scholarly journals Automatic real analyticity and a regal proof of a commutative multivariate Löwner theorem

2020 ◽  
pp. 1
Author(s):  
J. E. Pascoe ◽  
Ryan Tully-Doyle
Keyword(s):  
Real Analyticity

scholarly journals Cuspidal edges with the same first fundamental forms along a knot

2020 ◽  
Vol 29 (07) ◽  
pp. 2050047
Author(s):  
Atsufumi Honda ◽  
Kosuke Naokawa ◽  
Kentaro Saji ◽  
Masaaki Umehara ◽  
Kotaro Yamada
Keyword(s):  
Fundamental Form ◽  
Local Existence ◽  
Reasonable Assumption ◽  
Real Analyticity ◽  
Cuspidal Edge ◽  
First Fundamental Form

Letting [Formula: see text] be a compact [Formula: see text]-curve embedded in the Euclidean [Formula: see text]-space ([Formula: see text] means real analyticity), we consider a [Formula: see text]-cuspidal edge [Formula: see text] along [Formula: see text]. When [Formula: see text] is non-closed, in the authors’ previous works, the local existence of three distinct cuspidal edges along [Formula: see text] whose first fundamental forms coincide with that of [Formula: see text] was shown, under a certain reasonable assumption on [Formula: see text]. In this paper, if [Formula: see text] is closed, that is, [Formula: see text] is a knot, we show that there exist infinitely many cuspidal edges along [Formula: see text] having the same first fundamental form as that of [Formula: see text] such that their images are non-congruent to each other, in general.

scholarly journals Real analyticity for random dynamics of transcendental functions

2018 ◽  
Vol 40 (2) ◽  
pp. 490-520
Author(s):  
VOLKER MAYER ◽  
MARIUSZ URBAŃSKI ◽  
ANNA ZDUNIK
Keyword(s):  
Meromorphic Functions ◽  
Random Systems ◽  
Simple Approach ◽  
Pressure Function ◽  
Random Dynamics ◽  
Real Analyticity ◽  
Frobenius Operator ◽  
Transcendental Functions ◽  
Invariant Densities ◽  
Parameter Dependency

Analyticity results of expected pressure and invariant densities in the context of random dynamics of transcendental functions are established. These are obtained by a refinement of work by Rugh [On the dimension of conformal repellors, randomness and parameter dependency. Ann. of Math. (2) 168(3) (2008), 695–748] leading to a simple approach to analyticity. We work under very mild dynamical assumptions. Just the iterates of the Perron–Frobenius operator are assumed to converge. We also provide Bowen’s formula expressing the almost sure Hausdorff dimension of the radial fiberwise Julia sets in terms of the zero of an expected pressure function. Our main application establishes real analyticity for the variation of this dimension for suitable hyperbolic random systems of entire or meromorphic functions.

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