scholarly journals Hausdorff dimension for fractals invariant under multiplicative integers

2011 ◽  
Vol 32 (5) ◽  
pp. 1567-1584 ◽  
Author(s):  
RICHARD KENYON ◽  
YUVAL PERES ◽  
BORIS SOLOMYAK
Keyword(s):  
Variational Principle ◽  
Hausdorff Dimension ◽  
Sequence Space ◽  
Symbolic Sequence

AbstractWe consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0–1 sequences (xk) such thatxkx2k=0 for allk. We compute the Hausdorff and Minkowski dimensions of these sets and show that they are typically different. The proof proceeds via a variational principle for multiplicative subshifts.

scholarly journals The Projection Pressure for Asymptotically Weak Separation Condition and Bowen's Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Chenwei Wang ◽  
Ercai Chen
Keyword(s):  
Variational Principle ◽  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Canonical Projection ◽  
Separation Condition ◽  
Weak Separation Condition ◽  
Iterated Function ◽  
Conformal Iterated Function System ◽  
Weak Separation ◽  
Projection Pressure

Let{Si}i=1lbe a weakly conformal iterated function system onRdwith attractorK. Letπbe the canonical projection. In this paper we define a new concept called “projection pressure”Pπ(φ)forφ∈C(Σ)and show the variational principle about the projection pressure under AWSC. Furthermore, we check that the zero of “projection pressure” still satisfies Bowen's equation. Using the root of Bowen's equation, we can get the Hausdorff dimension of the attractorK.

APPLICATIONS OF A RELATIVE VARIATIONAL PRINCIPLE TO DIMENSIONS OF NONCONFORMAL EXPANDING MAPS

2011 ◽  
Vol 11 (04) ◽  
pp. 643-679 ◽  
Author(s):  
YUKI YAYAMA
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Hausdorff Dimension ◽  
Ergodic Measure ◽  
Sierpinski Carpet ◽  
3 Dimensional ◽  
Sofic Shift ◽  
Full Shift ◽  
Sierpiński Carpet ◽  
Invariant Ergodic Measure

Zhao and Cao (2008) showed the relative variational principle for subadditive potentials in random dynamical systems. Applying their result, we find the Hausdorff dimension of an n (≥3)-dimensional general Sierpiński carpet which has an irreducible sofic shift in symbolic representation and study an invariant ergodic measure of full Hausdorff dimension. These generalize the results of Kenyon and Peres (1996) on the Hausdorff dimension of an n-dimensional general Sierpiński carpet represented by a full shift.

ON HAUSDORFF DIMENSION AND TOPOLOGICAL ENTROPY

Fractals ◽  
2010 ◽  
Vol 18 (03) ◽  
pp. 363-370 ◽  
Author(s):  
DONGKUI MA ◽  
MIN WU
Keyword(s):  
Variational Principle ◽  
Hausdorff Dimension ◽  
Topological Space ◽  
Topological Entropy ◽  
Continuous Map ◽  
Compact Topological Space ◽  
Metric Function

Let f: X → X be a continuous map of a compact topological space. If there exists a metric function on X and it satisfies some restricted conditions, we obtain some relationships between Hausdorff dimension and topological entropy for any Z ⊆ X. Using those results, we also obtain a variational principle of dimensions, generalize some known results and give some examples.

HAUSDORFF DIMENSION OF CERTAIN RANDOM SELF-AFFINE FRACTALS

2011 ◽  
Vol 11 (04) ◽  
pp. 627-642 ◽  
Author(s):  
NUNO LUZIA
Keyword(s):  
Variational Principle ◽  
Hausdorff Dimension ◽  
Finite Number ◽  
The Self ◽  
Affine Transformations ◽  
Fractal Set ◽  
Bernoulli Measure ◽  
Sierpinski Carpets

In this work we are interested in the self-affine fractals studied by Gatzouras and Lalley [5] and by the author [11] who generalize the famous general Sierpinski carpets studied by Bedford [1] and McMullen [13]. We give a formula for the Hausdorff dimension of sets which are randomly generated using a finite number of self-affine transformations each generating a fractal set as mentioned before. The choice of the transformation is random according to a Bernoulli measure. The formula is given in terms of the variational principle for the dimension.

scholarly journals Dimensions of ‘self-affine sponges’ invariant under the action of multiplicative integers

2021 ◽  
pp. 1-43
Author(s):  
GUILHEM BRUNET
Keyword(s):  
Variational Principle ◽  
Similar Case ◽  
The Self ◽  
Probability Measures ◽  
Symbolic Sequence ◽  
Fixed Integer ◽  
Dimension Formula ◽  
Self Similar ◽  
Borel Probability ◽  
Sierpinski Carpets

Abstract Let $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\ldots ,m_1-1\} \times \{0,\ldots ,m_2-1\})^{\mathbb {N}^*}$ that are invariant under the action of the semigroup of multiplicative integers. These sets are defined following Kenyon, Peres, and Solomyak and using a fixed integer $q \geq 2$ . We compute the Hausdorff and Minkowski dimensions of the projection of these sets onto an affine grid of the unit square. The proof of our Hausdorff dimension formula proceeds via a variational principle over some class of Borel probability measures on the studied sets. This extends well-known results on self-affine Sierpiński carpets. However, the combinatoric arguments we use in our proofs are more elaborate than in the self-similar case and involve a new parameter, namely $j = \lfloor \log _q ( {\log (m_1)}/{\log (m_2)} ) \rfloor $ . We then generalize our results to the same subsets defined in dimension $d \geq 2$ . There, the situation is even more delicate and our formulas involve a collection of $2d-3$ parameters.

scholarly journals Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure

2012 ◽  
Vol 33 (3) ◽  
pp. 831-850 ◽  
Author(s):  
YONGLUO CAO ◽  
HUYI HU ◽  
YUN ZHAO
Keyword(s):  
Variational Principle ◽  
Hausdorff Dimension ◽  
Ergodic Measure ◽  
Topological Pressure ◽  
Compact Sets ◽  
Equivalent Definition ◽  
Ergodic Measures ◽  
Nonadditive Measure ◽  
Definition Of ◽  
Conformal Repeller

AbstractWithout any additional conditions on subadditive potentials, this paper defines subadditive measure-theoretic pressure, and shows that the subadditive measure-theoretic pressure for ergodic measures can be described in terms of measure-theoretic entropy and a constant associated with the ergodic measure. Based on the definition of topological pressure on non-compact sets, we give another equivalent definition of subadditive measure-theoretic pressure, and obtain an inverse variational principle. This paper also studies the superadditive measure-theoretic pressure which has similar formalism to the subadditive measure-theoretic pressure. As an application of the main results, we prove that an average conformal repeller admits an ergodic measure of maximal Hausdorff dimension. Furthermore, for each ergodic measure supported on an average conformal repeller, we construct a set whose dimension is equal to the dimension of the measure.

Sign in / Sign up

Export Citation Format

Share Document