scholarly journals Micromeasure distributions and applications for conformally generated fractals

2015 ◽  
Vol 159 (3) ◽  
pp. 547-566 ◽  
Author(s):  
JONATHAN M. FRASER ◽  
MARK POLLICOTT
Keyword(s):  
Finite Type ◽  
Julia Sets ◽  
Gibbs Measures ◽  
Schottky Groups ◽  
Limit Sets ◽  
Distance Set ◽  
Self Similar ◽  
Subshifts Of Finite Type

AbstractWe study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconer's distance set problem and dimensions of projections. Our analysis includes hyperbolic Julia sets, limit sets of Schottky groups and graph-directed self-similar sets.

scholarly journals Gibbs and equilibrium measures for some families of subshifts

2012 ◽  
Vol 33 (3) ◽  
pp. 934-953 ◽  
Author(s):  
TOM MEYEROVITCH
Keyword(s):  
Finite Type ◽  
Gibbs Measure ◽  
Equilibrium Measure ◽  
Gibbs Measures ◽  
Type Theorem ◽  
Equilibrium Measures ◽  
Strongly Irreducible ◽  
Subshifts Of Finite Type ◽  
Summable Variation

AbstractFor subshifts of finite type (SFTs), any equilibrium measure is Gibbs, as long as $f$ has $d$-summable variation. This is a theorem of Lanford and Ruelle. Conversely, a theorem of Dobrušin states that for strongly irreducible subshifts, shift-invariant Gibbs measures are equilibrium measures. Here we prove a generalization of the Lanford–Ruelle theorem: for all subshifts, any equilibrium measure for a function with $d$-summable variation is ‘topologically Gibbs’. This is a relaxed notion which coincides with the usual notion of a Gibbs measure for SFTs. In the second part of the paper, we study Gibbs and equilibrium measures for some interesting families of subshifts: $\beta $-shifts, Dyck shifts and Kalikow-type shifts (defined below). In all of these cases, a Lanford–Ruelle-type theorem holds. For each of these families, we provide a specific proof of the result.

Discrete Lyapunov exponents and Hausdorff dimension

2000 ◽  
Vol 20 (1) ◽  
pp. 145-172 ◽  
Author(s):  
SHMUEL FRIEDLAND
Keyword(s):  
Hausdorff Dimension ◽  
Finite Type ◽  
Lyapunov Exponents ◽  
Thermodynamic Formalism ◽  
Sufficient Conditions ◽  
Kleinian Groups ◽  
Limit Sets ◽  
Geometrically Finite ◽  
Subshifts Of Finite Type ◽  
Discrete Analogs

We study certain metrics on subshifts of finite type for which we define the discrete analogs of Lyapunov exponents. We prove Young's formula for $\mu$-Hausdorff dimension. We give sufficient conditions on the above metrics for which the Hausdorff dimension is given by thermodynamic formalism. We apply these results to the Hausdorff dimension of the limit sets of geometrically finite, purely loxodromic, Kleinian groups.

Subshifts of finite type and self-similar sets

Nonlinearity ◽  
2017 ◽  
Vol 30 (2) ◽  
pp. 659-686 ◽  
Author(s):  
Kan Jiang ◽  
Karma Dajani
Keyword(s):  
Finite Type ◽  
Self Similar ◽  
Subshifts Of Finite Type

Ruelle's transfer operator for random subshifts of finite type

1995 ◽  
Vol 15 (3) ◽  
pp. 413-447 ◽  
Author(s):  
Thomas Bogenschütz ◽  
Volker Mathias Gundlach
Keyword(s):  
Dynamical Systems ◽  
Finite Type ◽  
Invariant Measures ◽  
Gibbs Measures ◽  
Transfer Operator ◽  
Selection Procedure ◽  
Probability Measures ◽  
Random Functions ◽  
Subshifts Of Finite Type ◽  
Definition Of

AbstractWe consider a Ruelle—Perron—Frobenius type of selection procedure for probability measures that are invariant under random subshifts of finite type. In particular we prove that for a class of random functions this method leads to a unique probability exhibiting properties that justify the names Gibbs measure and equilibrium states. In order to do this we introduce the notion of bundle random dynamical systems and provide a theory for their invariant measures as well as give a precise definition of Gibbs measures.

Pressure and fractal indices for the Gibbs measures of hyperbolic Julia sets

1988 ◽  
Vol 21 (12) ◽  
pp. L639-L643 ◽  
Author(s):  
G Servizi ◽  
G Turchetti ◽  
S Vaienti
Keyword(s):  
Julia Sets ◽  
Gibbs Measures

Subshifts of finite type

1976 ◽  
pp. 117-130
Author(s):  
Manfred Denker ◽  
Christian Grillenberger ◽  
Karl Sigmund
Keyword(s):  
Finite Type ◽  
Subshifts Of Finite Type

scholarly journals SPACE-FILLING CURVES OF SELF-SIMILAR SETS (III): SKELETONS

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050028
Author(s):  
HUI RAO ◽  
SHU-QIN ZHANG
Keyword(s):  
Finite Type ◽  
Type Condition ◽  
Space Filling ◽  
Open Set Condition ◽  
Substitution Rule ◽  
Space Filling Curves ◽  
Neighbor Graph ◽  
Open Set ◽  
Self Similar

Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. In a previous paper by Dai and the authors [Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule, Nonlinearity 32(5) (2019) 1772–1809] it was shown that for all the connected self-similar sets with a skeleton satisfying the open set condition, space-filling curves can be constructed. In this paper, we give a criterion of existence of skeletons by using the so-called neighbor graph of a self-similar set. In particular, we show that a connected self-similar set satisfying the finite-type condition always possesses skeletons: an algorithm is obtained here.

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