Speiser class Julia sets with dimension near one
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SIMON ALBRECHT∗ AND CHRISTOPHER J. BISHOP† “Without deviation from normality, there can be no progress” - Frank Zappa For Larry Zalcman, who showed the converse is also true. Abstract. For any δ > 0 we construct an entire function f with three singular values whose Julia set has Hausdorff dimension at most 1 + δ. Stallard proved that the dimension must be strictly larger than 1 whenever f has a bounded singular set, but no examples with finite singular set and dimension strictly less than 2 were previously known.
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Introduction
Suppose f is an entire function. The Fatou set F(f ) is the union of all open disks on which the iterates f, f 2 , f 3 , . . . form a normal family and the Julia set J(f ) is the complement of this set. In 1975 Baker [2] proved that if f is transcendental (i.e., not a polynomial), then the Fatou set has no unbounded, multiply connected components. This implies the Julia set contains a non-trivial continuum and hence has Hausdorff dimension at least 1, but it is difficult to build examples that come close to attaining this minimum; constructing such examples is the transcendental counterpart of finding polynomial Julia sets with dimension near 2 (e.g., [16], [36], [43]). For transcendental entire functions, finding “large” Julia sets is easier: Misiurewicz [29] proved that the Julia set of f (z) = exp(z) is the whole plane, and McMullen [27] gave explicit families where the Julia set is not the whole plane, but still has dimension 2 (even positive area). Stallard [37], [39] proved that the Hausdorff dimension of a transcendental Julia set can attain every value in the interval (1, 2], and the second author [12] recently constructed a transcendental Julia set with dimension 1, Baker’s lower bound. The singular set of an entire function f is the closure of its critical values and finite asymptotic values (limits of f along a curve to ∞) and will be denoted S(f ). ∗ †
S. Albrecht is supported by the Deutsche Forschungsgemeinschaft, grant no. AL 2028/1-1 C. Bishop is partially supported by NSF Grant DMS 19-06259
49 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0128-1
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S. ALBRECHT AND C. J. BISHOP
The Eremenko–Lyubich class B consists of functions such that S(f ) is a bounded set (such functions are also called bounded-type). The Speiser class S ⊂ B consists of those functions for which S(f ) is a finite set. These are important classes in transcendental dynamics and it is an interesting problem to understand their differences and similarities. For example, functions in S cannot have wandering domains, whereas those in B can ([9], [18], [22]). Stallard’s examples with 1 < dim(J) < 2 are in the Eremenko–Lyubich class, and in this paper we show that such examples also exist in the Speiser class. Theorem 1.1. inf{dim(J(f )) : f ∈ S} = 1. Note that we do not claim that every dimension between 1 and 2 occurs; this remains an open problem. Theorem 1.1 is sharp in the sense that Stallard [38] proved that dim(J(f )) > 1 for any f ∈ B (her result has been extended beyo
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