Dimension of ergodic measures and currents on

Let $f$ be a holomorphic endomorphism of $\mathbb{P}^{2}$ of degree $d\geq 2$. We estimate the local directional dimensions of closed positive currents $S$ with respect to ergodic dilating measures $\unicode[STIX]{x1D708}$. We infer several applications. The first one is an upper bound for the lower pointwise dimension of the equilibrium measure, towards a Binder–DeMarco’s formula for this dimension. The second one shows that every current $S$ containing a measure of entropy $h_{\unicode[STIX]{x1D708}}>\log d$ has a directional dimension ${>}2$, which answers a question of de Thélin–Vigny in a directional way. The last one estimates the dimensions of the Green current of Dujardin’s semi-extremal endomorphisms.

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Smale endomorphisms over graph-directed Markov systems

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.42 ◽

2020 ◽

pp. 1-34

Author(s):

EUGEN MIHAILESCU ◽

MARIUSZ URBAŃSKI

Keyword(s):

Equilibrium Measure ◽

Parabolic Systems ◽

Skew Product ◽

Pointwise Dimension ◽

Markov Systems ◽

Large Classes ◽

Equilibrium Measures ◽

Skew Products ◽

Natural Extensions ◽

Induced Maps

We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$ -maps $T_{\unicode[STIX]{x1D6FD}}$ , for arbitrary $\unicode[STIX]{x1D6FD}>1$ .

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Relative equilibrium states and class degree

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2017.50 ◽

2017 ◽

Vol 39 (4) ◽

pp. 865-888

Author(s):

MAHSA ALLAHBAKHSHI ◽

JOHN ANTONIOLI ◽

JISANG YOO

Keyword(s):

Finite Type ◽

Upper Bound ◽

Relative Equilibrium ◽

Ergodic Measure ◽

Equilibrium States ◽

Shift Of Finite Type ◽

Sofic Shift ◽

Ergodic Measures ◽

Factor Code ◽

Special Case

Given a factor code $\unicode[STIX]{x1D70B}$ from a shift of finite type $X$ onto a sofic shift $Y$, an ergodic measure $\unicode[STIX]{x1D708}$ on $Y$, and a function $V$ on $X$ with sufficient regularity, we prove an invariant upper bound on the number of ergodic measures on $X$ which project to $\unicode[STIX]{x1D708}$ and maximize the measure pressure $h(\unicode[STIX]{x1D707})+\int V\,d\unicode[STIX]{x1D707}$ among all measures in the fiber $\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D708})$. If $\unicode[STIX]{x1D708}$ is fully supported, this bound is the class degree of $\unicode[STIX]{x1D70B}$. This generalizes a previous result for the special case of $V=0$ and thus settles a conjecture raised by Allahbakhshi and Quas.

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