|
Ergodic Theory and Dynamical Systems
Journal Prestige (SJR): 1.193  Citation Impact (citeScore): 1 Number of Followers: 3  Hybrid journal (It can contain Open Access articles)ISSN (Print) 0143-3857 - ISSN (Online) 1469-4417 Published by Cambridge University Press  [353 journals]
|
- ETS volume 45 issue 1 Cover and Front matter
-
Free pre-print version: Loading...
Pages: 1 - 2
PubDate: 2024-12-03
DOI: 10.1017/etds.2024.91
-
- ETS volume 45 issue 1 Cover and Back matter
-
Free pre-print version: Loading...
Pages: 1 - 2
PubDate: 2024-12-03
DOI: 10.1017/etds.2024.92
-
- Khintchine-type double recurrence in abelian groups
-
Free pre-print version: Loading...
Authors: ACKELSBERG; ETHAN
Pages: 1 - 33
Abstract: We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if
is a countable discrete abelian group, 
, and 
is an injective endomorphism with finite index image, then for any ergodic measure-preserving 
-system 
, any measurable set 
, and any 
, there is a syndetic set of 
such that 
. This generalizes the main results of Ackelsberg et al [Khintchine-type recurrence for 3-point configurations. Forum Math. Sigma 10 (2022), Paper no. e107] and essentially answers a question left open in that paper [Question 1.12; Khintchine-type recurrence for 3-point configurations. Forum Math. Sigma 10 (2022), Paper no. e107]. For the group 
, the result applies to pairs of endomorphisms given by matrices whose difference is non-singular. The key ingredients in the proof are: (1) a recent result obtained jointly with Bergelson and Shalom [Khintchine-type recurrence for 3-point configurations. Forum Math. Sigma 10 (2022), Paper no. e107] that says that the relevant ergodic averages are controlled by a characteristic factor closely related to the quasi-affine (or Conze–Lesigne) factor; (2) an extension trick to reduce to systems with well-behaved (with respect to 
PubDate: 2024-04-24
DOI: 10.1017/etds.2024.29
-
- Weighted topological pressure revisited
-
Free pre-print version: Loading...
Authors: ALIBABAEI; NIMA
Pages: 34 - 70
Abstract: Feng and Huang [Variational principle for weighted topological pressure. J. Math. Pures Appl. (9) 106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. Ergod. Th. & Dynam. Sys. 43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.
PubDate: 2024-05-08
DOI: 10.1017/etds.2024.35
-
- Schmidt games and Cantor winning sets
-
Free pre-print version: Loading...
Authors: BADZIAHIN; DZMITRY, HARRAP, STEPHEN, NESHARIM, EREZ, SIMMONS, DAVID
Pages: 71 - 110
Abstract: Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of study. We survey the definitions of the most common variants and connections between them. A new game called the Cantor game is invented and helps with presenting a unifying framework. We prove surprising new results such as the coincidence of absolute winning and
Cantor winning in metric spaces, and the fact that 
winning implies absolute winning for subsets of 
. We also suggest a prototypical example of a Cantor winning set to show the ubiquity of such sets in metric number theory and ergodic theory.
PubDate: 2024-04-19
DOI: 10.1017/etds.2024.23
-
- Asymptotic distribution for pairs of linear and quadratic forms at
integral vectors-
Free pre-print version: Loading...
Authors: HAN; JIYOUNG, LIM, SEONHEE, MALLAHI-KARAI, KEIVAN
Pages: 111 - 139
Abstract: We study the joint distribution of values of a pair consisting of a quadratic form
and a linear form 
over the set of integral vectors, a problem initiated by Dani and Margulis [Orbit closures of generic unipotent flows on homogeneous spaces of 
. Math. Ann. 286 (1990), 101–128]. In the spirit of the celebrated theorem of Eskin, Margulis and Mozes on the quantitative version of the Oppenheim conjecture, we show that if 
, then under the assumptions that for every 
, the form 
is irrational and that the signature of the restriction of 
to the kernel of 
is 
, where 
, the number of vectors 
for which 
, 
PubDate: 2024-04-17
DOI: 10.1017/etds.2024.30
-
- Symbolic dynamics for Hénon maps near the boundary of the horseshoe
locus-
Free pre-print version: Loading...
Authors: HIRONAKA; YUKI, ISHII, YUTAKA
Pages: 140 - 174
Abstract: Bedford and Smillie [A symbolic characterization of the horseshoe locus in the Hénon family. Ergod. Th. & Dynam. Sys. 37(5) (2017), 1389–1412] classified the dynamics of the Hénon map
defined on 
in terms of a symbolic dynamics when 
is close to the boundary of the horseshoe locus. The purpose of the current article is to generalize their results for all 
(including the case 
as well). The method of the proof is first to regard 
as a complex dynamical system in 
and second to introduce the new Markov-like partition in 
constructed by us [On parameter loci of the Hénon family. Comm. Math. Phys. 361(2) (2018), 343–414].
PubDate: 2024-05-23
DOI: 10.1017/etds.2024.34
-
- Complexity of non-abelian cut-and-project sets of polytopal type I:
special homogeneous Lie groups-
Free pre-print version: Loading...
Authors: KAISER; PETER
Pages: 175 - 217
Abstract: The aim of this paper is to determine the asymptotic growth rate of the complexity function of cut-and-project sets in the non-abelian case. In the case of model sets of polytopal type in homogeneous two-step nilpotent Lie groups, we can establish that the complexity function asymptotically behaves like
. Further, we generalize the concept of acceptance domains to locally compact second countable groups.
PubDate: 2024-05-13
DOI: 10.1017/etds.2024.38
-
- Poissonian pair correlation for directions in multi-dimensional affine
lattices and escape of mass estimates for embedded horospheres-
Free pre-print version: Loading...
Authors: KIM; WOOYEON, MARKLOF, JENS
Pages: 218 - 246
Abstract: We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author [The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. (2) 172 (2010), 1949–2033], and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded
-horospheres in the space of affine lattices.
PubDate: 2024-04-26
DOI: 10.1017/etds.2024.31
-
- Homology and K-theory of dynamical systems IV. Further structural results
on groupoid homology-
Free pre-print version: Loading...
Authors: PROIETTI; VALERIO, YAMASHITA, MAKOTO
Pages: 247 - 273
Abstract: We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math. 521 (2000), 25–46], with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for principal groupoids whose orbits are copies of an Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.
PubDate: 2024-05-15
DOI: 10.1017/etds.2024.37
-
- On invariant holonomies between centers
-
Free pre-print version: Loading...
Authors: SAGHIN; RADU
Pages: 274 - 293
Abstract: We prove that for
, 
-bunched, dynamically coherent partially hyperbolic diffeomorphisms, the stable and unstable holonomies between center leaves are 
, and the derivative depends continuously on the points and on the map. Also for 
, 
-bunched partially hyperbolic diffeomorphisms, the derivative cocycle restricted to the center bundle has invariant continuous holonomies which depend continuously on the map. This generalizes previous results by Pugh, Shub, and Wilkinson; Burns and Wilkinson; Brown; Obata; Avila, Santamaria, and Viana; and Marin.
PubDate: 2024-05-08
DOI: 10.1017/etds.2024.33
-
- On transversal Hölder regularity for flat Wieler solenoids
-
Free pre-print version: Loading...
Authors: TREVIÑO; RODRIGO
Pages: 294 - 320
Abstract: This paper studies various aspects of inverse limits of locally expanding affine linear maps on flat branched manifolds, which I call flat Wieler solenoids. Among the aspects studied are different types of cohomologies, the rates of mixing given by the Ruelle spectrum of the hyperbolic map acting on this space, and solutions of the cohomological equation in primitive substitution subshifts for Hölder functions. The overarching theme is that considerations of
-Hölder regularity on Cantor sets go a long way.
PubDate: 2024-09-10
DOI: 10.1017/etds.2024.41
-
