@article{MTMT:34767759, title = {Special families of piecewise linear iterated function systems}, url = {https://m2.mtmt.hu/api/publication/34767759}, author = {Prokaj, Rudolf Dániel and Simon, Károly}, journal-iso = {DYNAM SYST APPL}, journal = {DYNAMIC SYSTEMS AND APPLICATIONS}, volume = {accepted}, unique-id = {34767759}, issn = {1056-2176}, abstract = {This paper investigates the dimension theory of some families of continuous piecewise linear iterated function systems. For one family, we show that the Hausdorff dimension of the attractor is equal to the exponential growth rate obtained from the most natural covering system. We also prove that for Lebesgue typical parameters, the 1-dimensional Lebesgue measure of the underlying attractor is positive, if this number is bigger than 1 and all the contraction ratios are positive.}, year = {2024}, pages = {&} } @article{MTMT:34109543, title = {Fractal dimensions of continuous piecewise linear iterated function systems}, url = {https://m2.mtmt.hu/api/publication/34109543}, author = {Prokaj, Rudolf Dániel and Raith, Peter and Simon, Károly}, doi = {10.1090/proc/16430}, journal-iso = {P AM MATH SOC}, journal = {PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY}, volume = {151}, unique-id = {34109543}, issn = {0002-9939}, abstract = {We consider iterated function systems on the real line that consist of continuous, piecewise linear functions. Under a mild separation condition, we show that the Hausdorff and box dimensions of the attractor are equal to the minimum of 1 and the exponent which comes from the most natural system of covers of the attractor.}, keywords = {Mathematics, Applied; Key words and phrases; Piecewise linear iterated function system}, year = {2023}, eissn = {1088-6826}, pages = {4703-4719}, orcid-numbers = {Simon, Károly/0000-0003-2397-3828} } @article{MTMT:32916675, title = {Dimension estimates for C-1 iterated function systems and repellers. Part I}, url = {https://m2.mtmt.hu/api/publication/32916675}, author = {Feng, De-Jun and Simon, Károly}, doi = {10.1017/etds.2022.41}, journal-iso = {ERGOD THEOR DYN SYST}, journal = {ERGODIC THEORY AND DYNAMICAL SYSTEMS}, volume = {43}, unique-id = {32916675}, issn = {0143-3857}, abstract = {This is the first paper in a two-part series containing some results on dimension estimates for C-1 iterated function systems and repellers. In this part, we prove that the upper box-counting dimension of the attractor of any C-1 iterated function system (IFS) on R-d is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Similar results are obtained for the repellers for C-1 expanding maps on Riemannian manifolds.}, keywords = {PRESSURE; INVARIANT-MEASURES; Hausdorff dimension; Hausdorff dimension; THERMODYNAMIC FORMALISM; packing dimension; Mathematics, Applied; Equilibrium states; ITERATED FUNCTION SYSTEMS; singularity dimension; repellers of expanding maps; FULL DIMENSION}, year = {2023}, eissn = {1469-4417}, pages = {2673-2706}, orcid-numbers = {Simon, Károly/0000-0003-2397-3828} } @article{MTMT:33168630, title = {Dimension and measure of sums of planar sets and curves}, url = {https://m2.mtmt.hu/api/publication/33168630}, author = {Simon, Károly and Taylor, Krystal}, doi = {10.1112/mtk.12168}, journal-iso = {MATHEMATIKA}, journal = {MATHEMATIKA}, volume = {68}, unique-id = {33168630}, issn = {0025-5793}, abstract = {Considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the measure and dimension of A+Gamma:={a+v:a is an element of A,v is an element of Gamma}$A+\Gamma :=\lbrace a+v:a\in A, v\in \Gamma \rbrace$ when A subset of R2$A\subset \mathbb {R}<^>2$ and Gamma is a piecewise C2$\mathcal {C}<^>2$ curve. Assuming Gamma has non-vanishing curvature, we verify that: if dimHA <= 1$\dim _{\rm H} A \leqslant 1$, then dimH(A+Gamma)=dimHA+1$\dim _{\rm H} (A+\Gamma )=\dim _{\rm H} A +1$; if dimHA>1$\dim _{\rm H} A>1$, then L2(A+Gamma)>0$\mathcal {L}_2(A+\Gamma )>0$; if dimHA=1$\dim _{\rm H} A=1$ and H1(A)1(A) < \infty$, then L2(A+Gamma)=0$\mathcal {L}_2(A+\Gamma )=0$ if and only if A is an irregular (purely unrectifiable) 1-set. (a):(b):(c):In this article, we develop an approach using nonlinear projection theory which gives new proofs of (a) and (b) and the first proof of (c). Item (c) has a number of consequences: if a circle is thrown randomly on the plane, it will almost surely not intersect the four corner Cantor set. Moreover, the pinned distance set of an irregular 1-set has 1-dimensional Lebesgue measure equal to zero at almost every pin t is an element of R2$t\in \mathbb {R}<^>2$.}, keywords = {LOWER BOUNDS; Self-similar sets; Mathematics, Applied}, year = {2022}, eissn = {2041-7942}, pages = {1364-1392} } @article{MTMT:32857595, title = {Typical absolute continuity for classes of dynamically defined measures}, url = {https://m2.mtmt.hu/api/publication/32857595}, author = {Bárány, Balázs and Simon, Károly and Solomyak, Boris and Spiewak, Adam}, doi = {10.1016/j.aim.2022.108258}, journal-iso = {ADV MATH}, journal = {ADVANCES IN MATHEMATICS}, volume = {399}, unique-id = {32857595}, issn = {0001-8708}, abstract = {We consider one-parameter families of smooth uniformly contractive iterated function systems {f(j)(lambda)} on the real line. Given a family of parameter dependent measures {mu(lambda)} on the symbolic space, we study geometric and dimensional properties of their images under the natural projection maps Pi(lambda.) The main novelty of our work is that the measures mu(lambda) depend on the parameter, whereas up till now it has been usually assumed that the measure on the symbolic space is fixed and the parameter dependence comes only from the natural projection. This is especially the case in the question of absolute continuity of the projected measure (Pi(lambda))*mu(lambda), where we had to develop a new approach in place of earlier attempt which contains an error. Our main result states that if mu(lambda) are Gibbs measures for a family of Holder continuous potentials phi(lambda), with Holder continuous dependence on lambda and {Pi(lambda)} satisfy the transversality condition, then the projected measure (Pi(lambda))*mu(lambda & nbsp;)is absolutely continuous for Lebesgue a.e. lambda, such that the ratio of entropy over the Lyapunov exponent is strictly greater than 1. We deduce it from a more general almost sure lower bound on the Sobolev dimension for families of measures with regular enough dependence on the parameter. Under less restrictive assumptions, we also obtain an almost sure formula for the Hausdorff dimension. As applications of our results, we study stationary measures for iterated function systems with place-dependent probabilities (place-dependent Bernoulli convolutions and the Blackwell measure for binary channel) and equilibrium measures for hyperbolic IFS with overlaps (in particular: natural measures for non-homogeneous self-similar IFS and certain systems corresponding to random continued fractions). (C)& nbsp;2022 The Author(s). Published by Elsevier Inc.& nbsp;}, keywords = {SERIES; INVARIANT-MEASURES; Hausdorff dimension; Absolute continuity; ITERATED FUNCTION SYSTEMS; analyticity; Overlaps; Transversality; Self-similar measures; PARABOLIC IFS; Place-dependent measures; BERNOULLI CONVOLUTIONS}, year = {2022}, eissn = {1090-2082}, orcid-numbers = {Bárány, Balázs/0000-0002-0129-8385} } @article{MTMT:32524421, title = {Piecewise linear iterated function systems on the line of overlapping construction}, url = {https://m2.mtmt.hu/api/publication/32524421}, author = {Prokaj, Rudolf Dániel and Simon, Károly}, doi = {10.1088/1361-6544/ac355e}, journal-iso = {NONLINEARITY}, journal = {NONLINEARITY}, volume = {35}, unique-id = {32524421}, issn = {0951-7715}, year = {2022}, eissn = {1361-6544}, pages = {245-277} } @article{MTMT:32185828, title = {Dimension estimates for C^1 iterated function systems and repellers. Part II}, url = {https://m2.mtmt.hu/api/publication/32185828}, author = {FENG, DE-JUN and Simon, Károly}, doi = {10.1017/etds.2021.92}, journal-iso = {ERGOD THEOR DYN SYST}, journal = {ERGODIC THEORY AND DYNAMICAL SYSTEMS}, volume = {42}, unique-id = {32185828}, issn = {0143-3857}, abstract = {This is the second part of our study on the dimension theory of C-1 iterated function systems (IFSs) and repellers on R-d. In the first part [D.-J. Feng and K. Simon. Dimension estimates for C-1 iterated function systems and repellers. Part I. Preprint, 2020, arXiv:2007.15320], we proved that the upper box-counting dimension of the attractor of every C-1 IFS on R-d is bounded above by its singularity dimension, and the upper packing dimension of every ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Here we introduce a generalized transversality condition (GTC) for parameterized families of C-1 IFSs, and show that if the GTC is satisfied, then the dimensions of the IFS attractor and of the ergodic invariant measures are given by these upper bounds for almost every (in an appropriate sense) parameter. Moreover, we verify the GTC for some parameterized families of C-1 IFSs on R-d}, keywords = {PRESSURE; INVARIANT-MEASURES; Hausdorff dimension; THERMODYNAMIC FORMALISM; Variational principle; Mathematics, Applied; self-affine sets; ITERATED FUNCTION SYSTEMS; BOX; Hausdorff and box-counting dimensions; singularity dimension; Lyapunov dimension; NONCONFORMAL REPELLERS; PARABOLIC IFS}, year = {2022}, eissn = {1469-4417}, pages = {3357-3392} } @article{MTMT:32949662, title = {Dimension Estimates for C 1 Iterated Function Systems and C 1 Repellers, a Survey}, url = {https://m2.mtmt.hu/api/publication/32949662}, author = {Feng, D.-J. and Simon, Károly}, doi = {10.1007/978-3-030-74863-0_13}, journal-iso = {LECT NOTES MATH}, journal = {LECTURE NOTES IN MATHEMATICS}, volume = {2290}, unique-id = {32949662}, issn = {0075-8434}, abstract = {In this note we give a survey about some of the results related to fractal dimensions of attractors and ergodic measures of non-linear and non-conformal Iterated Function Systems (IFS) and the repellers of expanding maps on ℝd. The only new result in this note is the proof of the fact that Theorem 13.1.1 implies Theorem 13.1.2. © 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.}, year = {2021}, eissn = {1617-9692}, pages = {421-467} } @inproceedings{MTMT:32240128, title = {Dimension Theory of Some Non-Markovian Repellers Part II: Dynamically Defined Function Graphs}, url = {https://m2.mtmt.hu/api/publication/32240128}, author = {Bárány, Balázs and Rams, Michał and Simon, Károly}, booktitle = {Topological Dynamics and Topological Data Analysis}, doi = {10.1007/978-981-16-0174-3_3}, unique-id = {32240128}, year = {2021}, pages = {49-66}, orcid-numbers = {Bárány, Balázs/0000-0002-0129-8385} } @article{MTMT:32240123, title = {Dimension Theory of Some Non-Markovian Repellers Part I: A Gentle Introduction}, url = {https://m2.mtmt.hu/api/publication/32240123}, author = {Bárány, Balázs and Rams, Michał and Simon, Károly}, doi = {10.1007/978-981-16-0174-3_2}, journal-iso = {SPRINGER PROC MATH STAT}, journal = {SPRINGER PROCEEDINGS IN MATHEMATICS AND STATISTICS}, volume = {350}, unique-id = {32240123}, issn = {2194-1009}, year = {2021}, eissn = {2194-1017}, pages = {15-48}, orcid-numbers = {Bárány, Balázs/0000-0002-0129-8385} }