studied were the one-dimensional slices of the exponential family, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="exp left-parenthesis z right-parenthesis plus a"> <mml:semantics> <mml:mrow> <mml:mi>exp</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\exp (z) + a</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the tangent family <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda tangent z"> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mi>tan</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda \tan z</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The exponential case exhibited phenomena not seen for rational maps: Cantor bouquets in both the dynamic and parameter spaces, and no bounded hyperbolic components. The tangent case, with its two finite asymptotic values <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="plus-or-minus lamda i"> <mml:semantics> <mml:mrow> <mml:mo>±</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>i</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\pm \lambda i</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is closer to the rational case, a kind of infinite degree version of the latter. In this paper, we consider a general family that interpolates between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="exp left-parenthesis z right-parenthesis plus a"> <mml:semantics> <mml:mrow> <mml:mi>exp</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\exp (z) + a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda tangent z"> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mi>tan</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda \tan z</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our new family has two asymptotic values and a one-dimensional slice for which one of the asymptotic values is constrained to be a pole, the “polar asymptotic value” of the title. We show how the dynamic and parameter planes for this slice exhibit behavior that is a surprisingly delicate interplay between that of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="exp left-parenthesis z right-parenthesis plus a"> <mml:semantics> <mml:mrow> <mml:mi>exp</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\exp (z) + a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda tangent z"> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mi>tan</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda \tan z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> families.

Tao Chen

Linda Keen

Meromorphic functions with a polar asymptotic value

In the paper [<italic>The Teichmüller theory of harmonic maps</italic>, J. Differential Geom. 29 (1989), no. 2, 449–479], Wolf provided a global coordinate system of the Teichmüller space of a closed oriented surface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the vector space of holomorphic quadratic differentials on a Riemann surface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> homeomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This coordinate system is via harmonic maps from the Riemann surface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to hyperbolic surfaces. Moreover, he gave a compactification of the Teichmüller space by adding a point at infinity to each endpoint of harmonic map rays starting from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the space. Wolf also showed this compactification coincides with the Thurston compactification. In this paper, we extend the harmonic map ray compactification to the case of punctured Riemann surfaces and show that it still coincides with the Thurston compactification.

Kento Sakai

The Harmonic Map Compactification of Teichmüller Spaces for Punctured Riemann Surfaces

We consider moduli spaces of dynamical systems of correspondences over the projective line as a generalization of moduli spaces of dynamical systems of endomorphisms on the projective line. We define the moduli space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D y n Subscript d comma e"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mi>y</mml:mi> <mml:msub> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">Dyn_{d,e}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d comma e right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>e</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(d,e)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> correspondences. We construct a family <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript c Baseline colon upper D y n Subscript d comma e Baseline right dasheD arrow upper D y n Subscript 1 comma d plus e minus 1 Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>D</mml:mi> <mml:mi>y</mml:mi> <mml:msub> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">⇢</mml:mo> <mml:mi>D</mml:mi> <mml:mi>y</mml:mi> <mml:msub> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mi>e</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\rho _c : Dyn_{d,e} \dashrightarrow Dyn_{1,d+e-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of rational maps representation-theoretically. Here we note that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D y n Subscript 1 comma d plus e minus 1"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mi>y</mml:mi> <mml:msub> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mi>e</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">Dyn_{1,d+e-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is identical to the moduli space of the usual dynamical systems of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d plus e minus 1"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mi>e</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d+e-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that the moduli space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D y n Subscript d comma e"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mi>y</mml:mi> <mml:msub> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">Dyn_{d,e}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is rational by using <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript c"> <mml:semantics> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\rho _c</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, the multiplier maps for the fixed points factor through <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript c"> <mml:semantics> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\rho _c</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Furthermore, we show the Woods Hole formulae for correspondences of different degrees are also related by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript c"> <mml:semantics> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\rho _c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and obtain another representation-theoretically simplified form of the formula.

Rin Gotou

Dynamical systems of correspondences on the projective line I: Moduli spaces and multiplier maps

To investigate the degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> connectedness locus, Thurston [<italic>On the geometry and dynamics of iterated rational maps</italic>, Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137] studied <italic><inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript d"> <mml:semantics> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\sigma _d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant laminations</italic>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript d"> <mml:semantics> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\sigma _d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals z squared plus c"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>c</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f(z) = z^2 +c</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the spirit of Thurston’s work, we consider the space of all <italic>cubic symmetric polynomials</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript lamda Baseline left-parenthesis z right-parenthesis equals z cubed plus lamda squared z"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f_\lambda (z)=z^3+\lambda ^2 z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a series of three articles. In the present paper, the first in the series, we construct a lamination <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript s Baseline upper C upper L"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mi>C</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">C_sCL</mml:annotation> </mml:semantics> </mml:math> </inline-formula> together with the induced factor space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S slash upper C Subscript s Baseline upper C upper L"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mi>C</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {S}/C_sCL</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the unit circle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {S}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As will be verified in the third paper of the series, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S slash upper C Subscript s Baseline upper C upper L"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mi>C</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {S}/C_sCL</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a monotone model of the <italic>cubic symmetric connectedness locus</italic>, i.e. the space of all cubic symmetric polynomials with connected Julia sets.

Alexander Blokh

Lex Oversteegen

Nikita Selinger

Vladlen Timorin

Sandeep Vejandla

Symmetric cubic laminations

This paper, which is the second of a series of three papers, studies dynamical properties of elements of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O u t left-parenthesis upper F Subscript monospace n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Out</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="monospace">n</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Out}(F_{\mathtt n})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the outer automorphism group of a nonabelian free group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript monospace n"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="monospace">n</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">F_{\mathtt n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that, for every exponentially growing outer automorphism of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript monospace n"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="monospace">n</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">F_{\mathtt n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists a preferred compact topological space, the space of currents relative to a malnormal subgroup system, on which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ</mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts by homeomorphism with a North-South dynamics behavior.

yassine Guerch

North-South type dynamics of relative atoroidal automorphisms of free groups on a relative space of currents

Since the 1980s, much progress has been done in completely determining which functions share a Julia set. The polynomial case was completely solved in 1995, and it was shown that the symmetries of the Julia set play a central role in answering this question. The rational case remains open, but it was already shown to be much more complex than the polynomial one. Here, we offer partial extensions to Beardon’s results on the symmetry group of Julia sets, and discuss them in the context of singularly perturbed maps.

Gustavo Ferreira

Symmetries of Julia sets for rational maps

We analyze the existence and types of unbounded Fatou components for elliptic functions and other meromorphic functions with doubly periodic Julia sets. We show that apart from Herman rings and Siegel disks, all types of dynamics can occur in these domains, which are called toral bands. We show that toral bands are not necessarily periodic, and we give results about the number of distinct residue classes of critical points in each toral band.

Jane Hawkins

Lorelei Koss

Single and double toral band Fatou components in meromorphic dynamics

In this paper we study properties of endomorphisms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P Superscript k"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {P}^k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> using a symmetric product construction <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-struck upper P Superscript 1 Baseline right-parenthesis Superscript k Baseline slash German upper S Subscript k Baseline approximately-equals double-struck upper P Superscript k"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>k</mml:mi> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">S</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>≅</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {P}^1)^k/\mathfrak {S}_k \cong \mathbb {P}^k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Symmetric products have been used to produce examples of endomorphisms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P Superscript k"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {P}^k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with certain characteristics, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k\geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the present note, we discuss the use of these maps to enlighten stability phenomena in parameter spaces. In particular, we study <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-deep post-critically finite maps and characterize families of Lattès maps.

Thomas Gauthier

Benjamin Hutz

Scott Kaschner

Symmetrization of rational maps: Arithmetic properties and families of Lattès maps of ℙ^{𝕜}

We discuss problems concerning the conformal condenser capacity of “hedgehogs”, which are compact sets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the unit disk <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper D equals StartSet z colon StartAbsoluteValue z EndAbsoluteValue greater-than 1 EndSet"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">D</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>z</mml:mi> <mml:mo>:</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {D}=\{z:\,|z|&gt;1\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consisting of a central body <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 0"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">E_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that is typically a smaller disk <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper D overbar Subscript r Baseline equals StartSet z colon StartAbsoluteValue z EndAbsoluteValue less-than-or-equal-to r EndSet"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">D</mml:mi> </mml:mrow> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mi>r</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>z</mml:mi> <mml:mo>:</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:mi>r</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\overline {\mathbb {D}}_r=\{z:\,|z|\le r\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than r greater-than 1"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>r</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0&gt;r&gt;1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and several spikes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript k"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">E_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that are compact sets lying on radial intervals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis alpha Subscript k Baseline right-parenthesis equals StartSet t e Superscript i alpha Super Subscript k Superscript Baseline colon 0 less-than-or-equal-to t greater-than 1 EndSet"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>t</mml:mi> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:msup> <mml:mo>:</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">I(\alpha _k)=\{te^{i\alpha _k}:\,0\le t&gt;1\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The main questions we are concerned with are the following: (1) How does the conformal capacity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal c normal a normal p left-parenthesis upper E right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">c</mml:mi> <mml:mi mathvariant="normal">a</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {cap}(E)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E equals union upper E Subscript k"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mo>∪</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">E=\cup _{k=0}^n E_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> behave when the spikes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript k"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">E_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals 1"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, move along the intervals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis alpha Subscript k Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">I(\alpha _k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> toward the central body if their hyperbolic lengths are preserved during the motion? (2) How does the capacity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal c normal a normal p left-parenthesis upper E right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">c</mml:mi> <mml:mi mathvariant="normal">a</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {cap}(E)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depend on the distribution of angles between the spikes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript k"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">E_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>? We prove several results related to these questions and discuss methods of applying symmetrization type transformations to study the capacity of hedgehogs. Several open problems, including problems on the capacity of hedgehogs in the three-dimensional hyperbolic space, will also be suggested.

Dimitrios Betsakos

Alexander Solynin

Matti Vuorinen

Conformal capacity of hedgehogs

According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose limit set is a generalized Apollonian gasket <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript upper H"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We design a surgery that relates <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a rational map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose Julia set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper J Subscript g"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">J</mml:mi> </mml:mrow> <mml:mi>g</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {J}_g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is (non-quasiconformally) homeomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript upper H"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show for a large class of triangulations, however, the groups of quasisymmetries of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript upper H"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper J Subscript g"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">J</mml:mi> </mml:mrow> <mml:mi>g</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {J}_g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, this group is equal to the group of Möbius symmetries of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript upper H"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which is the semi-direct product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> itself and the group of Möbius symmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript upper H"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the classical Apollonian gasket), we give a quasiregular model for the above actions which is quasiconformally equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and produces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by a David surgery. We also construct a mating between the group and the map coexisting in the same dynamical plane and show that it can be generated by Schwarz reflections in the deltoid and the inscribed circle.

Russell Lodge

Mikhail Lyubich

Sergei Merenkov

Sabyasachi Mukherjee

On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections

In this short note we observe that within the family of slowly recurrent rational maps on the Riemann sphere, the Collet–Eckmann, second Collet–Eckmann, and topological Collet–Eckmann conditions are equivalent and also invariant under topological conjugacy.

Mats Bylund

Equivalence of Collet–Eckmann conditions for slowly recurrent rational maps

Totally ramified rational maps and regularly ramified rational maps are defined and studied in this paper. We first give a complete classification of regularly ramified rational maps and show that our definition of a regularly ramified rational map is equivalent to a much stronger definition of a map of this kind given by Milnor [<italic>Dynamics in one complex variable</italic>, Princeton University Press, Princeton, NJ, 2006]. Then we show that (1) any totally ramified rational map of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d less-than-or-equal-to 6"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d\leq 6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must be regularly ramified; (2) for any integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d greater-than 6"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d&gt;6</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists a totally ramified rational map of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is not regularly ramified. Furthermore, we count totally ramified rational maps up to degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="10"> <mml:semantics> <mml:mn>10</mml:mn> <mml:annotation encoding="application/x-tex">10</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Finally, we present explicit formulas for all totally but not regularly ramified rational maps of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="7"> <mml:semantics> <mml:mn>7</mml:mn> <mml:annotation encoding="application/x-tex">7</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="8"> <mml:semantics> <mml:mn>8</mml:mn> <mml:annotation encoding="application/x-tex">8</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, up to pre- and post-composition by Möbius transformations.

Weiwei Cui

Jun Hu

Totally ramified rational maps

Let R(N,G) be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected compact Lie group G, and let X(N,G) be the corresponding moduli space. We show that there exists a natural Out(N)-invariant measure on X(N,G) and that whenever Out(N) has at least one hyperbolic element, the action of Out(N) on X(N,G) is mixing with respect to this measure.

Jean-Philippe Burelle

Sean Lawton

Dynamics on nilpotent character varieties

For 3-dimensional hyperbolic cone structures with cone angles $\theta$, local rigidity is known for $0 \leq \theta \leq 2\pi$, but global rigidity is known only for $0 \leq \theta \leq \pi$. The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most $\pi$ do not degenerate in deformations decreasing cone angles to zero. In this paper, we give an example of a degeneration of hyperbolic cone structures with decreasing cone angles less than $2\pi$. These cone structures are constructed on a certain alternating link in the thickened torus by gluing four copies of a certain polyhedron. For this construction, we explicitly describe the isometry types on such a hyperbolic polyhedron.

Ken’ichi Yoshida

Degeneration of 3-dimensional hyperbolic cone structures with decreasing cone angles

The Speiser class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the set of all entire functions with finitely many singular values. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript q Baseline subset-of upper S"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">S_q\subset S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the set of all transcendental entire functions with exactly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> distinct singular values. The Fatou-Shishikura inequality for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper S Subscript q"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">f\in S_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> gives an upper bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the sum of the numbers of its Cremer cycles and its cycles of immediate attractive basins, parabolic basins, and Siegel disks. In this paper, we show that the inequality for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper S Subscript q"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">f\in S_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is best possible in the following sense: For any combination of the numbers of these cycles which satisfies the inequality, some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T element-of upper S Subscript q"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">T\in S_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> realizes it. In our construction, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a structurally finite transcendental entire function.

Masashi Kisaka

Hiroto Naba

Best possibility of the Fatou-Shishikura inequality for transcendental entire functions in the Speiser class

ln this paper, a generalized cusp is a properly convex manifold with strictly convex boundary that is diffeomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M times left-bracket 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>×</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M\times [0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a closed Euclidean manifold. These are classified by Ballas, Cooper, and Leitner [J. Topol. 13 (2020), pp. 1455-1496]. The marked moduli space is homeomorphic to a subspace of the space of conjugacy classes of representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 1 upper M"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi _1M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It has one description as a generalization of a trace-variety, and another description involving weight data that is similar to that used to describe semi-simple Lie groups. It is also a bundle over the space of Euclidean similarity (conformally flat) structures on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the fiber is a closed cone in the space of cubic differentials. For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional orientable generalized cusps, the fiber is homeomorphic to a cone on a solid torus.

Samuel Ballas

Daryl Cooper

Arielle Leitner

The moduli space of marked generalized cusps in real projective manifolds

In this note we consider the entropy by Dankwart [<italic>On the large-scale geometry of flat surfaces</italic>, 2014, PhD thesis. https://bib.math.uni-bonn.de/downloads/bms/BMS-401.pdf] of unit area translation surfaces in the<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L left-parenthesis 2 comma double-struck upper R right-parenthesis"><mml:semantics><mml:mrow><mml:mi>S</mml:mi><mml:mi>L</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">SL(2, \mathbb R)</mml:annotation></mml:semantics></mml:math></inline-formula>orbits of square tiled surfaces that are the union of squares, where the singularities occur at the vertices and the singularities have a common cone angle. We show that the entropy over such orbits is minimized at those surfaces tiled by equilateral triangles where the singularities occur precisely at the vertices. We also provide a method for approximating the entropy of surfaces in the orbits.

Paul Colognese

Mark Pollicott

Minimizing entropy for translation surfaces

For a proper geodesic metric space, the Morse boundary focuses on the hyperbolic-like directions in the space. The Morse boundary is a quasi-isometry invariant. That is, a quasi-isometry between two spaces induces a homeomorphism on their Morse boundaries. In this paper, we investigate additional structures on the Morse boundary which determine the space up to a quasi-isometry. We prove that a homeomorphism between the Morse boundaries of two proper, cocompact spaces is induced by a quasi-isometry if and only if both the homeomorphism and its inverse are bihölder, or quasi-symmetric, or strongly quasi-conformal.

Qing Liu

Maps on the Morse boundary

We derive an integral inequality between the mean curvature and the scalar curvature of the boundary of any scalar flat conformal compactifications of Poincaré-Einstein manifolds. As a first consequence, we obtain a sharp lower bound for the first eigenvalue of the conformal half-Laplacian of the boundary of such manifolds. Secondly, a new upper bound for the renormalized volume is given in the four dimensional setting. Finally, some estimates on the first eigenvalues of Dirac operators are also deduced.

Simon Raulot

Scalar flat compactifications of Poincaré-Einstein manifolds and applications

Our goal of this paper is to research the completeness of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Weil-Petersson distance, which is induced by the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Weil-Petersson metric on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -integrable Teichmüller space of hyperbolic Riemann surfaces. As a result, we see that the metric is incomplete for all the hyperbolic Riemann surfaces with Lehner’s condition except for the ones that are conformally equivalent to either the unit disk or the punctured unit disk. The proof is based on the one by Wolpert’s original paper, which is given in the case of compact Riemann surfaces.

Masahiro Yanagishita

Completeness of 𝑝-Weil-Petersson distance

We study endomorphisms constructed by Sarah Koch in [Teichmüller theory and critically finite endomorphisms,<italic> Advances in Mathematics</italic> <bold>248</bold> (2013), 573–617] and we focus on the eigenvalues of the differential of such maps at its fixed points. To each post-critically finite unicritical polynomial, Koch associated a post-critically algebraic endomorphism of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C double-struck upper P Superscript k"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>k</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathbb {CP}}^k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Koch showed that the eigenvalues of the differentials of such maps along periodic cycles outside the post-critical sets have modulus strictly greater than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this article, we show that the eigenvalues of the differentials at fixed points are either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or have modulus strictly greater than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This confirms a conjecture proposed by the author in his thesis. We also provide a concrete description of such values in terms of the multiplier of a unicritical polynomial.

短名	Conform. Geom. Dyn.
Journal Impact	0.16
国际分区	MATHEMATICS(Q4)

ISSN	1088-4173
h-index	18

涉及主题	计算机科学数学纯数学数学分析几何学物理
出版信息	出版商: American Mathematical Society，出版周期: ，期刊类型: journal
基本数据	创刊年份: 1997，原创研究文献占比: 100.00%，自引率:0.00%， Gold OA占比: 81.25%

Conformal Geometry and Dynamics

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Completeness of 𝑝-Weil-Petersson distance

Fixed points of Koch maps

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Conformal Geometry and Dynamics

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Completeness of 𝑝-Weil-Petersson distance

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