Abstract
We consider f:widehat{I}rightarrow mathbb {R} being a C^3 (or C^2 with bounded distortion) real-valued multimodal map with non-flat critical points, defined on widehat{I} being the union of closed intervals, and its restriction to the maximal forward invariant subset Ksubset widehat{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:widehat{mathbb {C}}rightarrow widehat{mathbb {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_{{{mathrm{mathrm{spanning}}}}}(t) for the potential function varphi _t=-tlog |f'| for t>0 and iteration of f defined in a standard way using (n,varepsilon )-spanning sets. Despite of phi _t=infty at critical points of f, this definition makes sense (unlike the standard definition using (n,varepsilon )-separated sets) and we prove that P_{{{mathrm{mathrm{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_{{{mathrm{mathrm{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).
Highlights
Let us start with the classicalDefinition 1.1 (Topological pressure via separated sets) Let f : X → X be a continuous map of a compact metric space (X, ρ) and φ : X → R be a real continuous function
We assume f ∈ C2, is non-flat at all its turning and inflection critical points, has bounded distortion property for its iterates, f |K is topologically transitive (that is for every U, V open in K there exists n > 0 such that f n(U ) ∩ V = ∅) and has positive topological entropy on K
In particular in the complex case Ptree(z, t) does not depend on z safe; it is constant except z in a set of Hausdorff dimension 0
Summary
Definition 1.1 (Topological pressure via separated sets) Let f : X → X be a continuous map of a compact metric space (X, ρ) and φ : X → R be a real continuous function. This pressure depends on topology, but does not depend on metric. 2. (Real) f is a real generalized multimodal map It is defined on a neighbourhood U ⊂ R of its compact invariant subset K. We assume f ∈ C2, is non-flat at all its turning and inflection critical points, has bounded distortion property for its iterates (see the definition below), f |K is topologically transitive (that is for every U, V open in K there exists n > 0 such that f n(U ) ∩ V = ∅) and has positive topological entropy on K. A set X is said to be hyperbolic, uniformly hyperbolic or expanding if there is a constant λX > 1 such that for all n large enough and all x ∈ X we have |( f n) (x)| ≥ λnX
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