Real analyticity for random dynamics of transcendental functions

VOLKER MAYER; MARIUSZ URBAŃSKI; ANNA ZDUNIK

doi:10.1017/etds.2018.42

Real analyticity for random dynamics of transcendental functions

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.42 ◽

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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METASTABILITY, LYAPUNOV EXPONENTS, ESCAPE RATES, AND TOPOLOGICAL ENTROPY IN RANDOM DYNAMICAL SYSTEMS

Stochastics and Dynamics ◽

10.1142/s0219493713500044 ◽

2013 ◽

Vol 13 (04) ◽

pp. 1350004 ◽

Cited By ~ 6

Author(s):

GARY FROYLAND ◽

OGNJEN STANCEVIC

Keyword(s):

Dynamical Systems ◽

Finite Type ◽

Topological Entropy ◽

Lower Bounds ◽

Random Systems ◽

Random Dynamical Systems ◽

Upper And Lower Bounds ◽

Random System ◽

Random Maps ◽

Frobenius Operator

We explore the concept of metastability in random dynamical systems, focusing on connections between random Perron–Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron–Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint random systems with rigorous upper and lower bounds on (i) the escape rate in the setting of random maps, and (ii) topological entropy in the setting of random shifts of finite type, respectively.

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Random dynamics of transcendental functions

Journal d Analyse Mathématique ◽

10.1007/s11854-018-0007-1 ◽

2018 ◽

Vol 134 (1) ◽

pp. 201-235 ◽

Cited By ~ 2

Author(s):

Volker Mayer ◽

Mariusz Urbański

Keyword(s):

Random Dynamics ◽

Transcendental Functions

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Invariant densities for random systems of the interval

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.127 ◽

2020 ◽

pp. 1-39

Author(s):

CHARLENE KALLE ◽

MARTA MAGGIONI

Keyword(s):

Linear Systems ◽

Invariant Measures ◽

Piecewise Linear ◽

Random Systems ◽

Density Functions ◽

Expanding Maps ◽

Absolutely Continuous ◽

Random System ◽

Piecewise Linear Systems ◽

Invariant Densities

Abstract For random piecewise linear systems T of the interval that are expanding on average we construct explicitly the density functions of absolutely continuous T-invariant measures. If the random system uses only expanding maps our procedure produces all invariant densities of the system. Examples include random tent maps, random W-shaped maps, random $\beta $ -transformations and random Lüroth maps with a hole.

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

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385707000648 ◽

2008 ◽

Vol 28 (3) ◽

pp. 915-946 ◽

Cited By ~ 14

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.

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Real analyticity of Jacobian of invariant measures for hyperbolic meromorphic functions

Bulletin of the London Mathematical Society ◽

10.1112/blms/bdn083 ◽

2008 ◽

Vol 40 (6) ◽

pp. 1017-1024 ◽

Cited By ~ 2

Author(s):

Agnieszka Badeńska

Keyword(s):

Invariant Measures ◽

Meromorphic Functions ◽

Real Analyticity

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Influence of X-Ray Induced Fluorescence on Energy Dispersive X-Ray Analysis of Thin Foils

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100079164 ◽

1977 ◽

Vol 35 ◽

pp. 328-329

Author(s):

E. A. Kenik ◽

J. Bentley

Keyword(s):

Thin Foil ◽

Electron Excitation ◽

Simple Approach ◽

Foil Thickness ◽

X Rays ◽

X Ray ◽

Hole Spectrum ◽

Illumination System ◽

Thin Foils ◽

Intensity Ratios

Cliff and Lorimer (1) have proposed a simple approach to thin foil x-ray analy sis based on the ratio of x-ray peak intensities. However, there are several experimental pitfalls which must be recognized in obtaining the desired x-ray intensities. Undesirable x-ray induced fluorescence of the specimen can result from various mechanisms and leads to x-ray intensities not characteristic of electron excitation and further results in incorrect intensity ratios.In measuring the x-ray intensity ratio for NiAl as a function of foil thickness, Zaluzec and Fraser (2) found the ratio was not constant for thicknesses where absorption could be neglected. They demonstrated that this effect originated from x-ray induced fluorescence by blocking the beam with lead foil. The primary x-rays arise in the illumination system and result in varying intensity ratios and a finite x-ray spectrum even when the specimen is not intercepting the electron beam, an ‘in-hole’ spectrum. We have developed a second technique for detecting x-ray induced fluorescence based on the magnitude of the ‘in-hole’ spectrum with different filament emission currents and condenser apertures.

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