Some remarks about self-products and entropy
arxiv
Irreducibility and Periodicity in Z^2 Symbolic Systems
arxiv.
Analytic Self-Maps of Cantor Sets in the Line
arxiv.
On self-similar sets with overlaps and inverse theorems for entropy in R^d
To appear in Memoires of the AMS, preprint, arxiv.
A short proof of Host's equidistribution theorem
Israel Journal of Mathematics 2022, vol 251, 527-539, journal, arxiv
Hausdorff dimension of planar self-affine sets and measures with overlaps
With Ariel Rapaport.
Journal of the EMS 2022, Vol 24, No. 7, pp. 2361-2441, journal, arxiv
Hausdorff dimension of planar self-affine sets and measures
With Balázs Bárány and Ariel Rapaport.
Inventiones Mathematicae 2019, Volume 216, Issue 3, pp 601–659, journal, arxiv
Every Borel automorphism without finite invariant measures admits a two-set generator
Journal of the European Mathematical Society, Volume 21, Issue 1, 2019, pp. 271–317, journal, arxiv.
Smooth symmetries of xa invariant sets
Journal of Modern Dynamics, 2018, vol 13, pp 187-197. journal, arxiv.
Self embeddings of Bedford-McMullen carpets
With Amir Algom.
Ergodic Theory Dynamical Systems, Volume 39, Issue 3, March 2019, pp. 577-603, journal, arxiv.
On the dimension of Furstenberg measure for SL(2,R) random matrix products
With Boris Solomyak.
Inventiones Mathematicae 2017, Volume 210, Issue 3, pp 815–875, journal, arxiv ,
Clarification.
On the Turing Degree Spectra of Minimal Subshifts
With Pascal Vanier.
CSR 2017, pp 154-161, journal, arxiv.
Some problems on the boundary of fractal geometry and additive combinatorics
Recent Developments in Fractals and Related Fields, 2017, pp 129-174, journal, arxiv.
Equidistribution from fractal measures
With Pablo Shmerkin.
Inventiones Mathematicae Volume 202 (2015), Issue 1, pp 427-479, journal,
arxiv.
Self-similar sets, entropy and additive combinatorics
In Geometry and Analysis of Fractals,
Springer Proceedings in Mathematics & Statistics Volume 88, 2014, pp 225-252,
book,
arxiv,
Remark about print version.
On self-similar sets with overlaps and inverse theorems for entropy
Annals of Mathematics 180 (2014), no. 2, 773–822.
journal,
arxiv.
On the ratio ergodic theorem for group actions
Journal of the London Mathematical Society (2) 88 (2013), no. 2, 465-482 journal, arxiv.
Isomorphism and embedding into Markov shifts off universally null sets
Acta Applicandae Mathematicae, Volume 126 (Hang Kim Memorial Volume), Issue 1 (2013), Page 187-201.
journal,
arxiv,
erratum.
Examples of non-polygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models
With Michael Damron.
Annals of Applied Probability 2013, Vol. 23, No. 3, 1074-1085, journal,
arxiv.
Geometric rigidity of times-m invariant measures
Journal of the European Mathematical Society, Volume 14, Issue 5, 2012, pp. 1539–1563,
journal,
arxiv,
erratum.
On the finite-dimensional marginals of shift-invariant measures
With
J.-R. Chazottes , J.-M. Gambaudo , and E.
Ugalde .
Ergodic Theory and Dynamical Systems, Volume 32, Issue 5, 2012 , pp 1485-1500, journal, arxiv
Local entropy averages and projections of fractal measures.
With Pablo Shmerkin.
Annals of Mathematics, Volume 175 Number 3 (2012), 1001-1059, journal,
arxiv.
Slow entropy and differentiable models for infinite measure preserving Z^d actions
Ergodic Theory and Dynamical Systems 32 (2012), 653--674 (special volume in memory of Dan Rudolph), journal,
arxiv
Inverses, products and cartesian powers of topologically
deterministic maps
With Artur Siemaszko
Topological Methods in Nonlinear Analysis 39 (2012) No. 1,
journal,
arxiv
Rohlin properties for $Z^d$-actions on the Cantor set.
Transactions of the AMS 364 (2012) No. 3, 1127--1143
journal,
arxiv
On notions of determinism in topological dynamics.
Ergodic Theory Dynamical Systems, 32 (2012) no. 1, 119--140,
journal,
arxiv.
Non-expansive directions for $Z^2$ actions.
Ergodic Theory Dynamical Systems 31 (2011), no. 1, 91--112,
journal,
arxiv.
A Characterization of the Entropies of Multidimensional Shifts of Finite Type.
With Tom Meyerovitch,
Annals of Mathematics Vol. 171 (2010), No. 3, 2011-2038,
journal,
arxiv.
On zero-temperature limits of Gibbs states.
With J.R. Chazottes,
Communications in Mathematical Physics Volume 297, Number 1 / July, 2010, 265-281.
journal,
arxiv,
Clarification.
On the automorphism group of multidimensional SFTs.
Ergodic Theory Dynamical Systems (2010), 30, 809--840.
journal,
preprint.
A ratio ergodic theorem for multiparameter actions.
Journal of the European Mathematical Society 12 (2010), no. 2, 365--383.
journal,
arxiv.
Upcrossing inequalities for stationary sequences and applications to entropy and complexity
Ann. Probab. Volume 37, Number 6 (2009), 2135-2149.
journal,
arxiv.
On universality in multidimensional symbolic dynamics,
Discrete and Continuous Dynamical Systems S, Volume: 2, No. 2, 2009.
journal,
arxiv.
On the dynamics and recursive properties of multidimensional symbolic systems .
Inventiones Mathematicae, Volume 176, Number 1, April, 2009.
journal,
preprint
On processes which cannot be distinguished by finitary observation.
With Yonatan Gutman,
Israel journal of Mathematics, Volume 164, Number 1 / March, 2008,
journal,
arxiv.
Genericity in Topological Dynamics.
Ergodic Theory Dynamical Systems (2008), 28 : 125-165,
journal
arxiv.
Averaging sequences and abelian rank in amenable groups.
Israel Journal of Mathematics, Volume 158, Number 1 / March, 2007,
journal,
arxiv.
Return times, recurrence densities and entropy for actions of some discrete amenable groups,
Journal d'Analyse Mathematique, Volume 100, Number 1 / December, 2006,
journal,
arxiv.
Dimension theory of self-similar sets and measures
Proceedings of the ICM 2018. pdf
Lectures on dynamics, fractal geometry and metric number theory.
Lecture notes frmo the 2011 Bedlewo summer schoold on “Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory”. The notes cover Furstenebrg's theory of CP chains, and applications to intersections and projections of fractals (including Furstenberg's 1970 theorem on intersections of Cantor sets and the Peres-Shmerkin and Hochman-Shmerkin theorems on projections. pdf, journal version.
On one-sided and two sided ergodic averages of non-integrable functions.
This note answers a question raised by Yves Derriennic in the Fall 2008 semester on Ergodic Theory at MSRI. In retrospect it is a pretty simple thing. A similar proof was given independently by Benjamin Weiss. pdf.
Probability measures on R^d are amenable
This is the measure-theoretic analogue of the main lemma in my paper on
ratio ergodic theorems. It is apparentl;y not known (let me know if you
have seen it before).. pdf.
Recurrence densities for ergodic processes
This unbulished note is a short (6 pages, incl. intro & bib) version of the main result from the paper above, formulated for stationary processes (indexed by Z rather than a general amenable group).pdf
Fractal Geometry (2023)
A graduate course in fractal geometry. About 2/3 of these notes are similar to the 2012 notes, but with many correcetions. The course covers box and Hausdorff dimension of sets, dimension of measures, iterated functions systems, density theorems, Marstrand's theorem, entropy dimension and some discussion fo additive combinatorics (the notes are not complete at the end of the course). pdf.
Topological dynamics (2022)
A graduate course in topological dynamics. We covered minimality, equicontinuity and group translations, expansive and symbolic systems, the enveloping semigroup and application to partition regularity of IP-sets, transitivity and weak mixing, continuous spectrum in minimal non-wek-mixing systems, distal systems, topological entropy, mixing SFTs, symbolic coding of toral automorphisms, Van der Waerden's theorem and Furstenberg's x2,x3 theorem. pdf
Ergodic theory (2017 / 2012)
Introductory course in ergodic theory, covering recurrence, th ergodic theorem, ergodicity, weak mixing and pure point spectrum. In the 2017 course we covered entropy, in 2012 a little less entropy but some disjointness and Rohlin's lemma. 2017, 2012.
Fractal geometry (2012)
We covered box and Hausdorff dimension of sets, dimension of measures, iterated functions systems, density theorems, Marstrand's theorem, Bernoulli convolutions and other projection-type examples, Furstenberg's theorem of microsets and CP-chains. pdf
Notes on information theory (source coding).
These are rough notes from a short (3 lecture) course I gave for advanced undergraduates on information theory, focusing on source coding. I would recommend some background in elementary probability, athough the notes include a brief review. Topics include: Kraft's ineuqality and Shannon's coding theorems, definition and properties of entropy, mean entropy for stationary processes, Markov chains and LLN for them, Shannon-McMillan theorem for Markov chains, and the Lempel-Ziv universal coding algorithm. pdf.
Together with my friends Yonatan Harel, Ittay Weiss and Ofek Shilon, I have written a calculus textbook to accompany the first year course given at the Hebrew University (the book is in Hebrew, of course). Here is the table of contents. The book is available from Academon Press. Corrections can be found on the book site.