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About the Algebraic Yuzvinski Formula

Anna Giordano Bruno, Simone Virili
2015 Topological Algebra and its Applications  
AbstractThe 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  ... 
doi:10.1515/taa-2015-0008 fatcat:3pryqfi2mnhojkb3xwiq3736re

Adjoint entropy vs Topological entropy [article]

Anna Giordano Bruno
2011 arXiv   pre-print
Recently the adjoint 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.  ... 
arXiv:1007.0346v5 fatcat:qqn26utis5b6rn3t46xi7t2ari

Adjoint entropy vs topological entropy

Anna Giordano Bruno
2012 Topology and its Applications  
In particular, the topological adjoint 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.  ... 
doi:10.1016/j.topol.2011.07.032 fatcat:grf3h6d2l5btholq4bgazgatne

Topological Entropy and Algebraic Entropy for group endomorphisms [article]

Dikran Dikranjan, Anna Giordano Bruno
2013 arXiv   pre-print
We are mainly concerned with the topological 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  ... 
arXiv:1308.4019v1 fatcat:ocyqkeklmbhfhf63f3c6fcyupq

Discrete dynamical systems in group theory [article]

Dikran Dikranjan, Anna Giordano Bruno
2013 arXiv   pre-print
In the last part we recall the definition 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.  ... 
arXiv:1308.4035v1 fatcat:go2gv73j4zhpxogzkpeyccbb2q

Adjoint algebraic entropy

Dikran Dikranjan, Anna Giordano Bruno, Luigi Salce
2010 Journal of Algebra  
The new notion 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.  ... 
doi:10.1016/j.jalgebra.2010.03.025 fatcat:jjobgsl4yjfi7jxh3w4jrvcgde

Algebraic Yuzvinski Formula [article]

Anna Giordano Bruno, Simone Virili
2013 arXiv   pre-print
Topological 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  ... 
arXiv:1111.1287v3 fatcat:zb54t265hvbitkxhlb5xups6vu

When the intrinsic algebraic entropy is not really intrinsic

Brendan Goldsmith, Luigi Salce
2015 Topological Algebra and its Applications  
AbstractThe intrinsic 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  ... 
doi:10.1515/taa-2015-0005 fatcat:jbgfs6uvsvgrdnj7jca4fvltyq

Entropy of Endomorphisms of Lie Groups [article]

André Caldas, Mauro Patrão
2012 arXiv   pre-print
Since every compact 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] ).  ... 
arXiv:1105.4344v2 fatcat:2mmi7srtozannfysu6atsmntsm

Noncommutative topological entropy of endomorphisms of Cuntz algebras II [article]

Adam Skalski
2010 arXiv   pre-print
A study 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.  ... 
arXiv:1002.2276v1 fatcat:3ar5oflzbnh2hngilhz2le7xkq

Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian Groups

Brendan Goldsmith, Ketao Gong
2015 Topological Algebra and its Applications  
AbstractNecessary 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.  ... 
doi:10.1515/taa-2015-0007 fatcat:t7bvbobv6ze2lbvg6zkuiiavga

Noncommutative Topological Entropy of Endomorphisms of Cuntz Algebras II

Adam Skalski
2011 Publications of the Research Institute for Mathematical Sciences  
A study 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.  ... 
doi:10.2977/prims/54 fatcat:4caly66j6nbajhdd5vmm2nonbe

Algebraic entropy for Abelian groups

Dikran Dikranjan, Brendan Goldsmith, Luigi Salce, Paolo Zanardo
2009 Transactions of the American Mathematical Society  
Here we study the 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.  ... 
doi:10.1090/s0002-9947-09-04843-0 fatcat:dgu5ynml6jdlteyrfim5lmxkr4

Endomorphisms and automorphisms from subfactors illustrating non-commutative entropy

Anne Louise Svendsen
2005 Mathematica Scandinavica  
We present a series 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.  ... 
doi:10.7146/math.scand.a-14975 fatcat:fcfckbjiwjbizcwzyl6sj5377m

Entropy for endomorphisms of LCA groups

Simone Virili
2012 Topology and its Applications  
We study some 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.  ... 
doi:10.1016/j.topol.2011.02.017 fatcat:eidl35cfbjblxc3rspyoqhoyxa
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