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

TL;DR

This paper investigates geometric pressure in one-dimensional real and complex dynamics, establishing its equivalence with other pressure definitions using spanning sets and trees, under certain conditions.

Contribution

It introduces a new approach to defining and equating geometric pressure via spanning sets and trees in both real and complex dynamics, even at critical points.

Findings

Pressure $P_{spanning}(t)$ equals other geometric pressures under mild conditions.

$P_{spanning}(t)$ is finite for rational maps but may be infinite in the real case.

Tree pressure is consistent across safe points, excluding a set of Hausdorff dimension zero.

Abstract

We consider $f:\hat I\to \R$ being a $C^3$ (or $C^2$ with bounded distortion) real-valued multimodal map with non-flat critical points, defined on $\hat I$ being the union of closed intervals, and its restriction to the maximal forward invariant subset $K\subset I$. We assume that $f|_K$ is topologically transitive. We call this setting the (generalized multimodal) real case. We consider also $f:\C\to \C$ 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 $\varphi_t=-t\log |f'|$ for $t>0$ and iteration of $f$ defined in a standard way using $(n,\e)$-spanning sets. Despite of $\phi_t=\infty$ at critical points of $f$, this definition makes sense (unlike the standard definition using $(n,\e)$-separated sets) and we prove that $P_{\spanning}(t)$ is equal to other pressure quantities, called for this potential {\it 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.