261 research outputs found
Borel Isomorphism of SPR Markov Shifts
We show that strongly positively recurrent Markov shifts (in particular
shifts of finite type) are classified up to Borel conjugacy by their entropy,
period and their numbers of periodic points
Good potentials for almost isomorphism of countable state Markov shifts
Almost isomorphism is an equivalence relation on countable state Markov
shifts which provides a strong version of Borel conjugacy; still, for mixing
SPR shifts, entropy is a complete invariant of almost isomorphism. In this
paper, we establish a class of potentials on countable state Markov shifts
whose thermodynamic formalism is respected by almost isomorphism
Almost isomorphism for countable state Markov shifts
Countable state Markov shifts are a natural generalization of the well-known
subshifts of finite type. They are the subject of current research both for
their own sake and as models for smooth dynamical systems. In this paper, we
investigate their almost isomorphism and entropy conjugacy and obtain a
complete classification for the especially important class of strongly positive
recurrent Markov shifts. This gives a complete classification up to entropy
conjugacy of the natural extensions of smooth entropy expanding maps, including
all smooth interval maps with non-zero topological entropy
Flow Equivalence of G-SFTs
In this paper, a G-shift of finite type (G-SFT) is a shift of finite type
together with a free continuous shift-commuting action by a finite group G. We
reduce the classification of G-SFTs up to equivariant flow equivalence to an
algebraic classification of a class of poset-blocked matrices over the integral
group ring of G. For a special case of two irreducible components with
G, we compute explicit complete invariants. We relate our matrix
structures to the Adler-Kitchens-Marcus group actions approach. We give
examples of G-SFT applications, including a new connection to involutions of
cellular automata.Comment: The paper has been augmented considerably and the second version is
now 81 pages long. This version has been accepted for publication in
Transactions of the American Mathematical Societ
Path methods for strong shift equivalence of positive matrices
In the early 1990's, Kim and Roush developed path methods for establishing
strong shift equivalence (SSE) of positive matrices over a dense subring U of
the real numbers R. This paper gives a detailed, unified and generalized
presentation of these path methods. New arguments which address arbitrary dense
subrings U of R are used to show that for any dense subring U of R, positive
matrices over U which have just one nonzero eigenvalue and which are strong
shift equivalent over U must be strong shift equivalent over U_+. In addition,
we show positive real matrices on a path of shift equivalent positive real
matrices are SSE over R_+; positive rational matrices which are SSE over R_+
must be SSE over Q_+; and for any dense subring U of R, within the set of
positive matrices over U which are conjugate over U to a given matrix, there
are only finitely many SSE-U_+ classes.Comment: This version adds a 3-part program for studying SEE over the reals.
One part is handled by the arxiv post "Strong shift equivalence and algebraic
K-theory". This version is the author version of the paper published in the
Kim memorial volume. From that, my short lifestory of Kim (and more) is on my
web page http://www.math.umd.edu/~mboyle/papers/index.htm
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