POISSON PROCESSES FOR SUBSYSTEMS OF FINITE TYPE IN SYMBOLIC DYNAMICS

Let Δ ⊊ V be a proper subset of the vertices V of the defining graph of an irreducible and aperiodic shift of finite type [Formula: see text]. Let ΣΔ be the subshift of allowable paths in the graph of [Formula: see text] which only passes through the vertices of Δ. For a random point x chosen with respect to an equilibrium state μ of a Hölder potential φ on [Formula: see text], let τn be the point process defined as the sum of Dirac point masses at the times k > 0, suitably rescaled, for which the first n-symbols of Tkx belong to Δ. We prove that this point process converges in law to a marked Poisson point process of constant parameter measure. The scale is related to the pressure of the restriction of φ to ΣΔ and the parameters of the limit law are explicitly computed.

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Perturbations of multidimensional shifts of finite type

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385710000040 ◽

2010 ◽

Vol 31 (2) ◽

pp. 483-526 ◽

Cited By ~ 4

Author(s):

RONNIE PAVLOV

Keyword(s):

Finite Type ◽

Lower Bounds ◽

Symbolic Dynamics ◽

Upper And Lower Bounds ◽

Shift Of Finite Type ◽

Strongly Irreducible ◽

Dimensional Cube

AbstractIn this paper, we study perturbations of multidimensional shifts of finite type. Specifically, for any ℤd shift of finite type X with d>1 and any finite pattern w in the language of X, we denote by Xw the set of elements of X not containing w. For strongly irreducible X and patterns w with shape a d-dimensional cube, we obtain upper and lower bounds on htop (X)−htop (Xw) dependent on the size of w. This extends a result of Lind for d=1 . We also apply our methods to an undecidability question in ℤd symbolic dynamics.

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Concave Majorants of Random Walks and Related Poisson Processes

Combinatorics Probability Computing ◽

10.1017/s0963548311000307 ◽

2011 ◽

Vol 20 (5) ◽

pp. 651-682

Author(s):

JOSH ABRAMSON ◽

JIM PITMAN

Keyword(s):

Random Walks ◽

Point Process ◽

Finite Length ◽

Poisson Point Process ◽

Complete Information ◽

Arithmetic Mean ◽

Poisson Processes ◽

Infinite Length ◽

Unified Approach ◽

Geometric Length

We offer a unified approach to the theory of concave majorants of random walks, by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst providing complete information about the concave majorant. This leads to a description of a walk of random geometric length as a Poisson point process of excursions away from its concave majorant, which is then used to find a complete description of the concave majorant of a walk of infinite length. In the case where subsets of increments may have the same arithmetic mean, we investigate three nested compositions that naturally arise from our construction of the concave majorant.

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