
	Hdim Version 2.0 - April 1997

	Hausdorff Dimension of Julia Sets and
		Schottky Limit Sets

The program hdim computes the Hausdorff dimension of the limit
set of a Kleinian group generated by reflections in the
boundaries of a family of *disjoint* round 
disks in the plane.

INSTALLATION:  
a) type "cd src; make" to create the executable "hdim".
b) type "hdim" for a test run.
c) see the directory examples/ for usage examples.

USAGE: hdim [options]
	-s [n ang] symmetric Schottky group
	-S [n alow ahigh astep] sequence of Schottky groups
	-c [c.x c.y] Julia set for z^2+c
	-C [clow chigh cstep] sequence of quadratics
	-b [p.x p.y] Julia set for Blaschke product
	-B [plow phigh pstep] sequence of B. products
	-h [r] Limit set for Hecke group; 0<r<1
	-H [rllow rhigh rstep] sequence of H. groups
	-a Apollonian packing
	-i Input from stdin
	-e [epsilon] pruning epsilon
	-E [emin emax] range of epsilons
        -v non-verbose
	-p [filename] postscript image

-s:  prints dimensions for the n-generator Schottky group
generated by reflections in circles symmetrically 
arrayed around the origin, each with visual angle=ang degrees
as seen from z=0.  One should set ang <= 360/n.

-S:  computes a sequence of Schottky groups

-c:  computes the dimension of the Julia set
     for z^2+c.  

-C:  computes a series of Julia sets with c real

-b:  computes the dimension of the Julia set of
     f[z_]:=((p+2)z^2+(2-p))/((p+2)+(2-p)z^2).
     This map has a fixed point at z=1 with
     multiplier p.  For p real, this map is
     a Blaschke product.

-B:  computes a series of Blaschke products.

-a:  initializes the 4 circles that
generate the Apollonian gasket, and performs
a sequence of dimension calculation up to the
indicated depth.

-i:  reads (disjoint) generating circles for a Schottky
group from stdin.  A circle is specified by 
three numbers: cx,cy,r.

-e:  prunes Markov partition by stopping subdivision of
     blocks of size less than epsilon.

-E:  calculates dimension for a range of epsilons.

-v:  print only parameter and dimension

-p:  writes a postscript image of covering
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Copyright (c) April 1997

Curtis T McMullen

email:  ctm@math.harvard.edu
