Invariants for finitary isomorphisms with finite expected code lengths

1984 ◽  
Vol 76 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Klaus Schmidt
Keyword(s):  
Finitary Isomorphisms

scholarly journals A return time invariant for finitary isomorphisms

1984 ◽  
Vol 4 (2) ◽  
pp. 225-231 ◽  
Author(s):  
U. R. Fiebig
Keyword(s):  
Finite Measure ◽  
Measurable Subset ◽  
Return Time ◽  
Time Function ◽  
Markov Shifts ◽  
Measure Preserving Transformation ◽  
Recurrence Theorem ◽  
First Return Time ◽  
Time Invariant ◽  
Finitary Isomorphisms

AbstractPoincare's recurrence theorem says that, given a measurable subset of a space on which a finite measure-preserving transformation acts, almost every point of the subset returns to the subset after a finite number of applications of the transformation. Moreover, Kac's recurrence theorem refines this result by showing that the average of the first return times to the subset over the subset is at most one, with equality in the ergodic case. In particular, the first return time function to any measurable set is integrable. By considering the supremum over all p ≥ 1 for which the first return time function is p-integrable for all open sets, we obtain a number for each almost-topological dynamical system, which we call the return time invariant. It is easy to show that this invariant is non-decreasing under finitary homomorphism. We use the invariant to construct a continuum number of countable state Markov shifts with a given entropy (and hence measure-theoretically isomorphic) which are pairwise non-finitarily isomorphic.

scholarly journals Finitary isomorphisms of Poisson point processes

2019 ◽  
Vol 47 (5) ◽  
pp. 3055-3081
Author(s):  
Terry Soo ◽  
Amanda Wilkens
Keyword(s):  
Point Processes ◽  
Poisson Point Processes ◽  
Finitary Isomorphisms

scholarly journals Finitary isomorphisms of Brownian motions

2020 ◽  
Vol 48 (4) ◽  
pp. 1966-1979
Author(s):  
Zemer Kosloff ◽  
Terry Soo
Keyword(s):  
Brownian Motions ◽  
Finitary Isomorphisms

scholarly journals Generating stationary random graphs on ℤ with prescribed independent, identically distributed degrees

2006 ◽  
Vol 38 (02) ◽  
pp. 287-298 ◽  
Author(s):  
Maria Deijfen ◽  
Ronald Meester
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Random Graphs ◽  
Random Number ◽  
Random Variables ◽  
Nearest Neighbors ◽  
Expected Length ◽  
Vertex Set ◽  
Finitary Isomorphisms ◽  
Independent Identically Distributed

Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

Finitary isomorphisms of irreducible Markov shifts

1979 ◽  
Vol 34 (4) ◽  
pp. 281-286 ◽  
Author(s):  
Michael Keane ◽  
Meir Smorodinsky
Keyword(s):  
Markov Shifts ◽  
Finitary Isomorphisms

scholarly journals Generating stationary random graphs on ℤ with prescribed independent, identically distributed degrees

2006 ◽  
Vol 38 (2) ◽  
pp. 287-298 ◽  
Author(s):  
Maria Deijfen ◽  
Ronald Meester
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Random Graphs ◽  
Random Number ◽  
Random Variables ◽  
Nearest Neighbors ◽  
Expected Length ◽  
Vertex Set ◽  
Finitary Isomorphisms ◽  
Independent Identically Distributed

Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

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