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1. Index divisibility in dynamical sequences and cyclic orbits modulo p

   Annie S. Chen, T. Alden (August 2017, New York journal of mathematics)

   Let ϕ(x) = xd + c be an integral polynomial of degree at least 2, and consider the sequence (ϕn(0))n=0∞, which is the orbit of 0 under iteration by ϕ. Let Dd,c denote the set of positive integers n for which n | ϕn(0). We give a characterization of Dd,c in terms of a directed graph and describe a number of its properties, including its cardinality and the primes contained therein. In particular, we study the question of which primes p have the property that the orbit of 0 is a single p-cycle modulo p. We show that the set of such primes is finite when d is even, and conjecture that it is infinite when d is odd. 

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2. Algebraic stability of meromorphic maps descended from Thurston’s pullback maps

   https://doi.org/10.1090/tran/8221  

   Ramadas, Rohini (January 2021, Transactions of the American Mathematical Society)

   null (Ed.)

   Let ϕ : S 2 → S 2 \phi :S^2 \to S^2 be an orientation-preserving branched covering whose post-critical set has finite cardinality n n . If ϕ \phi has a fully ramified periodic point p ∞ p_{\infty } and satisfies certain additional conditions, then, by work of Koch, ϕ \phi induces a meromorphic self-map R ϕ R_{\phi } on the moduli space M 0 , n \mathcal {M}_{0,n} ; R ϕ R_{\phi } descends from Thurston’s pullback map on Teichmüller space. Here, we relate the dynamics of R ϕ R_{\phi } on M 0 , n \mathcal {M}_{0,n} to the dynamics of ϕ \phi on S 2 S^2 . Let ℓ \ell be the length of the periodic cycle in which the fully ramified point p ∞ p_{\infty } lies; we show that R ϕ R_{\phi } is algebraically stable on the heavy-light Hassett space corresponding to ℓ \ell heavy marked points and ( n − ℓ ) (n-\ell ) light points. 

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3. Reduction of dynatomic curves

   https://doi.org/10.1017/etds.2017.140  

   DOYLE, JOHN R.; KRIEGER, HOLLY; OBUS, ANDREW; PRIES, RACHEL; RUBINSTEIN-SALZEDO, SIMON; WEST, LLOYD (October 2019, Ergodic Theory and Dynamical Systems)

   null (Ed.)

   In this paper, we make partial progress on a function field version of the dynamical uniform boundedness conjecture for certain one-dimensional families $${\mathcal{F}}$$ of polynomial maps, such as the family $$f_{c}(x)=x^{m}+c$$ , where $$m\geq 2$$ . We do this by making use of the dynatomic modular curves $$Y_{1}(n)$$ (respectively $$Y_{0}(n)$$ ) which parametrize maps $$f$$ in $${\mathcal{F}}$$ together with a point (respectively orbit) of period $$n$$ for $$f$$ . The key point in our strategy is to study the set of primes $$p$$ for which the reduction of $$Y_{1}(n)$$ modulo $$p$$ fails to be smooth or irreducible. Morton gave an algorithm to construct, for each $$n$$ , a discriminant $$D_{n}$$ whose list of prime factors contains all the primes of bad reduction for $$Y_{1}(n)$$ . In this paper, we refine and strengthen Morton’s results. Specifically, we exhibit two criteria on a prime $$p$$ dividing $$D_{n}$$ : one guarantees that $$p$$ is in fact a prime of bad reduction for $$Y_{1}(n)$$ , yet this same criterion implies that $$Y_{0}(n)$$ is geometrically irreducible. The other guarantees that the reduction of $$Y_{1}(n)$$ modulo $$p$$ is actually smooth. As an application of the second criterion, we extend results of Morton, Flynn, Poonen, Schaefer, and Stoll by giving new examples of good reduction of $$Y_{1}(n)$$ for several primes dividing $$D_{n}$$ when $n=7,8,11$ , and $$f_{c}(x)=x^{2}+c$$ . The proofs involve a blend of arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set. 

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4. Examples of non-Dini domains with large singular sets

   https://doi.org/10.1515/ans-2022-0058  

   Kenig, Carlos; Zhao, Zihui (January 2023, Advanced Nonlinear Studies)

   Abstract Let u u be a nontrivial harmonic function in a domain D ⊂ R d D\subset {{\mathbb{R}}}^{d} , which vanishes on an open set of the boundary. In a recent article, we showed that if D D is a C 1 {C}^{1} -Dini domain, then, within the open set, the singular set of u u , defined as { X ∈ D ¯ : u ( X ) = 0 = ∣ ∇ u ( X ) ∣ } \left\{X\in \overline{D}:u\left(X)=0=| \nabla u\left(X)| \right\} , has finite ( d − 2 ) \left(d-2) -dimensional Hausdorff measure. In this article, we show that the assumption of C 1 {C}^{1} -Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose singular sets have infinite ℋ d − 2 {{\mathcal{ {\mathcal H} }}}^{d-2} -measures. 

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5. On the three ball theorem for solutions of the Helmholtz equation

   https://doi.org/10.1007/s40627-021-00070-3  

   Berge, Stine Marie; Malinnikova, Eugenia (June 2021, Complex Analysis and its Synergies)

   null (Ed.)

   Abstract Let $$u_{k}$$ u k be a solution of the Helmholtz equation with the wave number k , $$\varDelta u_{k}+k^{2} u_{k}=0$$ Δ u k + k 2 u k = 0 , on (a small ball in) either $${\mathbb {R}}^{n}$$ R n , $${\mathbb {S}}^{n}$$ S n , or $${\mathbb {H}}^{n}$$ H n . For a fixed point p , we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$ M u k ( r ) = max d ( x , p ) ≤ r | u k ( x ) | . The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$ M u k ( 2 r ) ≤ C ( k , r , α ) M u k ( r ) α M u k ( 4 r ) 1 - α is well known, it holds for some $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and $$C(k,r,\alpha )>0$$ C ( k , r , α ) > 0 independent of $$u_{k}$$ u k . We show that the constant $$C(k,r,\alpha )$$ C ( k , r , α ) grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds. 

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