Real analyticity of Jacobian of invariant measures for hyperbolic meromorphic functions

2008 ◽  
Vol 40 (6) ◽  
pp. 1017-1024 ◽  
Author(s):  
Agnieszka Badeńska
Keyword(s):  
Invariant Measures ◽  
Meromorphic Functions ◽  
Real Analyticity

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.

Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order

2008 ◽  
Vol 28 (3) ◽  
pp. 915-946 ◽  
Author(s):  
VOLKER MAYER ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Finite Order ◽  
Exponential Family ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Gibbs Measures ◽  
Thermodynamic Formalism ◽  
Riemannian Metrics ◽  
Elliptic Functions ◽  
Real Analyticity ◽  
Conformal Measures

AbstractWorking with well chosen Riemannian metrics and employing Nevanlinna’s theory, we make the thermodynamic formalism work for a wide class of hyperbolic meromorphic functions of finite order (including in particular exponential family, elliptic functions, cosine, tangent and the cosine–root family and also compositions of these functions with arbitrary polynomials). In particular, the existence of conformal (Gibbs) measures is established and then the existence of probability invariant measures equivalent to conformal measures is proven. As a geometric consequence of the developed thermodynamic formalism, a version of Bowen’s formula expressing the Hausdorff dimension of the radial Julia set as the zero of the pressure function and, moreover, the real analyticity of this dimension, is proved.

Further investigations on Fujimoto type strong uniqueness polynomials

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5203-5216
Author(s):  
Abhijit Banerjee ◽  
Bikash Chakraborty ◽  
Sanjay Mallick
Keyword(s):  
Meromorphic Functions ◽  
Future Research ◽  
Strong Uniqueness ◽  
Open Questions

Taking the question posed by the first author in [1] into background, we further exhaust-ably investigate existing Fujimoto type Strong Uniqueness Polynomial for Meromorphic functions (SUPM). We also introduce a new kind of SUPM named Restricted SUPM and exhibit some results which will give us a new direction to discuss the characteristics of a SUPM. Moreover, throughout the paper, we pose a number of open questions for future research.

On Fixed Points of Meromorphic Functions f(z) and f(z + c), Δcf(z)

2019 ◽  
Vol 39 (5) ◽  
pp. 1277-1289
Author(s):  
Shuangting Lan ◽  
Zongxuan Chen
Keyword(s):  
Fixed Points ◽  
Meromorphic Functions

Extending Quasi-Invariant Measures by Using Subgroups of a Given Group

2003 ◽  
Vol 10 (2) ◽  
pp. 247-255
Author(s):  
A. Kharazishvili
Keyword(s):  
Invariant Measures ◽  
Uncountable Group

Abstract A method of extending σ-finite quasi-invariant measures given on an uncountable group, by using a certain family of its subgroups, is investigated.

scholarly journals Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 80
Author(s):  
Sergey Kryzhevich ◽  
Viktor Avrutin ◽  
Nikita Begun ◽  
Dmitrii Rachinskii ◽  
Khosro Tajbakhsh
Keyword(s):  
Risk Management ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Management Model ◽  
Closing Lemma ◽  
Symbolic Model ◽  
Metric Properties ◽  
Ergodic Measures ◽  
Maximal Invariant

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

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