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About the Algebraic Yuzvinski Formula

2015
*
Topological Algebra and its Applications
*

AbstractThe

doi:10.1515/taa-2015-0008
fatcat:3pryqfi2mnhojkb3xwiq3736re
*Algebraic*Yuzvinski Formula expresses the*algebraic**entropy**of*an*endomorphism**of*a finitedimensional rational vector space as the Mahler measure*of*its characteristic polynomial. ... Finally,we give a new*and*purely*algebraic*proof*of*the*Algebraic*Yuzvinski Formula for the intrinsic*algebraic**entropy*. ... Simone Virili was partially supported by Fondazione Cassa di Risparmio di Padova e Rovigo (Progetto di Eccellenza "*Algebraic*structures*and*their applications")*and*the projects DGI MINECO MTM2011-28992 ...##
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Adjoint entropy vs Topological entropy
[article]

2011
*
arXiv
*
pre-print

Recently the adjoint

arXiv:1007.0346v5
fatcat:qqn26utis5b6rn3t46xi7t2ari
*algebraic**entropy**of**endomorphisms**of*abelian*groups*was introduced*and*studied. ... In particular, the topological adjoint*entropy**and*the topological*entropy*coincide on continuous*endomorphisms**of*totally disconnected compact abelian*groups*. ... McAndrew defined the*algebraic**entropy**of**endomorphisms**of*abelian*groups*. ...##
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Adjoint entropy vs topological entropy

2012
*
Topology and its Applications
*

In particular, the topological adjoint

doi:10.1016/j.topol.2011.07.032
fatcat:grf3h6d2l5btholq4bgazgatne
*entropy**and*the topological*entropy*coincide on continuous*endomorphisms**of*totally disconnected compact abelian*groups*. ... We generalize the notion*of*adjoint*entropy*to continuous*endomorphisms**of*topological abelian*groups*. ... Acknowledgements I would like to thank Professor Dikran Dikranjan for his very useful comments*and*suggestions,*and*Simone Virili for checking the computations in Section 6. ...##
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Topological Entropy and Algebraic Entropy for group endomorphisms
[article]

2013
*
arXiv
*
pre-print

We are mainly concerned with the topological

arXiv:1308.4019v1
fatcat:ocyqkeklmbhfhf63f3c6fcyupq
*and*the*algebraic**entropy*in the context*of*continuous*endomorphisms**of*locally compact*groups*, paying special attention to the case*of*compact*and*discrete ... Furthermore we give some connections with*other*topics in Mathematics as Mahler measure*and*Lehmer Problem from Number Theory,*and*the growth rate*of**groups**and*Milnor Problem from Geometric*Group*Theory ... Moreover in the*algebraic*context*other**entropies*were defined for*endomorphisms**of*modules by means*of*real-valued module invariants in [90] ; these*entropies*generalize the*algebraic**entropy*ent in ...##
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Discrete dynamical systems in group theory
[article]

2013
*
arXiv
*
pre-print

In the last part we recall the definition

arXiv:1308.4035v1
fatcat:go2gv73j4zhpxogzkpeyccbb2q
*and*the fundamental properties*of*the*algebraic**entropy*for*group**endomorphisms*, noting how its deeper properties depend on the specific setting. ... First we recall the notion*of*semigroup*entropy*h_S in the category S*of*normed semigroups*and*contractive homomorphisms, recalling also its properties. ... Virili for his kind permission to anticipate here some*of*the main results from [27] . Thanks are due also to J. Spevák for letting us insert his example in item (b)*of*Example 2.12,*and*to L. ...##
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Adjoint algebraic entropy

2010
*
Journal of Algebra
*

The new notion

doi:10.1016/j.jalgebra.2010.03.025
fatcat:jjobgsl4yjfi7jxh3w4jrvcgde
*of*adjoint*algebraic**entropy**of**endomorphisms**of*Abelian*groups*is introduced. Various examples*and*basic properties are provided. ... As applications, we compute the adjoint*algebraic**entropy**of*the shift*endomorphisms**of*direct sums,*and*we prove an Addition Theorem for the adjoint*algebraic**entropy**of*bounded Abelian*groups*. ... the*algebraic**entropy**of*continuous*endomorphisms**of*compact Abelian*groups*: Corollary 7.8. ...##
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Algebraic Yuzvinski Formula
[article]

2013
*
arXiv
*
pre-print

Topological

arXiv:1111.1287v3
fatcat:zb54t265hvbitkxhlb5xups6vu
*entropy*is very well-understood for*endomorphisms**of*compact Abelian*groups*. ... In two papers*of*1979*and*1981 Peters gave a different definition*of**entropy*for automorphisms*of*locally compact Abelian*groups*. ... On the*other*hand, it is again the*Algebraic*Yuzvinski Formula that allows one to prove the fundamental connection between the*algebraic**entropy**of**endomorphisms**of*discrete Abelian*groups**and*the topological ...##
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When the intrinsic algebraic entropy is not really intrinsic

2015
*
Topological Algebra and its Applications
*

AbstractThe intrinsic

doi:10.1515/taa-2015-0005
fatcat:jbgfs6uvsvgrdnj7jca4fvltyq
*algebraic**entropy*ent(ɸ)*of*an*endomorphism*ɸ*of*an Abelian*group*G can be computed using fully inert subgroups*of*ɸ-invariant sections*of*G, instead*of*the whole family*of*ɸ-inert ... For a class*of**groups*containing the*groups**of*finite rank, aswell as those groupswhich are trajectories*of*finitely generated subgroups, it is proved that only fully inert subgroups*of*the*group*itself ... Acknowledgement: The authors would like to thank Anna Giordano Bruno*and*Simone Virili for their useful comments given during the preparation*of*the paper,*and*Paolo Zanardo for some discussions on the ...##
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Entropy of Endomorphisms of Lie Groups
[article]

2012
*
arXiv
*
pre-print

Since every compact

arXiv:1105.4344v2
fatcat:2mmi7srtozannfysu6atsmntsm
*group*is reductive, the formula for the*entropy**of*a*endomorphism**of*a compact*group*reduces to the formula for the*entropy**of*an*endomorphism**of*a torus. ... We show, when G is a nilpotent or reductive Lie*group*, that the*entropy**of*any surjective*endomorphism*coincides with the*entropy**of*its restriction to the toral component*of*the center*of*G. ... In fact, the formula for the*entropy**of*a*endomorphism**of*a compact*group*reduces to the formula for the*entropy**of*an*endomorphism**of*a torus (see [12] ). ...##
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Noncommutative topological entropy of endomorphisms of Cuntz algebras II
[article]

2010
*
arXiv
*
pre-print

A study

arXiv:1002.2276v1
fatcat:3ar5oflzbnh2hngilhz2le7xkq
*of*noncommutative topological*entropy**of*gauge invariant*endomorphisms**of*Cuntz*algebras*began in our earlier work with Joachim Zacharias is continued*and*extended to*endomorphisms*which are not ... In particular it is shown that if H is an N-dimensional Hilbert space, V is an irreducible multiplicative unitary on the tensor product*of*H with itself*and*F is the tensor flip, then the Voiculescu*entropy*... The work on this note was started during a visit*of*the author to University*of*Tokyo in October-November 2009 funded by a JSPS Short Term Postdoctoral Fellowship. ...##
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Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian Groups

2015
*
Topological Algebra and its Applications
*

AbstractNecessary

doi:10.1515/taa-2015-0007
fatcat:t7bvbobv6ze2lbvg6zkuiiavga
*and*sufficient conditions to ensure that the direct sum*of*two Abelian*groups*with zero*entropy*is again*of*zero*entropy*are still unknown; interestingly the same problem is also unresolved ... for direct sums*of*Hopfian*and*co-Hopfian groups.We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the*groups*. ... Definitions*and*Basic results In this section, we recall the definitions*of**algebraic**entropy**and*adjoint*entropy**of**groups*,*and*list some useful properties*of*such*entropies*. ...##
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Noncommutative Topological Entropy of Endomorphisms of Cuntz Algebras II

2011
*
Publications of the Research Institute for Mathematical Sciences
*

A study

doi:10.2977/prims/54
fatcat:4caly66j6nbajhdd5vmm2nonbe
*of*noncommutative topological*entropy**of*gauge invariant*endomorphisms**of*Cuntz*algebras*begun in our earlier work with J. ... Zacharias is continued*and*extended to*endomorphisms*which are not necessarily*of*permutation type. ... Acknowledgements The work on this note was started during a visit*of*the author to University*of*Tokyo in October-November 2009 funded by a JSPS Short Term Postdoctoral Fellowship. ...##
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Algebraic entropy for Abelian groups

2009
*
Transactions of the American Mathematical Society
*

Here we study the

doi:10.1090/s0002-9947-09-04843-0
fatcat:dgu5ynml6jdlteyrfim5lmxkr4
*algebraic**entropy**of*the*endomorphisms**of*Abelian*groups*, introduced in 1965 by Adler, Konheim*and*McAndrew. ...*and**of*the*endomorphism*induced on the quotient*group*. ...*algebraic**entropy**of**endomorphisms**of*discrete torsion Abelian*groups*. ...##
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Endomorphisms and automorphisms from subfactors illustrating non-commutative entropy

2005
*
Mathematica Scandinavica
*

We present a series

doi:10.7146/math.scand.a-14975
fatcat:fcfckbjiwjbizcwzyl6sj5377m
*of*examples*of**endomorphisms**and*automorphisms arising from subfactors, which illustrate some*of*the recent theorems in non-commutative*entropy*theory. ... Moreover it is shown that for these examples the Connes-Størmer*entropy**of*the automorphism is maximal*and*coincides with the topological*entropy*. ... The author would like to thank Professors Erling Størmer*and*Sergey Neshveyev, both University*of*Oslo, Norway, for many fruitful conversations. ...##
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Entropy for endomorphisms of LCA groups

2012
*
Topology and its Applications
*

We study some

doi:10.1016/j.topol.2011.02.017
fatcat:eidl35cfbjblxc3rspyoqhoyxa
*of*the basic properties*of*this new*entropy**and*we give direct proofs*of*the formulae that allow one to compute the*entropy**of**endomorphisms**of*Z N , R N*and*C N , for every positive integer ...*entropy*defined for locally compact Abelian*groups*by Peters. ...*and*found a counterexample that helped me in understanding how to use the convex hull in computing*entropy*,*and*Prof. ...
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