Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Intrinsic metric on graded graphs,standardness,and invariant measures   We define a general notion of a smooth invariant (central) ergodic measure on
the space of paths of an $N$-graded graph (Bratteli diagram). It is based on
the notion of standardness of the tail filtration in the space of paths, and
the smoothness criterion uses the so-called intrinsic metric which can be
canonically defined on the set of vertices of these graphs. In many cases known
to the author, like the Pascal graph, the Young graph, the space of
configurations, all ergodic central measures are smooth (in this case, we say
that the graph is smooth). But even in these cases, the intrinsic metric is far
from being obvious and does not coincide with the "natural" metric. We apply
and generalize the theory of filtrations developed by the author during the
last forty years to the case of tail filtrations and, in particular, introduce
the notion of a standard filtration as a generalization to the case of
semi-homogeneous filtrations of the notion of a standard homogeneous (dyadic)
filtration in the sense of that theory. The crucial role is played by the new
notion of \textbf{intrinsic semi-metric on the set of vertices of a graph} and
the notion of regular paths, which allows us to refine the ergodic method for
the case of smooth measures. In future, we will apply this new approach to the
theory of invariant measures in combinatorics, ergodic theory, and the theory
of C$^*$-algebras.

25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
1
The categories of this paper are: cs