TY - JOUR AU - Prokaj, Rudolf Dániel AU - Simon, Károly TI - Special families of piecewise linear iterated function systems JF - DYNAMIC SYSTEMS AND APPLICATIONS J2 - DYNAM SYST APPL VL - accepted PY - 2024 SP - & SN - 1056-2176 UR - https://m2.mtmt.hu/api/publication/34767759 ID - 34767759 AB - 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. LA - English DB - MTMT ER - TY - JOUR AU - Prokaj, Rudolf Dániel AU - Raith, Peter AU - Simon, Károly TI - Fractal dimensions of continuous piecewise linear iterated function systems JF - PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY J2 - P AM MATH SOC VL - 151 PY - 2023 IS - 11 SP - 4703 EP - 4719 PG - 17 SN - 0002-9939 DO - 10.1090/proc/16430 UR - https://m2.mtmt.hu/api/publication/34109543 ID - 34109543 N1 - Funding Agency and Grant Number: National Research, Development and Innovation Office-NKFIH [K142169]; Stiftung Aktion Osterich Ungarn [103ou6] Funding text: & nbsp;The research of the first and third authors was partially supported by National Research, Development and Innovation Office-NKFIH, Project K142169. This work was partially supported by the grant Stiftung Aktion Osterich Ungarn 103ou6. AB - 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. LA - English DB - MTMT ER - TY - JOUR AU - Feng, De-Jun AU - Simon, Károly TI - Dimension estimates for C-1 iterated function systems and repellers. Part I JF - ERGODIC THEORY AND DYNAMICAL SYSTEMS J2 - ERGOD THEOR DYN SYST VL - 43 PY - 2023 IS - 8 SP - 2673 EP - 2706 PG - 34 SN - 0143-3857 DO - 10.1017/etds.2022.41 UR - https://m2.mtmt.hu/api/publication/32916675 ID - 32916675 N1 - Funding Agency and Grant Number: Hong Kong Research Grant Council [CUHK14301218]; Direct Grant for Research in CUHK; [OTKA K104745] Funding text: The research of Feng was partially supported by the General Research Fund CUHK14301218 from the Hong Kong Research Grant Council, and by a Direct Grant for Research in CUHK. The research of Simon was partially supported by the grant OTKA K104745. The authors are grateful to Ching-Yin Chan for reading an early version of this paper and catching some typos. AB - 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. LA - English DB - MTMT ER - TY - JOUR AU - Simon, Károly AU - Taylor, Krystal TI - Dimension and measure of sums of planar sets and curves JF - MATHEMATIKA J2 - MATHEMATIKA VL - 68 PY - 2022 IS - 4 SP - 1364 EP - 1392 PG - 29 SN - 0025-5793 DO - 10.1112/mtk.12168 UR - https://m2.mtmt.hu/api/publication/33168630 ID - 33168630 N1 - Funding Agency and Grant Number: MTA-BME Stochastics Research Group [OTKA 123782]; Simons Foundation [523555]; MRI at the Ohio State Funding text: MRI at the Ohio State; MTA-BME Stochastics Research Group, Grant/Award Number: OTKA 123782; Simons Foundation, Grant/Award Number: 523555; ICERM: this work came out of a collaboration initiated at ICERM at Brown University in Rhode Island AB - 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$t\in \mathbb {R}<^>2$. LA - English DB - MTMT ER - TY - JOUR AU - Bárány, Balázs AU - Simon, Károly AU - Solomyak, Boris AU - Spiewak, Adam TI - Typical absolute continuity for classes of dynamically defined measures JF - ADVANCES IN MATHEMATICS J2 - ADV MATH VL - 399 PY - 2022 PG - 73 SN - 0001-8708 DO - 10.1016/j.aim.2022.108258 UR - https://m2.mtmt.hu/api/publication/32857595 ID - 32857595 N1 - Export Date: 16 October 2023 AB - 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; LA - English DB - MTMT ER - TY - JOUR AU - Prokaj, Rudolf Dániel AU - Simon, Károly TI - Piecewise linear iterated function systems on the line of overlapping construction JF - NONLINEARITY J2 - NONLINEARITY VL - 35 PY - 2022 IS - 1 SP - 245 EP - 277 PG - 33 SN - 0951-7715 DO - 10.1088/1361-6544/ac355e UR - https://m2.mtmt.hu/api/publication/32524421 ID - 32524421 N1 - Export Date: 29 April 2022 LA - English DB - MTMT ER - TY - JOUR AU - FENG, DE-JUN AU - Simon, Károly TI - Dimension estimates for C^1 iterated function systems and repellers. Part II JF - ERGODIC THEORY AND DYNAMICAL SYSTEMS J2 - ERGOD THEOR DYN SYST VL - 42 PY - 2022 IS - 11 SP - 3357 EP - 3392 PG - 36 SN - 0143-3857 DO - 10.1017/etds.2021.92 UR - https://m2.mtmt.hu/api/publication/32185828 ID - 32185828 N1 - Export Date: 22 September 2021 Funding details: Hungarian Scientific Research Fund, OTKA, K104745 Funding details: Chinese University of Hong Kong, CUHK Funding text 1: The authors would like to thank the referee for helpful comments and suggestions. They also thank Zhou Feng for catching some typos. The research of D.-J.F. was partially supported by a HKRGC GRF grant and the Direct Grant for Research in CUHK. The research of K.S. was partially supported by the grant OTKA K104745. A significant part of the research was done during the visit of K.S. to CUHK, which was supported by a HKRGC GRF grant. Export Date: 23 September 2021 Funding details: Hungarian Scientific Research Fund, OTKA, K104745 Funding details: Chinese University of Hong Kong, CUHK Funding text 1: The authors would like to thank the referee for helpful comments and suggestions. They also thank Zhou Feng for catching some typos. The research of D.-J.F. was partially supported by a HKRGC GRF grant and the Direct Grant for Research in CUHK. The research of K.S. was partially supported by the grant OTKA K104745. A significant part of the research was done during the visit of K.S. to CUHK, which was supported by a HKRGC GRF grant. AB - 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 LA - English DB - MTMT ER - TY - JOUR AU - Feng, D.-J. AU - Simon, Károly TI - Dimension Estimates for C 1 Iterated Function Systems and C 1 Repellers, a Survey JF - LECTURE NOTES IN MATHEMATICS J2 - LECT NOTES MATH VL - 2290 PY - 2021 SP - 421 EP - 467 PG - 47 SN - 0075-8434 DO - 10.1007/978-3-030-74863-0_13 UR - https://m2.mtmt.hu/api/publication/32949662 ID - 32949662 N1 - Megjelent a 'Thermodynamic Formalism' kötetben, Softcover ISBN: 978-3-030-74862-3 Export Date: 14 July 2022 Correspondence Address: Simon, K.; Department of Stochastics, Hungary; email: simonk@math.bme.hu AB - 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. LA - English DB - MTMT ER - TY - CHAP AU - Bárány, Balázs AU - Rams, Michał AU - Simon, Károly ED - Vinod Kumar, P.B. ED - Chan, Kit C. ED - Devaney, Robert L. TI - Dimension Theory of Some Non-Markovian Repellers Part II: Dynamically Defined Function Graphs T2 - Topological Dynamics and Topological Data Analysis PB - Springer-Verlag Singapore CY - Singapore SN - 9789811601736 T3 - Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009 ; 350. PY - 2021 SP - 49 EP - 66 PG - 18 DO - 10.1007/978-981-16-0174-3_3 UR - https://m2.mtmt.hu/api/publication/32240128 ID - 32240128 N1 - Department of Stochastics, MTA-BME Stochastics Research Group, Budapest University of Technology and Economics, P.O. Box 91, Budapest, 1521, Hungary Polish Academy of Sciences, Institute of Mathematics, ul. Śniadeckich 8, Warszawa, 00-656, Poland Conference code: 266019 Export Date: 14 July 2022 Correspondence Address: Bárány, B.; Department of Stochastics, P.O. Box 91, Hungary; email: balubsheep@gmail.com LA - English DB - MTMT ER - TY - JOUR AU - Bárány, Balázs AU - Rams, Michał AU - Simon, Károly TI - Dimension Theory of Some Non-Markovian Repellers Part I: A Gentle Introduction JF - SPRINGER PROCEEDINGS IN MATHEMATICS AND STATISTICS J2 - SPRINGER PROC MATH STAT VL - 350 PY - 2021 SP - 15 EP - 48 PG - 34 SN - 2194-1009 DO - 10.1007/978-981-16-0174-3_2 UR - https://m2.mtmt.hu/api/publication/32240123 ID - 32240123 N1 - Megjelent a Topological Dynamics and Topological Data Analysis c. kötetben, Hardcover ISBN: 978-981-16-0173-6 Department of Stochastics, MTA-BME Stochastics Research Group, Budapest University of Technology and Economics, P.O. Box 91, Budapest, 1521, Hungary Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, Warszawa, 00-656, Poland Conference code: 266019 Export Date: 14 July 2022 Correspondence Address: Simon, K.; Department of Stochastics, P.O. Box 91, Hungary; email: simonk@math.bme.hu LA - English DB - MTMT ER -