The Hausdorff dimension of the Julia sets concerning generated renormalization transformation

Tingting Li; Junyang Gao; Tingting Li; Junyang Gao

doi:10.3934/math.2022056

AIMS Mathematics

2022, Volume 7, Issue 1: 939-956. doi: 10.3934/math.2022056

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Research article

The Hausdorff dimension of the Julia sets concerning generated renormalization transformation

Tingting Li ,
Junyang Gao ^,

School of Science, China University of Mining and Technology, Beijing 100083, China

Received: 04 June 2021 Accepted: 09 October 2021 Published: 19 October 2021
MSC : 37F10, 37F45

Considering a family of rational map $ {U_{mn\lambda }} $ of the renormalization transformation of the generalized diamond hierarchical Potts model, we give the asymptotic formula of the Hausdorff dimension of the Julia sets of $ {U_{mn\lambda }} $ as the parameter $ \lambda $ tends to infinity, here

$ {U_{mn\lambda }} = {\left[ {\frac{{{{\left( {z + \lambda - 1} \right)}^n} + \left( {\lambda - 1} \right){{\left( {z - 1} \right)}^n}}}{{{{\left( {z + \lambda - 1} \right)}^n} - {{\left( {z - 1} \right)}^n}}}} \right]^m}, $

where $ m \ge 2 $, $ n \ge 2 $ are two natural numbers, $ \lambda \in {{\mathbb{C}} } $.
- renormlization transformation,
- Potts model,
- asymptotic formula,
- Hausdorff dimension,
- Julia set
Citation: Tingting Li, Junyang Gao. The Hausdorff dimension of the Julia sets concerning generated renormalization transformation[J]. AIMS Mathematics, 2022, 7(1): 939-956. doi: 10.3934/math.2022056

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Abstract

Considering a family of rational map $ {U_{mn\lambda }} $ of the renormalization transformation of the generalized diamond hierarchical Potts model, we give the asymptotic formula of the Hausdorff dimension of the Julia sets of $ {U_{mn\lambda }} $ as the parameter $ \lambda $ tends to infinity, here

$ {U_{mn\lambda }} = {\left[ {\frac{{{{\left( {z + \lambda - 1} \right)}^n} + \left( {\lambda - 1} \right){{\left( {z - 1} \right)}^n}}}{{{{\left( {z + \lambda - 1} \right)}^n} - {{\left( {z - 1} \right)}^n}}}} \right]^m}, $

where $ m \ge 2 $, $ n \ge 2 $ are two natural numbers, $ \lambda \in {{\mathbb{C}} } $.

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