A panorama of specification-like properties and their consequences

Kwietniak, Dominik; Łącka, Martha; Oprocha, Piotr

doi:10.48550/arxiv.1503.07355

Cited by 1 publication

(1 citation statement)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Even a weaker periodic specification does not hold, where N = N (ε) is replaced by N (n, ε) for ε small enough (see the survey [6]). Namely for every function N (n, ε) there exists a such that for g a with an appropriate kneading sequence I the specification with N (ε, n) does not hold.…”

Section: Geometric Pressure Via Spanning Sets the Real Casementioning

confidence: 99%

Geometric pressure in real and complex 1-dimensional dynamics via trees of pre-images and via spanning sets

Przytycki

2017

Monatsh Math

View full text Add to dashboard Cite

We consider f : I → R being a C 3 (or C 2 with bounded distortion) realvalued multimodal map with non-flat critical points, defined on I being the union of closed intervals, and its restriction to the maximal forward invariant subset K ⊂ I . We assume that f | K is topologically transitive and, usually, of positive topological entropy. We call this setting the generalized real multimodal case. We consider also f : C → C a rational map on the Riemann sphere and its restriction to K = J ( f ) being Julia set, the complex case. We consider topological pressure P spanning (t) for the potential function ϕ t = −t log | f | for t > 0 and iteration of f defined in a standard way using (n, ε)-spanning sets. Despite of φ t = ∞ at critical points of f , this definition makes sense (unlike the standard definition using (n, ε)-separated sets) and we prove that P spanning (t) is equal to other pressure quantities, called for this potential geometric pressure, in the real case under mild additional assumptions, and in the complex case provided there is at most one critical point with forward trajectory accumulating in J ( f ). P spanning (t) is proved to be finite for general rational maps, but it may occur infinite in the real case. We also prove that geometric tree pressure in the real case is the same for trees rooted at all safe points, in particular at all points except the set of Hausdorff dimension 0, the fact missing in Przytycki and Rivera-Letelier (Geometric pressure for multimodal maps of the interval, arXiv:1405.2443) proved in the complex case in Przytycki (Trans Am Math Soc 351:2081-2099, 1999 Preface I dedicate this paper to the memory of Marian Smoluchowski 1 , one of founders of Statistical Physics, on 100 anniversary of his premature death.In Markov chain models in Statistical Physics, the spaces of configurations of states (symbols, spins) over lattices, e.g. over Z d , see e.g. [21] or [22], are considered. One can restrict considerations to Z, where the configurations are just sets of trajectories under an action of a function f on a state space. Free energy is replaced by so-called topological pressure P(φ) = P( f, φ), depending on a potential function φ, replacing Hamiltonian. Equilibrium measures are considered, for which measure entropy + integral of potential attains the pressure.In this paper, which is a complement to [18] and [16], we just study the pressure itself, various definitions and relations between them, in the case φ = − t log | f | in the one-dimensional settings, real or complex. The difficulties are caused by the singularities of the derivative f at critical points of f . As usual t is a parameter, inverse of temperature. We call the pressure geometric since the potential f is related to local geometry, in particular for many (so-called: hyperbolic) points x we have |( f n ) (x)| −1 ≈ diam B n (x), where B n (x) is the connected component of the set f −n (B ( f n (x), )) containing x, for a constant > 0. It corresponds to an n-th cylinder (a configuration fixed over n +1 consec...

show abstract

Section: Geometric Pressure Via Spanning Sets the Real Casementioning

confidence: 99%

Geometric pressure in real and complex 1-dimensional dynamics via trees of pre-images and via spanning sets

Przytycki

2017

Monatsh Math

View full text Add to dashboard Cite

show abstract

A panorama of specification-like properties and their consequences

Cited by 1 publication

References 0 publications

Geometric pressure in real and complex 1-dimensional dynamics via trees of pre-images and via spanning sets

Geometric pressure in real and complex 1-dimensional dynamics via trees of pre-images and via spanning sets

Contact Info

Product

Resources

About