Conformal measures and Hausdorff dimension for infinitely renormalizable quadratic polynomials

1999 ◽  
Vol 30 (1) ◽  
pp. 31-52
Author(s):  
Eduardo A. Prado
Keyword(s):  
Hausdorff Dimension ◽  
Quadratic Polynomials ◽  
Conformal Measures

Local dimension for piecewise monotonic maps on the interval

1995 ◽  
Vol 15 (6) ◽  
pp. 1119-1142 ◽  
Author(s):  
Franz Hofbauer
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measures ◽  
Positive Characteristic ◽  
Characteristic Exponent ◽  
Local Dimension ◽  
Pressure Function ◽  
Piecewise Monotonic ◽  
Almost Everywhere ◽  
Conformal Measures

AbstractThe local dimension of invariant and conformal measures for piecewise monotonic transformations on the interval is considered. For ergodic invariant measures m with positive characteristic exponent χm we show that the local dimension exists almost everywhere and equals hm/χm For certain conformal measures we show a relation between a pressure function and the Hausdorff dimension of sets, on which the local dimension is constant.

Conformal trace theorem for Julia sets of quadratic polynomials

2017 ◽  
Vol 39 (9) ◽  
pp. 2481-2506 ◽  
Author(s):  
A. CONNES ◽  
E. MCDONALD ◽  
F. SUKOCHEV ◽  
D. ZANIN
Keyword(s):  
Hausdorff Dimension ◽  
Jordan Curve ◽  
Hausdorff Measure ◽  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Trace Theorem ◽  
Quadratic Polynomials ◽  
Full Proof

If $c$ is in the main cardioid of the Mandelbrot set, then the Julia set $J$ of the map $\unicode[STIX]{x1D719}_{c}:z\mapsto z^{2}+c$ is a Jordan curve of Hausdorff dimension $p\in [1,2)$. We provide a full proof of a formula for the Hausdorff measure on $J$ in terms of singular traces announced by the first named author in 1996.

THE AVERAGE DENSITY OF SELF-CONFORMAL MEASURES

2001 ◽  
Vol 63 (3) ◽  
pp. 721-734 ◽  
Author(s):  
M. ZÄHLE
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Iterated Function System ◽  
Main Tool ◽  
Average Density ◽  
Energy Integral ◽  
Almost Everywhere ◽  
Iterated Function ◽  
Conformal Measures ◽  
Conformal Iterated Function System

The paper calculates the average density of the normalized Hausdorff measure on the fractal set generated by a conformal iterated function system. It equals almost everywhere a positive constant given by a truncated generalized s-energy integral, where s is the corresponding Hausdorff dimension. As a main tool a conditional Gibbs measure is determined. The appendix proves an appropriate extension of Birkhoff's ergodic theorem which is also of independent interest.

HAUSDORFF DIMENSION OF JULIA SETS OF QUADRATIC POLYNOMIALS

Fractals ◽  
2018 ◽  
Vol 26 (03) ◽  
pp. 1850020
Author(s):  
LUIS MANUEL MARTÍNEZ ◽  
GAMALIEL BLÉ
Keyword(s):  
Hausdorff Dimension ◽  
Julia Sets ◽  
Quadratic Polynomial ◽  
Small Oscillation ◽  
Expanding Maps ◽  
Quadratic Polynomials

The Hausdorff dimension of Julia sets of expanding maps can be computed by the eigenvalue algorithm. In this work, an implementation of this algorithm for quadratic polynomial, that allows the calculation of the Hausdorff dimension of Julia sets for complex parameters, is done. In particular, the parameters in a neighborhood of the parabolic parameter [Formula: see text] are analyzed and a small oscillation in Hausdorff dimension is shown.

On t-conformal measures and Hausdorff dimension for a family of non-uniformly hyperbolic horseshoes

2009 ◽  
Vol 29 (6) ◽  
pp. 1917-1950 ◽  
Author(s):  
RENAUD LEPLAIDEUR ◽  
ISABEL RIOS
Keyword(s):  
Hausdorff Dimension ◽  
Equilibrium State ◽  
Unstable Manifold ◽  
Markov Partition ◽  
Limit Set ◽  
Unique Equilibrium ◽  
Conformal Measures ◽  
Continuous Potential ◽  
Homoclinic Tangencies ◽  
Unique Equilibrium State

AbstractIn this paper we consider horseshoes with homoclinic tangencies inside the limit set. For a class of such maps, we prove the existence of a unique equilibrium state μt, associated to the (non-continuous) potential −tlog Ju. We also prove that the Hausdorff dimension of the limit set, in any open piece of unstable manifold, is the unique number t0 such that the pressure of μt0 is zero. To deal with the discontinuity of the jacobian, we introduce a countable Markov partition adapted to the dynamics, and work with the first return map defined in a rectangle of it.

Hausdorff dimension 2 for Julia sets of quadratic polynomials

2001 ◽  
Vol 237 (3) ◽  
pp. 571-583 ◽  
Author(s):  
Stefan-M. Heinemann ◽  
Bernd O. Stratmann
Keyword(s):  
Hausdorff Dimension ◽  
Julia Sets ◽  
Quadratic Polynomials ◽  
Dimension 2

scholarly journals Hausdorff dimension of biaccessible angles for quadratic polynomials

2017 ◽  
Vol 238 (3) ◽  
pp. 201-239 ◽  
Author(s):  
Henk Bruin ◽  
Dierk Schleicher
Keyword(s):  
Hausdorff Dimension ◽  
Quadratic Polynomials

Parabolic iterated function systems

2000 ◽  
Vol 20 (5) ◽  
pp. 1423-1447 ◽  
Author(s):  
R. D. MAULDIN ◽  
M. URBAŃSKI
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measures ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Topological Pressure ◽  
Limit Set ◽  
Iterated Function ◽  
Conformal Measures ◽  
Conformal Iterated Function System ◽  
Dynamical Features

In this paper we introduce and explore conformal parabolic iterated function systems. We define and study topological pressure, Perron–Frobenius-type operators, semiconformal and conformal measures and the Hausdorff dimension of the limit set. With every parabolic system we associate an infinite hyperbolic conformal iterated function system and we employ it to study geometric and dynamical features (properly defined invariant measures for example) of the limit set.

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