What was the dominant theory of atonement shown in the Apostolic Fathers's writings? Most articles I read about the theory of atonement in the early church say that Ransom Theory was the prevailing one, although some disagree.  This thinking was promoted by a landmark book on the topic: Christus Victor by Gustav Aulen (1879-1978).

When examining writings from a narrower period belonging to the Apostolic Fathers era (before Irenaeus) is there evidence one way or the other?
I think the clearest statement to explicitly deal with atonement would be this one from the Epistle of Clement to the Corinthians:


  Through charity did the Lord join us unto himself; whilst, for the love that he bore towards us, our Lord Jesus Christ gave his own blood for us, by the will of God: his flesh for our flesh, his soul for our souls. [1 Clement, XLIX]


This notion is suggestive of a (not necessarily penal) substitutionary atonement ("join us unto himself", "his for ours"). I have not found anything else which addressed the issue with that kind of clarity, but there is a general tendency to consider Christ's death with reference to its efficacy in bringing life, forgiving sins, etc. Consider the Epistle of Barnabas, chapter VII:


  Understand then, my beloved children, that the good God hath before manifested all things unto us, that we might know to whom we ought always to give thanks and praise. If, therefore, the Son of God, who is the Lord of all, and shall come to judge both the quick and the dead, has suffered, that by his stripes we might live, let us believe that the Son of God could not have suffered but for us. But, being crucified, they gave him vinegar and gall to drink. Hear therefore, how the priests of the temple did foreshow this also: the Lord, by his command which was written, declared, that whosoever did not fast the appointed fast he should die the death: because he also was himself one day to offer up his body for our sins; so that the type of what was done in Isaac might be fulfilled, who was offered upon the altar.


He goes on to further consider a way Jesus could be typologically foreshadowed in the temple sacrificial system, and concludes:


  Wherefore you here again see a type of Jesus who was to suffer for us.


This focus on the the fact that the death of Christ is "for us" is perhaps very vaguely suggestive of a substitutionary atonement, but I wouldn't read too much into it, since after all, a Ransom or Victor theory is compatible with this kind of language, insofar as the benefits of Christ's death can indeed be considered to be "for" the Christian. For completeness though, we have the following:


  ... the root of the faith, which was preached from ancient times, remains firm in you to this day, and brings forth fruit to our Lord Jesus Christ, who suffered himself to be brought even to the death for our sins. [Polycarp, I]
  
  They abstain from the eucharist, and from the public offices, because they confess not the eucharist to be the flesh of our Saviour Jesus Christ, which suffered for our sins, and which the Father, of his goodness, raised from the dead. [Ignatius to the Smyrnaeans, VII]
  
  Let us, therefore, without ceasing, hold steadfastly to him who is our hope, and the earnest of our righteousness, even Jesus Christ, "who his own self bare our sins in his own body on the tree; who did no sin, neither was guile found in his
  mouth," but suffered all for us, that we might live through him. [Polycarp, VIII; quoting 1 Peter 2:24]
  
  For whereas you are subject to your Bishop as to Jesus Christ, you appear to me to live not after the manner of Men, but according to Jesus Christ; who died for us, that for believing in his Death, you might escape Death. [Ignatius to the Trallians, II]
  
  He was also truly crucified by Pontius Pilate and Herod the tetrarch, being nailed for us in the flesh, by the fruits of which we are saved, even by the most blessed passion, that he might set up a token for all ages, through his resurrection, to all his holy and faithful servants, whether they be Jews or Gentiles, in one body of his church. Now all these things he suffered for us, that we might be saved. [Ignatius to the Smyrnaeans, I-II]
  
  Let us look stedfastly to the blood of Christ, and see how precious his blood is in the sight of God; which, being shed for our salvation, has obtained the grace of repentance for all the world. [1 Clement, VII]
  
  Wherefore I exhort all of you that you obey the word of righteousness, and exercise all patience, which you have seen set forth before your eyes, not only in the blessed Ignatius, and Zozimus, and Rufus, but in others among yourselves, and in Paul himself, and the rest of the apostles. Being confident of this, that all these have not run in vain, but in faith and righteousness, and are gone to the place that was due to them from the Lord, with whom also they suffered; for they loved not this present world, but him who died, and was raised again by God for us. [Polycarp, IX]
  
  All the ends of the world, and the kingdoms of it, will profit me nothing: I would rather die for Jesus Christ, than rule to the utmost ends of the earth. Him I seek who died for us; Him I desire who rose again for us. [Ignatius to the Romans, VI]
  
  Let us reverence our Lord Jesus Christ, whose blood was given for us; let us honour those who are set over us; let us respect the aged that are amongst us; and let us instruct the younger men in the discipline and fear of the Lord. [1 Clement, XXI]


On the other hand, there is this from the Epistle to Diognetus. Depending on when you date that document, it may or may not be before Iraneus. In it, there is a reference to Christ's Ransom:


  He hated us not, neither rejected us, nor bore us malice, but was long-suffering and patient, and in pity for us took upon Himself our sins, and Himself parted with His own Son as a ransom for us, the holy for the lawless, the guileless for the evil, the just for the unjust, the incorruptible for the corruptible, the immortal for the mortal. For what else but His righteousness would have covered our sins? In whom was it possible for us lawless and ungodly men to have been justified, save only in the Son of God? O the sweet exchange, O the inscrutable creation, O the unexpected benefits; that the iniquity of many should be concealed in One Righteous Man, and the righteousness of One should justify many that are iniquitous! [Diognetus V]


Although the language of "ransom" is used, there is not a more developed notion of the ransom being paid to Satan characteristic of Origen's formulation, which is what we usually mean by "ransom theory". Instead, this is basically just in line with Matthew and Mark's notion of a ransom (Mark 10:45, Matthew 20:28), and there is considerable overlap between this idea and a subsitutionary atonement (the ransom-exchange being itself a kind of substitution.)

Conclusion

Unless I've missed something, there doesn't seem to be a focus of aspects of atonement that are not at least partially substitutionary. A discussion of Christ's victory or triumph seems to be absent.

Do bear in mind that Christus Victor atonement which Aulen advocates is rather distinct from ransom theory. The former emphasizes the conquering of death, and hence Christians' freedom. The latter has in view a kind of payment (usually to Satan), to retrieve Christians from Evil.

Also, note that subtitutionary atonement theories need not be penal. It doesn't seem like there's a notion of a penal substitution anywhere in these passages (that is, the idea that the cross served to appease the wrath of God on Christians' sin).