On the dimension of Furstenberg measure for $${ SL}_{2}(\mathbb {R})$$ S L 2 ( R ) random matrix products

Hochman, Michael; Solomyak, Boris

doi:10.1007/s00222-017-0740-6

Cited by 34 publications

(49 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The reader is referred to [18] for a discussion of more recent applications. We will recall the main result of [18], since it will be the main tool in proving Theorem 1.7.…”

Section: Ifs Of Linear Fractional Transformationsmentioning

confidence: 99%

Diophantine Property of Matrices and Attractors of Projective Iterated Function Systems in ℝℙ1

Solomyak

Takahashi

2020

International Mathematics Research Notices

Self Cite

View full text Add to dashboard Cite

We prove that almost every finite collection of matrices in GL d (R) and SL d (R) with positive entries is Diophantine. Next we restrict ourselves to the case d = 2. A finite set of SL 2 (R) matrices induces a (generalized) iterated function system on the projective line RP 1 . Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.

show abstract

“…The reader is referred to [18] for a discussion of more recent applications. We will recall the main result of [18], since it will be the main tool in proving Theorem 1.7.…”

Section: Ifs Of Linear Fractional Transformationsmentioning

confidence: 99%

Diophantine Property of Matrices and Attractors of Projective Iterated Function Systems in ℝℙ1

Solomyak

Takahashi

2020

International Mathematics Research Notices

Self Cite

View full text Add to dashboard Cite

show abstract

“…It follows that up to an O(1) error (a n ) is super-additive, so lim n→∞ 1 n a n exists, as desired. 8…”

Section: Cylinder Decomposition Of Stationary Measures and Entropy DImentioning

confidence: 99%

Some Problems on the Boundary of Fractal Geometry and Additive Combinatorics

Hochman

2017

Trends in Mathematics

Self Cite

View full text Add to dashboard Cite

This paper is an exposition, with some new applications, of our results from [5,6] on the growth of entropy of convolutions. We explain the main result on R, and derive, via a linearization argument, an analogous result for the action of the affine group on R. We also develop versions of the results for entropy dimension and Hausdorff dimension. The method is applied to two problems on the border of fractal geometry and additive combinatorics. First, we consider attractors X of compact families Φ of similarities of R. We conjecture that if Φ is uncountable and X is not a singleton (equivalently, Φ is not contained in a 1-parameter semigroup) then dim X = 1. We show that this would follow from the classical overlaps conjecture for self-similar sets, and unconditionally we show that if X is not a point and dim Φ > 0 then dim X = 1. Second, we study a problem due to Shmerkin and Keleti, who have asked how small a set ∅ = Y ⊆ R can be if at every point it contains a scaled copy of the middle-third Cantor set K. Such a set must have dimension at least dim K and we show that its dimension is at least dim K + δ for some constant δ > 0.

show abstract

“…Assumption (A3) follows from choice of E 1 (see (105)). Assumption (A4), (the transversality condition) follows from the fact that the associated 3-dimensional IFS S, defined in (14), satisfies SSP. Namely, the third coordinate of S is an IFS F on the line, defined in (9).…”

Section: Examplesmentioning

confidence: 99%

“…In the last two years there have been a very intensive development on this filed, partially due to the use of Furstenberg measure. See [1], [3], [5], [4], [19], [7], [14].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the dimension of triangular self-affine sets

Bárány

Rams

Simon

2017

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

On the dimension of Furstenberg measure for $${ SL}_{2}(\mathbb {R})$$ S L 2 ( R ) random matrix products

Abstract: Let µ be a measure on SL2(R) generating a non-compact and totally irreducible subgroup, and let ν be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if µ is supported on finitely many matrices with algebraic entries, then

Cited by 34 publications

References 24 publications

Diophantine Property of Matrices and Attractors of Projective Iterated Function Systems in ℝℙ1

Diophantine Property of Matrices and Attractors of Projective Iterated Function Systems in ℝℙ1

Some Problems on the Boundary of Fractal Geometry and Additive Combinatorics

On the dimension of triangular self-affine sets

Contact Info

Product

Resources

About