A thermodynamic definition of topological pressure for non-compact sets

Thompson, Daniel J.

doi:10.1017/s0143385709001151

Cited by 9 publications

(5 citation statements)

References 26 publications

(47 reference statements)

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“…Walters [33] then extended this concept to a compact space with the continuous transformation. Since topological pressure appeared to be useful in ergodic theory and dynamic systems, there were several attempts to find its suitable generalizations for other systems(see, for example, [2,4,12,13,19,23,24,26,30,35]).…”

Section: Introductionmentioning

confidence: 99%

On the measure-theoretic entropy and topological pressure of free semigroup actions

Lin¹,

Ma²,

Wang

2016

Ergod. Th. Dynam. Sys.

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Section: Introductionmentioning

confidence: 99%

On the measure-theoretic entropy and topological pressure of free semigroup actions

Lin¹,

Ma²,

Wang

2016

Ergod. Th. Dynam. Sys.

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“…The results in this paper are related and can be considered in some sense extensions of results in the higher dimensional multifractal analysis developed by Barreira, Saussol, Schmeling, Takens, Verbitskiy, and others (see for example [2,3,30]). For localizations using restrictions of the pressure to non-compact subsets we refer to [7,25,31] and the references therein. We will now describe our results in more detail.…”

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confidence: 99%

Localized Pressure and Equilibrium States

Kucherenko

Wolf

2015

J Stat Phys

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We introduce the notion of localized topological pressure for continuous maps on compact metric spaces. The localized pressure of a continuous potential ϕ is computed by considering only those (n, )-separated sets whose statistical sums with respect to an m-dimensional potential are "close" to a given value w ∈ R m . We then establish for several classes of systems and potentials ϕ and a local version of the variational principle. We also construct examples showing that the assumptions in the localized variational principle are fairly sharp. Next, we study localized equilibrium states and show that even in the case of subshifts of finite type and Hölder continuous potentials, there are several new phenomena that do not occur in the theory of classical equilibrium states. In particular, we construct an example with infinitely many ergodic localized equilibrium states. We also show that for systems with strong thermodynamic properties and w in the interior of the rotation set of there is at least one and at most finitely many localized equilibrium states.

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“…For the remaining arguments, we will consider an appropriate compact and invariant subset Y ⊂ X and study the entropy of f | Y on Z ∩ Y . Note that together with [33,Theorem 6.1], in this case it holds 4. Topological entropy on subsets.…”

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confidence: 99%

“…If the map f is clear from the context then we drop it in the notation. Following [33], given a nonempty Borel set…”

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confidence: 99%

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Exceptional sets for geodesic flows of noncompact manifolds

Gelfert,

Riquelme

2024

DCDS

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For a geodesic flow on a negatively curved Riemannian manifold M and some subset A ⊂ T 1 M , we study the limit A-exceptional set, that is, the set of points whose ω-limit does not intersect A. We show that if the topological * -entropy of A is smaller than the topological entropy of the geodesic flow, then the limit A-exceptional set has full topological entropy. This result follows as a consequence of the construction of basic sets, with arbitrarily large entropy, over which the dynamics is conjugated to a suspension flow over a subshift of finite type. As an application, we compute the entropy of limit exceptional sets of invariant compact subsets and proper submanifolds.

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A thermodynamic definition of topological pressure for non-compact sets

Cited by 9 publications

References 26 publications

On the measure-theoretic entropy and topological pressure of free semigroup actions

On the measure-theoretic entropy and topological pressure of free semigroup actions

Localized Pressure and Equilibrium States

Exceptional sets for geodesic flows of noncompact manifolds

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