scholarly journals Induced maps of hyperbolic Bernoulli systems

2001 ◽  
Vol 7 (1) ◽  
pp. 147-154
Author(s):  
Matthew Nicol ◽  
Keyword(s):  
Induced Maps ◽  
Bernoulli Systems

scholarly journals Homological epimorphisms, homotopy epimorphisms and acyclic maps

2020 ◽  
Vol 32 (6) ◽  
pp. 1395-1406
Author(s):  
Joseph Chuang ◽  
Andrey Lazarev
Keyword(s):  
Wide Class ◽  
Topological Spaces ◽  
Loop Spaces ◽  
Graded Algebras ◽  
Differential Graded Algebras ◽  
Algebraic Description ◽  
Acyclic Map ◽  
Induced Maps

AbstractWe show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.

A note on semi-open and almost open induced maps

2021 ◽  
pp. 107823
Author(s):  
Alvaro Andrade ◽  
Javier Camargo
Keyword(s):  
Almost Open ◽  
Induced Maps

scholarly journals Monotone maps on dendrites and their induced maps

2016 ◽  
Vol 204 ◽  
pp. 121-134 ◽  
Author(s):  
Haithem Abouda ◽  
Issam Naghmouchi
Keyword(s):  
Monotone Maps ◽  
Induced Maps

Lightness of Induced Maps and Homeomorphisms

2011 ◽  
Vol 54 (4) ◽  
pp. 607-618 ◽  
Author(s):  
Javier Camargo
Keyword(s):  
Positive Integer ◽  
Negative Answer ◽  
Arcwise Connected ◽  
Induced Maps

AbstractAn example is given of a map f defined between arcwise connected continua such that C(f) is light and 2f is not light, giving a negative answer to a question of Charatonik and Charatonik. Furthermore, given a positive integer n, we study when the lightness of the induced map 2f or Cn(f) implies that f is a homeomorphism. Finally, we show a result in relation with the lightness of C(C(f)).

scholarly journals Smale endomorphisms over graph-directed Markov systems

2020 ◽  
pp. 1-34
Author(s):  
EUGEN MIHAILESCU ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Equilibrium Measure ◽  
Parabolic Systems ◽  
Skew Product ◽  
Pointwise Dimension ◽  
Markov Systems ◽  
Large Classes ◽  
Equilibrium Measures ◽  
Skew Products ◽  
Natural Extensions ◽  
Induced Maps

We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$ -maps $T_{\unicode[STIX]{x1D6FD}}$ , for arbitrary $\unicode[STIX]{x1D6FD}>1$ .

Non-Bernoulli systems with completely positive entropy

2008 ◽  
Vol 28 (1) ◽  
pp. 87-124 ◽  
Author(s):  
A. H. DOOLEY ◽  
V. YA. GOLODETS ◽  
D. J. RUDOLPH ◽  
S. D. SINEL’SHCHIKOV
Keyword(s):  
Infinite Order ◽  
Amenable Group ◽  
Positive Entropy ◽  
Completely Positive ◽  
New Approach ◽  
Amenable Groups ◽  
Uncountable Family ◽  
Cutting And Stacking ◽  
Bernoulli Systems

AbstractA new approach to actions of countable amenable groups with completely positive entropy (cpe), allowing one to answer some basic questions in this field, was recently developed. The question of the existence of cpe actions which are not Bernoulli was raised. In this paper, we prove that every countable amenable groupG, which contains an element of infinite order, has non-Bernoulli cpe actions. In fact we can produce, for any$h \in (0, \infty ]$, an uncountable family of cpe actions of entropyh, which are pairwise automorphically non-isomorphic. These actions are given by a construction which we call co-induction. This construction is related to, but different from the standard induced action. We study the entropic properties of co-induction, proving that ifαGis co-induced from an actionαΓof a subgroup Γ, thenh(αG)=h(αΓ). We also prove that ifαΓis a non-Bernoulli cpe action of Γ, thenαGis also non-Bernoulli and cpe. Hence the problem of finding an uncountable family of pairwise non-isomorphic cpe actions of the same entropy is reduced to one of finding an uncountable family of non-Bernoulli cpe actions of$\mathbb Z$, which pairwise satisfy a property we call ‘uniform somewhat disjointness’. We construct such a family using refinements of the classical cutting and stacking methods.

scholarly journals Torsion in kernels of induced maps on divisor class groups

2015 ◽  
Vol 14 (08) ◽  
pp. 1550123
Author(s):  
Sean Sather-Wagstaff ◽  
Sandra Spiroff
Keyword(s):  
Projective Dimension ◽  
Class Groups ◽  
Divisor Class ◽  
Finite Projective Dimension ◽  
Induced Maps

We investigate torsion elements in the kernel of the map on divisor class groups of excellent local normal domains A and A/I, for an ideal I of finite projective dimension. The motivation for this work is a result of Griffith–Weston which applies when I is principal.

scholarly journals Lifting measures to inducing schemes

2008 ◽  
Vol 28 (2) ◽  
pp. 553-574 ◽  
Author(s):  
YA. B. PESIN ◽  
S. SENTI ◽  
K. ZHANG
Keyword(s):  
Metric Spaces ◽  
Borel Probability Measure ◽  
Equilibrium Measures ◽  
One Dimensional ◽  
Compact Metric Spaces ◽  
Borel Probability ◽  
One Dimensional Maps ◽  
Induced Maps ◽  
Inducing Schemes ◽  
Markov Extensions

AbstractIn this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [Y. Pesin and S. Senti. Thermodynamical formalism associated with inducing schemes for one-dimensional maps. Mosc. Math. J.5(3) (2005), 669–678; Y. Pesin and S. Senti. Equilibrium measures for maps with inducing schemes. Preprint, 2007]. We show that under some natural assumptions on the inducing schemes—which hold for many known examples—any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the inducing scheme. The argument uses the construction of connected Markov extensions due to Buzzi [J. Buzzi. Markov extensions for multi-dimensional dynamical systems. Israel J. Math.112 (1999), 357–380], his results on the liftability of measures of large entropy, and a generalization of some results by Bruin [H. Bruin. Induced maps, Markov extensions and invariant measures in one-dimensional dynamics. Comm. Math. Phys.168(3) (1995), 571–580] on relations between inducing schemes and Markov extensions. We apply our results to study the liftability problem for one-dimensional cusp maps (in particular, unimodal and multi-modal maps) and for some multi-dimensional maps.

Residual behavior of induced maps

1996 ◽  
Vol 93 (1) ◽  
pp. 387-398 ◽  
Author(s):  
Andres Del Junco ◽  
Daniel J. Rudolph
Keyword(s):  
Induced Maps
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